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%% BEGIN ITEM
\AN{05013668}{2006}
\AU{Hwang, Jun-Muk}
\TI{Log canonical thresholds of divisors on Grassmannians.}
\LA{English}
\DT{J}
\SO{Math. Ann. 334, No.2, 413-418 (2006).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s00208-005-0731-6}
\ISSN{025-583}
\AB{Summary: Let $L$ be the Pl\"ucker line bundle on the Grassmannian.
    Given $D \in |kL|$, we show that the log canonical threshold
    of $D$ is at least $\frac 1k$. The main ingredients of the proof
    are Kapranov's result on the derived category of coherent sheaves
    on the Grassmannian, Nadel's vanishing theorem for multiplier
    ideal sheaves, and Demailly's vanishing theorem for vector bundles.}
\CC{14M15
14F17}
%% END ITEM
%% BEGIN ITEM
\AN{1083.14010 (02190821)}{2005}
\AU{Zhang, Qi}
\TI{On projective varieties with nef anticanonical divisors.}
\LA{English}
\DT{J}
\SO{Math. Ann. 332, No.3, 697-703 (2005).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s00208-005-0649-z}
\ISSN{025-583}
\AB{The author proves a structure theorem for projective varieties
    with nef anti-canonical divisors. Namely, let $X$ be a projective
    variety and $D$ an effective $\Bbb Q$-divisor on $X$ such that
    the pair $(X,D)$ is log canonical and $-(K_{X}+D)$ is nef. Let
    $f:X\to Y$ be a dominant rational map, where $Y$ is a smooth
    variety. Then either (1) $Y$ is uniruled; or (2) the Kodaira
    dimension $\kappa(Y)=0$. Moreover in this case, $f$ is semistable
    in codimension $1$. \par
The tools the author uses in the proof
    of the theorem are the theory of weak (semi) positivity of the
    direct images of (log) relative dualizing sheaves, the notion
    of the divisorial Zariski decomposition, and some results in
    [{\it S.\ Boucksom, J.-P.\ Demailly, M.\ Paun} and {\it T.\ Peternell},
    ``The pseudo-effective cone of a compact K\"ahler manifold and
    varieties of negative Kodaira dimension", preprint, {\tt math.AG/0405285}].
    \par
As applications of this theorem, the author obtains some
    corollaries.}
\RV{Yoshiaki Fukuma (Kochi)}
\CC{14C20
14E30
14J17
14J40}
%% END ITEM
%% BEGIN ITEM
\AN{1082.32001 (02191968)}{2005}
\AU{Berman, Robert}
\TI{Holomorphic Morse inequalities on manifolds with boundary.}
\LA{English}
\DT{J}
\SO{Ann. Inst. Fourier 55, No.4, 1055-1105 (2005). 
\par http://annalif.ujf-grenoble.fr/Vol55/E554$\_1$/E554$\_1$.html}
\ISSN{373-095}
\AB{The holomorphic Morse inequalities of {\it J.-P.\ Demailly} [Ann.
    Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)] asymptotically
    estimate the dimension of a Dolbeault cohomology group of differential
    forms defined on a compact complex manifold and taking values
    in a high tensor power of a Hermitian holomorphic line bundle.
    Several authors have generalized these inequalities to the case
    of a complex manifold with boundary under various curvature assumptions.
    In this article, the author proves a version of the inequalities
    in the non-compact setting with a correction term (given by a
    boundary integral) that is valid for an arbitrary holomorphic
    line bundle. The only hypothesis is that the Levi form of the
    manifold is non-degenerate on the boundary. Examples show that
    the inequalities are sharp.}
\RV{Harold P.Boas (College Station)}
\CC{32A25
32L10
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{1082.53041 (02230734)}{2005}
\AU{Koll\'ar, J\'anos}
\TI{Einstein metrics on five-dimensional Seifert bundles.}
\LA{English}
\DT{J}
\SO{J. Geom. Anal. 15, No.3, 445-476 (2005). 
\par http://www.jga.wustl.edu/}
\ISSN{050-692}
\AB{Let $L$ be $2n+ 1$-dimensional manifold which admits a differentiable
    map $f: L\to X$ to a complex $n$-manifold $X$ such that every
    fiber is a circle. Such $L$ is called a higher-dimensional Seifert
    fibred manifold. For $n= 1$ there are Seifert fibred 3-manifolds.
    The author classified all such 5-dimensional manifolds which
    admit a positive Seifert bundle structure and in a few cases
    all Seifert bundle structures are classified. These results are
    then used to construct positive Ricci curvature Einstein metrics
    on this manifolds. The methods are not elementary. For example
    the Leray spectral sequence of the Seifert bundle is used. The
    construction of K\"ahler-Einstein metrics on Dell Pezzo orbifolds
    (it corresponds to $X$) uses the algebraic existence criterion
    of Demailly-Kollar. Then the lifting of the K\"ahler-Einstein
    metric on the base of a Seifert bundle to an Einstein metric
    on the total space uses the Kobayashi-Boyer-Galicki method. There
    is a special part related to Seifert bundles with a rational
    homology of the sphere and trivial first homology group over
    integers. Finally, we have to mention that the paper is very
    technical.}
\RV{Andrzej Szczepa\'nski (Gda\'nsk)}
\CC{53C25
57R22
55R20
55R40
55R55
57R20}
%% END ITEM
%% BEGIN ITEM
\AN{1075.14001 (02172229)}{2005}
\AU{Favre, Charles; Jonsson, Mattias}
\TI{Valuations and multiplier ideals.}
\LA{English}
\DT{J}
\SO{J. Am. Math. Soc. 18, No.3, 655-684 (2005).  ISSN 1088-6834]
\par http://www.ams.org/jams/2005-18-03/S0894-0347-05-00481-9/home.html}
\ISSN{894-034}
\AB{The authors present a new approach to the study of multiplier
    ideals in a regular two dimensional ring $(R,m)$. This allows
    to unify the definition on ideals, filtration of ideals or plurisubharmonic
    functions. This work is based on the introduction of the space
    ${\cal V}$ of all $\Bbb R\cup\{ +\infty \} $-valued valuations
    of $R$ centered at the maximal ideal $m$ and normalized by $\nu(m)=1$.
    ${\cal V}$ is a tree and encodes in a natural way all possible
    blow-ups of $R$ centered in $m$. The points of ${\cal V}$ that
    are not ends, form a subtree ${\cal V}_{qm}$. Multiplier ideals
    are defined in terms of some functions on ${\cal V}_{qm}$, called
    tree potentials. Among the applications are a formula for the
    singularity exponent of a plurisubharmonic function in terms
    of Kiselman numbers, and a proof of a conjecture by Demailly
    and Kollar. They give a new proof of the fact obtained by Lipman
    and Watanabe that any integrally closed ideal is a multiplier
    ideal.}
\RV{Marcel Morales (Saint-Martin-d'Heres)}
\CC{14B05
32U25
13H05}
%% END ITEM
%% BEGIN ITEM
\AN{1072.32010 (02185491)}{2005}
\AU{Biswas, Indranil; Subramanian, Swaminathan}
\TI{Numerically flat principal bundles.}
\LA{English}
\DT{J}
\SO{Tohoku Math. J., II. Ser. 57, No.1, 53-63 (2005). 
\par http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.tmj}
\ISSN{040-873}
\AB{In [{\it J.-P.\ Demailly}, {\it T. Peternell} and {\it M. Schneider},
    J. Algebraic Geom. 3, 295-345 (1994; Zbl 0827.14027)] it was
    introduced the notion of numerically effective holomorphic vector
    bundle and numerically flat vector bundle on a compact K\"ahler
    manifold. In the paper under review the authors introduce the
    same notions for a principal $G$-bundle $E_G$, where $G$ is a
    semisimple complex algebraic group. They prove that $E_G$ is
    numerically flat if and only if ad$(E_G)$ is numerically flat.
    Numerically flat implies semistable; the authors give the additional
    conditions needed to reverse this implication.}
\RV{Edoardo Ballico (Povo)}
\CC{32L05
53C05}
%% END ITEM
%% BEGIN ITEM
\AN{02235490}{2005}
\AU{Favre, Charles; Jonsson, Mattias}
\TI{Valuative analysis of planar plurisubharmonic functions.}
\LA{English}
\DT{J}
\SO{Invent. Math. 162, No.2, 271-311 (2005).  ISSN 1432-1297]
\par http://dx.doi.org/10.1007/s00222-005-0443-2}
\ISSN{020-991}
\AB{Summary: We show that valuations on the ring $R$ of holomorphic
    germs in dimension 2 may be naturally evaluated on plurisubharmonic
    functions, giving rise to generalized Lelong numbers in the sense
    of Demailly. Any plurisubharmonic function thus defines a real-valued
    function on the set $\cal V$ of valuations on $R$ and --- by
    way of a natural Laplace operator defined in terms of the tree
    structure on $\cal V$ --- a positive measure on $\cal V$. This
    measure contains a great deal of information on the singularity
    at the origin. Under mild regularity assumptions, it yields an
    exact formula for the mixed Monge-Amp\`ere mass of two plurisubharmonic
    functions. As a consequence, any generalized Lelong number can
    be interpreted as an average of valuations. Using our machinery
    we also show that the singularity of any positive closed $(1,1)$
    current $T$ can be attenuated in the following sense: there exists
    a finite composition of blowups such that the pull-back of $T$
    decomposes into two parts, the first associated to a divisor
    with normal crossing support, the second having small Lelong
    numbers.}
\CC{32U25
13A18
13H05}
%% END ITEM
%% BEGIN ITEM
\AN{02245718}{2005}
\AU{Pali, Nefton}
\TI{The Chern connection and curvature of the tangent bundle of an almost complex manifold. (La connexion et la courbure de Chern du fibr\'e tangent d'une vari\'et\'e presque complexe)}
\LA{French. English summary}
\DT{J}
\SO{New York J. Math. 11, 597-634 (2005). 
\par http://nyjm.albany.edu:8000/j/2005/Vol11nf.htm}
\ISSN{076-980}
\AB{Summary: The $\bar{\partial}_{_{J}}$ operator over an almost
    complex manifold induces canonical connections of type $(0,1)$
    over the bundles of $(p,0)$-forms. If the almost complex structure
    is integrable then the previous connections induce the canonical
    holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$
    we can extend the corresponding connection to all Schur powers
    of the bundle of $(1,0)$-forms. Moreover using the canonical
    $\Bbb C$-linear isomorphism betwen the bundle of $(1,0)$-forms
    and the complex cotangent bundle $T^\ast_{X,J}$ we deduce canonical
    connections of type $(0,1)$ over the Schur powers of the complex
    cotangent bundle $T^\ast_{X,J}$. If the almost complex structure
    is integrable then the previous $(0,1)$-connections induce the
    canonical holomorphic structures of those bundles. In the nonintegrable
    case those $(0,1)$-connections induce just the holomorphic canonical
    structures of the restrictions of the corresponding bundles to
    the images of smooth $J$-holomorphic curves. We introduce the
    notion of Chern curvature for those bundles. The geometrical
    meaning of this notion is a natural generalisation of the classical
    notion of Chern curvature for the holomophic vector bundles over
    a complex manifold. We have a particular interest for the case
    of the tangent bundle in view of applications concerning the
    regularisation of $J$-plurisubharmonic functions by means of
    the geodesic flow induced by a Chern connection on the tangent
    bundle. This method has been used by Demailly, 1994, in the complex
    integrable case. Our specific study in the case of the tangent
    bundle gives an asymptotic expanson of the Chern flow which relates
    in an optimal way the geometric obstructions caused by the torsion
    of the almost complex structure, and the nonsymplectic nature
    of the metric.}
\CC{32C35}
%% END ITEM
%% BEGIN ITEM
\AN{05001985}{2005}
\AU{Laytimi, F.; Nahm, W.}
\TI{On a vanishing problem of Demailly.}
\LA{English}
\DT{J}
\SO{Int. Math. Res. Not. 2005, No.47, 2877-2889 (2005).  ISSN 1687-0247]
\par http://www.hindawi.com/journals/imrn/}
\ISSN{073-792}
\AB{Review in preparation}
\CC{14-99
32-99}
%% END ITEM
%% BEGIN ITEM
\AN{1081.32013 (02152168)}{2004}
\AU{El Goul, Jawher}
\TI{Demailly's 2-jet negativity of certain hyperbolic fibrations.}
\LA{English}
\DT{CA}
\SO{Miyajima, Kimio (ed.) et al., Complex analysis in several variables - Memorial conference of Kiyoshi Oka's centennial birthday. Papers from the conference, Kyoto, Japan, October 30-November 5, 2001 and Nara, Japan, November 6-8, 2001. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 42, 85-93 (2004). [ISBN 4-931469-27-2/hbk]}
\AB{The concept of $k$-jet negativity was introduced by Demailly
    as a generalization to higher jets of the concept of negativity
    of holomorphic sectional curvature for Finsler metrics on tangent
    bundles. Demailly conjectured that the existence of a metric
    of negative curvature (in a weak sense) that he constructed on
    a $k$-jet bundle characterizes Kobayashi's hyperbolicity in the
    case of compact manifolds. In the paper under review, the author
    considers this conjecture in the case of a hyperbolic singular
    fibration on a hyperbolic curve with certain conditions on the
    singularities of special fibres. She proves the weak negativity
    property on the 2-jet bundle of such a fibration.}
\RV{Athanase Papadopoulos (Strasbourg)}
\CC{32Q05
32Q45
32Q99}
%% END ITEM
%% BEGIN ITEM
\AN{1077.14011 (02135197)}{2004}
\AU{Debarre, Olivier}
\TI{Seshadri constants of abelian varieties.}
\LA{English}
\DT{CA}
\SO{Collino, Alberto (ed.) et al., The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871-1952), Torino, Italy, September 29-October 5, 2002. Torino: Universit\`a di Torino, Dipartimento di Matematica. 379-394 (2004).}
\AB{The recent years have seen considerable interest and activity
    in understanding the local positivity of ample line bundles on
    complex algebraic varieties. Seshadri constants, introduced by
    {\it J.-P. Demailly} [in: Complex algebraic varieties (Proc.
    Conf. Bayreuth 1990), Lect. Notes Math. 1507, 87-104 (1992; Zbl
    0784.32024)], emerged as a natural measure of the local positivity
    of a line bundle. Although these numerical invariants are hard
    to calculate, in general, some deep results on their behavior
    have been obtained, in the meantime, mainly in regard of their
    lower bounds in special cases. The present state of art in the
    study of Seshadri constants and their geometric significance
    is thoroughly discussed in the first volume of {\it R. Lazarsfeld's}
    recent book [``Positivity in algebraic geometry'', Erg. Math.
    und ihrer Grenzgeb., 3. Folge, Vol. 48 (Berlin: Springer) (2004;
    Zbl 1066.14021)], where particular attention is paid to Seshadri
    constants of abelian varieties.\par
In the paper under review,
    the author advances the investigation of Seshadri constants of
    abelian varieties, with a particular view toward Jacobians of
    curves. As he points out, this note grew out of discussions with
    R. Lazarsfeld on the possibility of characterizing Jacobians
    among all principally polarized abelian varieties by means of
    their Seshadri constant with respect to the theta divisor. Namely,
    due to results of {\it P. Buser} and {\it P. Sarnak} [Invent.
    Math. 117, No. 1, 27-56 (1994; Zbl 0814.14033)], on the one hand,
    and by {\it R. Lazarsfeld} [Math. Res. Lett. 3, No. 4, 439-447
    (1996; Zbl 0890.14025)] on the other, it appears that the Seshadri
    constants of Jacobians take much smaller values than they do
    on general principally polarized abelian varieties, and this
    very fact motivates the above-mentioned characterization problem
    for Jacobians.\par
However, and unfortunately, the results of
    the present paper show that the picture is not as pretty as one
    could have hoped. There are in each dimension at least four indecomposable
    principally polarized abelian varieties that are not Jacobians
    but have Seshadri constant two, whereas general Jacobians of
    dimension $g$ have Seshadri constants that grow at least like
    $\log(g)$.\par
On the other hand, the author provides a detailed
    study of Seshadri constants of Jacobians and of principally polarized
    abelian varieties in low dimensions, thereby obtaining an almost
    complete list of their possible values. Also, his results and
    examples shed some more light on the still open problem of determining
    the rate of growth of the maximal value of Seshadri constants
    of Jacobians, and he touches upon R. Lazarsfeld's conjecture
    stating that indecomposable principally polarized abelian varieties
    with Seshadri constant less than two should be hyperelliptic
    Jacobians. This conjecture of Lazarsfeld is based on another
    long-standing conjecture, namely on the celebrated conjecture
    of {\it B. van Geemen} and {\it G. van der Geer} on indecomposable
    principally polarized abelian varieties that are not Jacobians
    [Am. J. Math. 108, 615-641 (1986; Zbl 0612.14044)].\par
In any
    case, the paper under review yields a lot of important new material
    on the behavior of Seshadri constants of abelian varieties and
    Jacobians of curves, which will serve as inspiration for further
    research in this contemporary area of complex algebraic geometry.}
\RV{Werner Kleinert (Berlin)}
\CC{14C20
14K05
14H40
14J10
14K10
14H42}
%% END ITEM
%% BEGIN ITEM
\AN{1077.14503 (02174525)}{2004}
\TI{Multiplier ideal sheaves in algebraic and complex geometry.}
\LA{English}
\DT{J}
\SO{Oberwolfach Rep. 1, No.2, Report 21, 1108-1166 (2004).  ISSN 1660-8941]
\par http://www.ems-ph.org/journals/show\_issue.php?issn=1660-8933\&vol=1\&iss=2}
\ISSN{660-893}
\AB{Contributions:\par
-- David Catlin, Estimates for the $\bar\partial$-Neumann
    problem p.1111\par
-- Thomas Peternell, Geometric stability
    of the cotangent bundle and the universal cover of a projective
    manifold p.1111\par
-- Luca Migliorini (joint with Mark A. de
    Cataldo), The Hodge theory of algebraic maps p.1113\par
-- Junjiro
    Noguchi (joint with J. Winkelmann and K. Yamanoi), Holomorphic
    Curves in Semi-Abelian Varieties and Applications p.1114\par
--
    Thomas Eckl, Numerically Trivial Foliations p.1119\par
-- Takeo
    Ohsawa, Recent applications of the twisted Nakano identity p.1120\par
--
    Bernard Shikman (joint with Mike Douglas and Steve Zelditch),
    Critical points of holomorphic sections and supersymmetric vacua
    p.1122\par
-- Lawrence Ein, Invariants from multiplier ideals
    p.1124\par
-- Peter Greiner, Subelliptic PDE's and SubRiemannian
    Geometry p.1124\par
-- Gordon Heier, On finite type and subelliptic
    estimates p.1127\par
-- Jekery D. McNeal (joint with Andr\'as
    N\'emethi), The order of contact of a holomorphic ideal in $\Bbb
    C^2$ p.1130\par
-- Hajime Tsuji, Geometry of varieties of general
    type p.1133\par
-- Mihai Paun, Currents associated to transcendental
    entire curves on compact K\"ahler manifolds p.1135\par
-- Stefan
    Helmke, Multiplier Ideals and Basepoint Freeness p.1137\par
--
    Song-Ying Li (joint with H.-S. Luk), Pseudo Hermitian Manifolds
    p.1139\par
-- Sai-Kee Yeung, Geometric and arithmetic properties
    of complex two ball quotients p.1140\par
-- Helene Esnault (joint
    with A. Beilinson, S. Bloch, P. Deligne), Periods for irregular
    connections on complex curves p.1143\par
-- Jun-Muk Hwang, Volumes
    of complex hyperbolic manifolds with cusps p.1145\par
-- Jacob
    Sturm (joint with D.H.Phong), Critical exponents of analytic
    functions p.1145\par
-- Stefano Trapani, Complexification of
    group actions p.1149\par
-- Philippe Eyssidieux, Linear coverings
    of complex manifolds p.1152\par
-- Daniel Barlet (joint with
    Jon Magnusson), Integration of meromorphic cohomology classes
    and applications p.1154\par
-- Sheng-Li Tan, Effective behaviour
    of multiple linear systems on an algebraic surface p.1155\par
--
    Toshiki Mabuchi, The Hitchin-Kobayashi correspondence p.1158\par
--
    Jean-Pierre Demailly (joint with S. Boucksom, M. Paun, Th. Peternell),
    Transcendental holomorphic Morse inequalities p.1159}
\CC{14-06
32-06
00B05}
%% END ITEM
%% BEGIN ITEM
\AN{1072.32003 (02170605)}{2004}
\AU{Qiu, Chunhui; Zhong, Tongde}
\TI{Integral formulas for differential forms of type $(p,q)$ on complex Finsler manifolds.}
\LA{English}
\DT{J}
\SO{Sci. China, Ser. A 47, No.2, 284-296 (2004). 
\par http://219.238.6.200/journal?code=02}
\ISSN{006-928}
\AB{The authors adapt the technique of {\it J.-P. Demailly} and {\it
    C. Laurent-Thi\'ebaut} [Ann. Sci. \'Ec. Norm. Sup\'er., IV. S\'er.
    20, No. 4, 579-598 (1987; Zbl 0632.32004)] to construct integral
    representation formulas of Koppelman type on complex manifolds
    having a strongly pseudoconvex Finsler metric (rather than a
    Hermitian metric).}
\RV{Harold P.Boas (College Station)}
\CC{32A25
32W05
53C56
53C60}
%% END ITEM
%% BEGIN ITEM
\AN{1071.14045 (02132941)}{2004}
\AU{Duval, Julien}
\TI{A hyperbolic sextic surface in $\Bbb P^3(\Bbb C)$. (Une sextique hyperbolique dans $\Bbb P^3(\Bbb C)$)}
\LA{French}
\DT{J}
\SO{Math. Ann. 330, No.3, 473-476 (2004).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s00208-004-0551-0}
\ISSN{025-583}
\AB{The aim of this paper is to construct a hyperbolic sextic in
    $\Bbb{P}^3(\Bbb{C})$, therefore showing the existence of such
    a surface. A subset of $\Bbb{P}^3(\Bbb{C})$ is called hyperbolic
    if it does not contain an entire curve, this is a non-constant
    holomorphic image of $\Bbb{C}$. The interest in the subject comes
    from Kobayashi's conjecture that states that a generic surface
    of degree greater than 4 in $\Bbb{P}^3(\Bbb{C})$ is hyperbolic.
    The result has been proved first for degree $\ge 36$ by {\it
    M. McQuillan} [Geom. Funct. Anal. 9, No. 2, 370-392 (1999; Zbl
    0951.14014)], and then for degree $\ge 21$ by {\it J.-P. Demailly}
    and {\it J. El Goul} [Am. J. Math. 122, No. 3, 515-546 (2000;
    Zbl 0966.32014)]. Parallel to this, an effort to construct examples
    of hyperbolic surfaces of lowest degree possible has been performed:
    prior to the article under review, the best known result corresponded
    to degree 8.}
\RV{Ernesto Girondo (Madrid)}
\CC{14J70
32Q45
14N05
14J29}
%% END ITEM
%% BEGIN ITEM
\AN{1066.32002 (02131981)}{2004}
\AU{Berman, Robert}
\TI{Bergman kernels and local holomorphic Morse inequalities.}
\LA{English}
\DT{J}
\SO{Math. Z. 248, No.2, 325-344 (2004).  ISSN 1432-1823]
\par http://dx.doi.org/10.1007/s00209-003-0630-z}
\ISSN{025-587}
\AB{{\it J.-P. Demailly's} holomorphic Morse inequalities [Ann. Inst.
    Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)] asymptotically
    estimate the dimension of the Dolbeault cohomology group associated
    to a high tensor power of a Hermitian holomorphic line bundle
    over a complex manifold. The author gives a relatively elementary
    proof of these inequalities. The new idea is to localize and
    to estimate the Bergman kernel function asymptotically by a model
    kernel in~$\Bbb{C}^n$.}
\RV{Harold P.Boas (College Station)}
\CC{32A25
32L10
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{1063.32002 (02057060)}{2004}
\AU{Popovici, Dan}
\TI{Effective estimate of positivity loss in current regularizations. (Estimation effective de la perte de positivit\'e dans la r\'egularisation des courants)}
\LA{French. Abridged English version}
\DT{J}
\SO{C. R., Math., Acad. Sci. Paris 338, No.1, 59-64 (2004). 
\par http://dx.doi.org/10.1016/j.crma.2003.11.011}
\ISSN{631-073}
\AB{Let $(X,\omega)$ be a compact complex Hermitian manifold, $\gamma$
    a real continuous $(1,1)$ form and $T \ge \gamma$ be a $d$-closed
    $(1,1)$ almost positive current on $X$. Then $T = \alpha + i
    \partial \bar{\partial} \varphi$, $\alpha$ a differentiable $(1,1)$
    form and $\varphi$ an almost plurisubharmonic function. A result
    of Demailly states the existence of a series of almost plurisubharmonic
    functions $\varphi_m$ with analytic singularities satisfying
    several nice properties, including $\alpha + i \partial \bar{\partial}\varphi_m
    \ge \gamma - \epsilon_m \omega$ with $\epsilon_m >0$ tending
    to zero. \par
In this paper the author gives, in case $\gamma$
    is $C^1$, the bound $C \root 4\of{ m}$ for $\epsilon_m$, where
    $C$ a constant not depending on $m$. The estimate is improved
    to $C/m$ in case $\gamma$ is $C^\infty$ and closed.}
\RV{Theo de Jong (Mainz)}
\CC{32C30}
%% END ITEM
%% BEGIN ITEM
\AN{1061.14019 (02111357)}{2004}
\AU{Sol\'a Conde, Luis Eduardo; Wi\'sniewski, Jaros{\l}aw A.}
\TI{On manifolds whose tangent bundle is big and 1-ample.}
\LA{English}
\DT{J}
\SO{Proc. Lond. Math. Soc., III. Ser. 89, No.2, 273-290 (2004).  ISSN 1460-244X]
\par http://dx.doi.org/10.1112/S0024611504014856}
\ISSN{024-611}
\AB{The article is a contribution to the classification of $n$-dimensional
    complex projective manifolds $X$ with respect to special positivity
    properties of their holomorphic tangent bundles $T(X)$, described
    in terms of the line bundle $L(X):={\cal O}_{\Bbb P(T(X))}(1)$
    over the projectivisation $\Bbb P(T(X))$ of $T(X)$. This theory
    is inspired by the famous result of {\it S. Mori} [Ann. Math.
    (2) 110, 593-606 (1979; Zbl 0423.14006)] that $L(X)$ is ample
    iff $X$ is isomorphic to $\Bbb P^n$. In the present paper it
    is assumed that $L(X)$ is big, $L^{\otimes m}(X)$ is generated
    by global sections for $m\gg 0$, and that the fibers of the holomorphic
    map $\Bbb P(T(X))\rightarrow \Bbb P^N$ induced by $H^0(\Bbb P(T(X)),
    L^{\otimes m})$ are at most onedimensional if $m$ is sufficiently
    big. The authors show: If $L$ is not ample, then $X$ is isomorphic
    to the smooth quadric $Q^n$ or, for $n=3$, to the flag manifold
    of full flags in $\Bbb C^3$, a $\Bbb P^2$-bundle over $\Bbb P^1$.
    The methods of the proof depend on the Picard number $\rho(X)$.
    In the case $\rho(X)=1$ they use (and reprove) a result of {\it
    J. Wierzba} [J. Algebr. Geom. 12, No. 3, 507-534 (2003; Zbl pre02064089)]
    about the existence of a symplectic form on the space parametrizing
    1-dimensional fibers of a symplectic contraction and results
    of {\it S. Kebekus, T. Peternell, A. J. Sommese} and {\it J.
    A. Wisniewski} [Invent. Math. 142, No. 1, 1-15 (2000; Zbl 0994.53024)]
    about contractions of contact manifolds, and in the case $\rho(X)>1$
    they apply (and reprove) a theorem of {\it J. Demailly, T. Peternell}
    and {\it M. Schneider} [J. Algebr. Geom. 3, No. 2, 295-345 (1994;
    Zbl 0827.14027)] about smoothness of Fano-Mori contractions of
    manifolds with nef tangent bundle.}
\RV{Eberhard Oeljeklaus (Bremen)}
\CC{14E30
32Q57
14J10
14J45
14J50}
%% END ITEM
%% BEGIN ITEM
\AN{1057.31003 (02127124)}{2004}
\AU{Toujani, Moncef; Ben Messaoud, H\`edi}
\TI{Extension of negative plurisubharmonic current with condition on the slices. (Prolongement d'un courant n\'egatif plurisousharmonique avec condition sur les tranches)}
\LA{French. Abridged English version}
\DT{J}
\SO{C. R., Math., Acad. Sci. Paris 339, No.8, 543-548 (2004). 
\par http://dx.doi.org/10.1016/j.crma.2004.07.021}
\ISSN{631-073}
\AB{Summary: We prove a theorem on the extension of a negative (or
    positive) plurisubharmonic current $T$ (i.e. such that $dd^cT\geqslant
    0$) with condition on the slices with respect to some coordinates.
    This theorem generalizes a result proved by El Mir-Ben Messaoud
    relative to $d$-closed positive currents with a condition on
    slices. The method consists first of proving a Chern-Levine-Nirenberg
    inequality for a positive (or negative) psh current, which is
    a generalization of results obtained by Bedford-Taylor, Demailly
    and Sibony for $d$-closed positive currents. Also we prove an
    Oka type inequality for positive psh currents, thereby generalizing
    former results by Ben Messaoud-El Mir concerning positive currents
    with a negative $dd^c$.}
\CC{31C10
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{1055.14007 (02116146)}{2004}
\AU{Bauer, Th.; K\"uronya, A.; Szemberg, T.}
\TI{Zariski chambers, volumes, and stable base loci.}
\LA{English}
\DT{J}
\SO{J. Reine Angew. Math. 576, 209-233 (2004).  ISSN 1435-5345]
\par http://www.degruyter.de/journals/crelle/2004/576\_209.html}
\ISSN{075-410}
\AB{Let $X$ be an $n$-dimensional smooth projective variety and $L$
    a line bundle on $X$. Let $\text{vol} _X(L):= \limsup _k h^0(X,L^{\otimes
    k})\cdot n!/k!$ be the asymptotic volume of $L$. This important
    invariant was introduced by {\it S. Cutkosky} [Duke Math. J.
    53, 149-156 (1986; Zbl 0604.14002)] and studied by Demailly,
    Ein, Lazarfeld, Nakamaye and others. It was recently proved [{\it
    R. Lazarsfeld}, ``Positivity in algebraic geometry'', Vol. I
    (2004; Zbl pre02106914), Vol. II (2004; Zbl pre02106915)] that
    $\text{vol} _X(L)$ is a continuous log-concave function on the
    N\'eron-Severi group. Here the authors prove that if $n=2$ the
    cone of big divisors of $X$ has a locally finite decomposition
    into locally polyhedral subcones such that the restriction of
    $\text{vol}_X(L)$ to each subcone is polynomial, but that this
    is not true for $n \ge 3$. They apply their results to the study
    of the stable loci of $L$ for varying $L$ (on the interiors of
    these subcones the stable base locus is constant). Among the
    application is a continuity theorem for the Zariski decomposition
    of $\bbfR$-divisors ($n=2$).}
\RV{Edoardo Ballico (Povo)}
\CC{14C22
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{02091228}{2004}
\AU{Takagi, Shunsuke; Watanabe, Kei-ichi}
\TI{When does the subadditivity theorem for multiplier ideals hold?.}
\LA{English}
\DT{J}
\SO{Trans. Am. Math. Soc. 356, No.10, 3951-3961 (2004).  ISSN 1088-6850]
\par http://www.ams.org/tran/2004-356-10/S0002-9947-04-03436-1/home.html}
\ISSN{002-994}
\AB{Summary (preliminary): Demailly, Ein and Lazarsfeld proved the
    subadditivity theorem for multiplier ideals on nonsingular varieties,
    which states the multiplier ideal of the product of ideals is
    contained in the product of the individual multiplier ideals.
    We prove that, in the two-dimensional case, the subadditivity
    theorem holds on log terminal singularities. However, in the
    higher dimensional case, we have several counterexamples. We
    consider the subadditivity theorem for monomial ideals on toric
    rings and construct a counterexample on a three-dimensional toric
    ring.}
\CC{13B22
14J17}
%% END ITEM
%% BEGIN ITEM
\AN{02210038}{2004}
\AU{Napier, Terrence; Ramachandran, Mohan}
\TI{Elementary construction of exhausting subsolutions of elliptic operators.}
\LA{English}
\DT{J}
\SO{Enseign. Math., II. S\'er. 50, No.3-4, 367-390 (2004). }
\ISSN{013-858}
\AB{A noncompact Riemannian manifold admits a $C^\infty$ strictly
    subharmonic exhausting function as it follows by a theorem due
    to {\it R.E. Greene} and {H. Wu} [Ann. Inst. Fourier 25, No.
    1, 215-235 (1975; Zbl 0307.31003)]. An elementary proof of this
    fact is given by {\it J.-P. Demailly} [Math. Z. 204, No. 2, 283-295
    (1990; Zbl 0682.32017)]. \par
In the paper under review, the
    authors propose a further simplification of Demailly's proof
    and describe some, mostly known, applications. These include
    for instance, the fact that the holomorphic line bundle associated
    to a nontrivial effective divisor on a compact connected complex
    manifold admits a $C^\infty$ Hermitian metric with positive scalar
    curvature.}
\RV{Dian K.Palagachev (Bari)}
\CC{58J05}
%% END ITEM
%% BEGIN ITEM
\AN{1077.32504 (02171928)}{2003}
\AU{Demailly, Jean-Pierre; Peternell, Thomas}
\TI{A Kawamata-Viehweg vanishing theorem on compact K\"ahler manifolds.}
\LA{English}
\DT{J}
\SO{J. Differ. Geom. 63, No.2, 231-277 (2003). 
\par http://www.intlpress.com/journals/JDG/}
\ISSN{022-040}
\AB{This paper appeared earlier under the same title in the proceedings
    of a conference. See {\it J.-P. Demailly} and {\it T. Peternell},
    Surv. Differ. Geom. 8, 139-169 (2003; Zbl 1053.32011).}
\RV{Imre Patyi (Atlanta)}
\CC{32L20
32J27}
%% END ITEM
%% BEGIN ITEM
\AN{1076.32035 (02145617)}{2003}
\AU{Ma, Zhongtai}
\TI{$C_{p,q}^{s+\alpha}$-forms solution of $\overline\partial_b$-equation and its $L^s_{p,q}$ estimates.}
\LA{English}
\DT{J}
\SO{Chin. Q. J. Math. 18, No.3, 234-241 (2003). }
\ISSN{002-046}
\AB{The author uses holomorphic support functions of strictly pseudoconvex
    domains in Stein manifolds and a kernel function defined by {\it
    J. P. Demailly} and {\it C. Laurent-Thiebaut} [Ann. Sci. \'Ec.
    Norm. Sup\'er., IV. S\'er. 20, No. 4, 579-598 (1987; Zbl 0632.32004)]
    in order to get an integral representation of the solution of
    the $(p,q)$-form $\overline{\partial}_b$-equation and estimates
    of the solution in Besov spaces.}
\RV{Fritz Haslinger (Wien)}
\CC{32W10
32W05
32A25}
%% END ITEM
%% BEGIN ITEM
\AN{1048.32016 (02002269)}{2003}
\AU{El Goul, Jawher}
\TI{Logarithmic jets and hyperbolicity.}
\LA{English}
\DT{J}
\SO{Osaka J. Math. 40, No.2, 469-491 (2003). }
\ISSN{030-612}
\AB{{\it S. Kobayashi} [Hyperbolic manifolds and holomorphic mappings
    (1970; Zbl 0207.37902)] conjectured that the complement of a
    generic plane curve of degree $d \geq 5$ is hyperbolic. In the
    author's previous work with {\it J. P. Demailly} [Am. J. Math.
    122, No. 3, 515-546 (2000; Zbl 0966.32014)] it was shown that
    the conjecture holds for very generic curves of degree $d \geq
    21$. In the present paper the bound is improved to $15$. \par
The
    basic strategy of proof is similar to that of the paper with
    Demailly, but using the logarithmic generalization by {\it G.
    Dethloff} and {\it S. Lu} [Osaka J. Math. 38, No. 1, 185-237
    (2001; Zbl 0982.32022)] of the projectivized jet bundles introduced
    by {\it J. P. Demailly} [Proc. Symp. Pure Math. 62, 285-360 (1997;
    Zbl 0919.32014)]. \par
The author also generalizes a result
    of {\it M. McQuillan} [Publ. Math., Inst. Hautes \'Etud. Sci.
    87, 121-174 (1998; Zbl 1006.32020)] to the logarithmic context,
    proving that for an algebraic surface $(\bar X,D)$ of log-general
    type, any entire curve contained in a leaf of a holomorphic foliation
    on $\bar X$ and omitting the divisor $D$ is algebraically degenerate.
    This result will certainly be of independant interest.}
\RV{Gerd Dethloff (Brest)}
\CC{32Q45
14J29
32S65
53C12}
%% END ITEM
%% BEGIN ITEM
\AN{1046.53028 (01922414)}{2003}
\AU{Boyer, Charles P.; Galicki, Krzysztof; Nakamaye, Michael}
\TI{On the geometry of Sasakian-Einstein 5-manifolds.}
\LA{English}
\DT{J}
\SO{Math. Ann. 325, No.3, 485-524 (2003).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s00208-002-0388-3}
\ISSN{025-583}
\AB{In this article the authors consider Sasaki-Einstein structures
    on compact 5-manifolds consisting of connected sums of $k$ copies
    of $S^2 \times S^3$ for some integer $k \ge 1$. In a previous
    paper the first two authors considered regular Sasaki-Einstein
    structures on these spaces, and they showed, among other results,
    that there exists a complex $2k-8$ parameter family of such structures
    for $5 \le k \le 8$. \par
In the present article the authors
    consider non-regular Sasaki-Einstein structures on the same 5-manifolds
    in the cases $1 \le k \le 7$. For $k = 1$ (i.e. $S^2 \times S^3$)
    they show that there are 14 inequivalent non-regular Sasaki-Einstein
    structures. For $2 \le k \le 7$ they prove the existence of $d$-parameter
    complex families of such structures, where $d$ is a positive
    integer (or integers) that depends on $k$. The case $k=8$ is
    treated in a separate article by the first two authors. The authors
    raise the question of whether there exist any Sasaki-Einstein
    structures on these spaces for $k \ge 9$. \par
The article uses
    methods based on recent work of {\it J.-P. Demailly} and {\it
    J. Koll\'ar} [Ann. Sci. \'Ec. Norm. Sup\'er., Iv. S\'er. 34,
    525-556 (2001; Zbl 0994.32021)] and {\it J. M. Johnson} and {\it
    J. Koll\'ar} [Ann. Inst. Fourier 51, 69-79 (2001; Zbl 0974.14023)].
    The authors of these two articles gave methods for constructing
    K\"ahler-Einstein metrics on log del Pezzo surfaces.}
\RV{Patrick Eberlein (Chapel Hill)}
\CC{53C25
53C20
53C12
32Q20}
%% END ITEM
%% BEGIN ITEM
\AN{1034.32017 (02019747)}{2003}
\AU{Rousseau, Erwan}
\TI{Hyperbolicit\'e du compl\'ementaire d'une courbe dans $\Bbb P^2$: Le cas de deux composantes (Hyperbolicity of the complements of plane algebraic curves: The two component case).}
\LA{French. Abridged English version}
\DT{J}
\SO{C. R., Math., Acad. Sci. Paris 336, No.8, 635-640 (2003). 
\par http://dx.doi.org/10.1016/S1631-073X(03)00136-5}
\ISSN{631-073}
\AB{Summary: We prove that the complement of a very generic complex
    curve with two components in $\Bbb P^2$ of degrees $d_1 \leqslant
    d_2$ is hyperbolic in the sense of Kobayashi in the following
    cases: $d_1 \geqslant 5$; $d_1=4$ and $d_2 \geqslant 7$; $d_1=d_2=4$;
    $d_1=3$ and $d_2 \geqslant 9$; $d_1=2$ and $d_2 \geqslant 12$.
    We consider logarithmic jets developed by {\it G. Dethloff} and
    {\it S. Lu} [Osaka J. Math. 38, 185-237 (2001; Zbl 0982.32022)],
    who generalized to the logarithmic situation {\it J.-P. Demailly}'s
    jet bundles [Proc. Symp. Pure Math. 62, 285-360 (1997; Zbl 0919.32014)],
    and used by El Goul to obtain results about the hyperbolicity
    of the complement of a very generic curve in $\Bbb P^2$ in the
    case of a single component.}
\CC{32Q45}
%% END ITEM
%% BEGIN ITEM
\AN{1029.32014 (01947255)}{2003}
\AU{Brunella, Marco}
\TI{Subharmonic variation of the leafwise Poincar\'e metric.}
\LA{English}
\DT{J}
\SO{Invent. Math. 152, No.1, 119-148 (2003).  ISSN 1432-1297]
\par http://dx.doi.org/10.1007/s00222-002-0269-0}
\ISSN{020-991}
\AB{Let $X$ be a complex algebraic surface which can have Hirzebruch-Jung
    singularities and let $F$ be a holomorphic foliation such that
    the singularities of $F$ are isolated and this singular locus
    sing$(F)$ is disjoint of that of $X$. \par
The author introduces
    a (singular) Hermitian metric on the restriction to $X \setminus
    $ sing$(F)$ of the canonical bundle $K_F$ of $F$, which is called
    the canonical Poincar\'e metric. He proves the following concretization
    of an existence result, due to Demailly, of Hermitian metrics
    on $K_F$ with positive curvature: let $F$ be a nef foliation
    on a $X$, suppose that there exists at least one hyperbolic leaf
    of $F$. Then, the canonical Poincar\'e metric has positive curvature.
    \par
Finally, in this interesting paper, the author proves that
    the only reduced nef foliations of Kodaira dimension $-\infty$
    are the Hilbert modular ones. This together with McQuillan's
    results achieves the classification of foliations of nongeneral
    type.}
\RV{Carlos Galindo (Castellon)}
\CC{32S65}
%% END ITEM
%% BEGIN ITEM
\AN{1029.53058 (01947258)}{2003}
\AU{Huybrechts, Daniel}
\TI{Erratum to: Compact hyperk\"ahler manifolds: basic results.}
\LA{English}
\DT{J}
\SO{Invent. Math. 152, No.1, 209-212 (2003).  ISSN 1432-1297]
\par http://dx.doi.org/10.1007/s00222-002-0280-5}
\ISSN{020-991}
\AB{Concerns the author's paper in ibid. 135, 63-113 (1999; Zbl 0953.53031).\par
It
    was pointed out by D. Kaledin that the proof of Proposition 3.8
    is wrong. Actually, the proposition itself cannot be true as
    we explain here. It was used to prove Corollary 3.10 and Theorem
    3.11. The latter is the so-called projectivity criterion for
    hyper-K\"ahler manifolds. Here we give a correct proof of these
    two results. The main input is a recent theorem of {\it J.-P.
    Demailly} and {\it M. Paun} [Ann. Math. (2) (to appear), see
    also math.AG/0105176]. All the other results of the paper remain
    unaffected.}
\CC{53C26
14J28}
%% END ITEM
%% BEGIN ITEM
\AN{1028.13003 (01910626)}{2003}
\AU{Hara, Nobuo; Yoshida, Ken-ichi}
\TI{A generalization of tight closure and multiplier ideals.}
\LA{English}
\DT{J}
\SO{Trans. Am. Math. Soc. 355, No.8, 3143-3174 (2003).  ISSN 1088-6850]
\par http://www.ams.org/journal-getitem?pii=S0002-9947-03-03285-9}
\ISSN{002-994}
\AB{The authors introduce a new variant of tight closure: For any
    fixed ideal $a$ in a ring $R$ of positive prime characteristic
    $p$, the $a$-tight closure $I^{sa}$ is the set of all $x \in
    R$ for which there exists $c \in R$ not in any minimal prime
    ideal, such that for all $q = p^e$, $c x^q a^q \subseteq I^{[q]}$,
    where $I^{[q]}$ is the $e$th Frobenius power of $I$. In case
    $a = R$, this is the definition of the usual tight closure of
    $I$, as defined by {\it M. Hochster} and {\it C. Huneke} [J.
    Am. Math. Soc. 3, 31-116 (1990; Zbl 0701.13002)]. The paper under
    review develops the theory of $a$-tight closure, and is rich
    in theory and applications. Hara and Yoshida prove that the $a$-tight
    closure is not a true closure operation, namely that $I^{sa}$
    in general need not equal $(I^{sa})^{sa}$, nevertheless, they
    prove that it is still very useful. In section~2 they prove that
    a criterion from a paper by {\it N. Hara, K.-I. Watanabe} and
    {\it K. Yoshida} [J. Algebra 247, 153-190 (2002; Zbl 1008.13001)]
    of F-rationality of a Rees algebra can be expressed using $a$-tight
    closure. Also, under the usual assumptions of the tight closure
    theory, the authors prove that $a$-test elements exist. Furthermore,
    the ideals $\tau(a)$ of $a$-test elements satisfy several interesting
    properties, including a form of the Brian\c{c}on-Skoda theorem
    [see {\it J. Lipman}, Math. Res. Lett. 1, 739-755 (1994; Zbl
    0844.13015) for a form using adjoint ideals]. In sections 3 and
    4 the authors prove some connections between $a$-test ideals
    and multiplier ideals. Some of such connections had been proved
    by the author and others [{\it N. Hara}, Trans. Am. Math. Soc.
    35, 1885-1906 (2001; Zbl 0976.13003), {\it K. E. Smith}, Commun.
    Algebra 28, 5915-5929 (2000; Zbl 0979.13007) and {\it S. Takagi},
    An interpretation of multiplier ideals via tight closure (preprint)].
    In the paper under review, theorem~4.5 asserts subadditivity
    of test ideals: $\tau(ab) \subseteq \tau(a) \tau(b)$. Compare
    this with subadditivity of multiplier ideals in the paper by
    {\it J.-P. Demailly, L. Ein} and {\it R. Lazarsfeld} [Mich. Math.
    J. 48, Spec. Vol., 137-156 (2000; Zbl pre01700866)]. Theorem~3.4
    more generally asserts that in a sense, the characteristic $p$
    proofs of $a$-test ideals prove the properties of multiplier
    ideals in characteristic $0$. Section~5 was motivated by {\it
    E. Hyry}'s papers [Manuscr. Math. 98, 377-390 (1999; Zbl 0933.13006)
    and Proc. Am. Math. Soc. 129, 1299-1308 (2001; Zbl 0986.13003)]:
    It shows connections between F-rationality of the Rees algebra
    of $I$ and test ideals of powers of $I$. In the last section
    the authors generalize $a$-test ideals to $a$-test ideals with
    rational coefficients. The motivation for this was the corresponding
    theory of multiplier ideals with rational coefficients, as in
    the paper by {\it L. Ein, R. Lazarsfeld} and {\it K. E. Smith}
    [Invent. Math. 144, 241-252 (2001; Zbl pre01655623) and {\it
    R. Lazarsfeld}, ``Positivity in algebraic geometry'' (preprint)].
    More on this and another form of the Brian\c{c}on-Skoda theorem
    can be found in the subsequent paper by {\it N. Hara} and {\it
    S. Takagi}, ``Some remarks on generalization of test ideals''
    (preprint).}
\RV{Irena Swanson (Las Cruces)}
\CC{13A35
14B05
13B22}
%% END ITEM
%% BEGIN ITEM
\AN{1023.32005 (01910572)}{2003}
\AU{Joita, Cezar}
\TI{Traces of convex domains.}
\LA{English}
\DT{J}
\SO{Proc. Am. Math. Soc. 131, No.9, 2721-2725 (2003).  ISSN 1088-6826]
\par http://www.ams.org/journal-getitem?pii=S0002-9939-03-07119-3}
\ISSN{002-993}
\AB{{\it K. Diederich} and {\it T. Ohsawa} [Tokyo J. Math. 21, 353-358
    (1998; Zbl 0922.32010)] proved that if $M$ is a complex manifold
    and $N$ is a complex submanifold, then any locally hyperconvex
    Stein open subset of $N$ is the trace of a locally hyperconvex
    Stein open subset of $M$. {\it Y. T. Siu} [Invent. Math. 38,
    89-100 (1976; Zbl 0343.32014)] argued that if $Y$ is a closed
    complex subspace of $X$ and $Y$ is Stein, then $Y$ has a Stein
    neighborhood. Also, it has been proved by {\it V. V\^aj\^aitu}
    [C. R. Acad. Sci., Paris, S\'er. I 322, 823-828 (1996; Zbl 0866.32008)]
    that if $Y$ is hyperconvex, then $Y$ has a hyperconvex neighborhood.\par
Using
    the methods of {\it J. P. Demailly} [Math. Z. 204, 283-295 (1990;
    Zbl 0682.32017)], the author sets up a general framework for
    the above theorems and generalizes Diederich and Ohsawa's results
    for reduced complex spaces.}
\RV{Vasily A.Chernecky (Odessa)}
\CC{32C15
32E10
32Q28}
%% END ITEM
%% BEGIN ITEM
\AN{1017.32018 (01821230)}{2003}
\AU{Campana, F.; Peternell, T.; Zhang, Qi}
\TI{On the Albanese maps of compact K\"ahler manifolds.}
\LA{English}
\DT{J}
\SO{Proc. Am. Math. Soc. 131, No.2, 549-553 (2003).  ISSN 1088-6826]
\par http://www.ams.org/journal-getitem?pii=S0002-9939-02-06702-3}
\ISSN{002-993}
\AB{The authors consider a K\"ahler manifold $X$ with numerically
    effective anticanonical bundle and are particularly interested
    in the Albanese map of $X$. In a case by case proof they show,
    based on results of former work [see: {\it J. P. Demailly, Th.
    Peternell} and {\it M. Schneider}, Comp. Math. 89, 217-240 (1993;
    Zbl 0884.32023)] and {\it Q. Zhang} [J. Reine Angew. Math. 478,
    57-60 (1996; Zbl 0855.14007)] that this map is surjective with
    connected fibers if $\dim X\le 4$. The same methods yield the
    following result: If there is a surjective holomorphic map from
    $X$ onto a normal projective algebraic variety $Y$ of general
    type, then dim $Y\le$ dim $X-3$.\par
Moreover the authors study
    holomorphic maps from a compact K\"ahler threefold onto a non-rational
    smooth curve $C$. Under the additional assumptions that the general
    fiber has Kodaira dimension $\kappa(F)=-\infty$, irregularity
    $q(F)=1$, and that the relative anticanonical bundle $K_{Z|C}$
    of $Z$ over $C$ is numerically effective it follows that the
    singular fibers of those maps are multiples of smooth surfaces.
    If $C$ is elliptic then the maps are already smooth and there
    is a factorization of such a map into a $\Bbb P^1$-bundle over
    a smooth minimal surface $Y$ and an elliptic fibration of $Y$
    over $C$ where the only singular fibers are multiple fibers.}
\RV{Eberhard Oeljeklaus (Bremen)}
\CC{32J27
32J17
14E30
14J35}
%% END ITEM
%% BEGIN ITEM
\AN{1016.53038 (01852948)}{2003}
\AU{Huybrechts, Daniel}
\TI{Compact hyperk\"ahler manifolds.}
\LA{English}
\DT{CA}
\SO{Gross, Mark (ed.) et al., Calabi-Yau manifolds and related geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001. Berlin: Springer. Universitext. 161-236 (2003). [ISBN 3-540-44059-3/pbk]}
\AB{These notes grew out of eight lectures delivered by the author
    at the Nordfjordeid summer school in June 2001. They form the
    third (and last) part of this Springer Universitext and consists
    of nine sections numbered by \S\S 20-28 (see also Zbl 1016.53041
    and Zbl 1014.14019).\par
The main theme of this part is the
    study of compact hyper-K\"ahler manifolds by means of their periods.
    From the differential geometric point of view, these objects
    are compact Riemannian manifolds with special holonomy $Sp(n)$.
    On the other hand, from an algebro-geometric point of view, compact
    hyper-K\"ahler manifolds occur as simply-connected compact K\"ahler
    manifolds that admit an essentially unique, everywhere non-degenerate
    holomorphic 2-form. A combination of the corresponding specific
    methods leads to a general theory of this type of manifolds which
    parallels, to a large extent, the classical theory of complex
    tori, abelian varieties, and $K3$ surfaces. The period of a compact
    hyper-K\"ahler manifold is given by the second cohomology endowed
    with its Hodge structure and the so-called Beauville-Bogomolov
    quadratic form. Finally, compact hyper-K\"ahler manifolds appear
    in the theory of Ricci-flat manifolds as one of the basic components,
    together with complex tori and Calabi-Yau manifolds, and this
    very fact makes them play a prominent role in modern mathematical
    physics, particularly in string theories and in the mysterious
    world of mirror symmetry. \par
Thus, in order to provide a sufficiently
    profound introduction to the geometry of compact hyper-K\"ahler
    manifolds, the author discusses the following selected topics:
    \S 20. (General) Introduction; \S 21. Holomorphic symplectic
    manifolds (including $K3$ surfaces, higher-dimensional examples,
    birational maps, and holomorphic symplectic manifolds as fiber
    spaces); \S 22. Deformations of complex structures (including
    a general account of deformations of K\"ahler manifolds, deformations
    of holomorphic symplectic manifolds, the Mukai flop, and Beauville's
    local Torelli theorem); \S 23. The Beauville-Bogomolov form (including
    the differential-geometric definition of a compact hyper-K\"ahler
    manifold, the construction of the Beauville-Bogomolov quadratic
    form, and the proof of the integrality of the latter by means
    of the Beauville-Fujiki relation); \S 24. Cohomology of compact
    hyper-K\"ahler manifolds (\`a la Verbitsky-Bogomolov, 1996);
    \S 25. Twistor space and moduli space (including the construction
    of the moduli space of marked irreducible holomorphic symplectic
    manifolds, the structure of the period map, and a counter-example
    to a global Torelli theorem for higher-dimensional holomorphic
    symplectic manifolds); \S 26. Projectivity of hyper-K\"ahler
    manifolds (including the projectivity criteria due to Demailly-Paun-Nakai-Moishezon
    and to the author himself, together with some finiteness theorems
    concerning the deformation types of holomorphic symplectic manifolds);
    \S 27. Birational hyper-K\"ahler manifolds; \S 28. The (birational)
    K\"ahler cone.\par
The last two sections (\S\S 27 and 28) touch
    upon those more special topics that are closely related to the
    author's own research work, which leads the reader directly to
    the forefront of current research in this field. Altogether,
    the author has tried to cover large parts of the existing theory
    on compact hyper-K\"ahler manifolds, and he has managed to provide
    a fairly comprehensive, detailed and vivid account of it. Results
    for which a proof is given (or sketched) are stated as ``Proposition'',
    whereas those results that are beyond the scope of these notes
    are labelled as ``Theorem''. Basically, the exposition is nearly
    self-contained and written in a very lucid, refined style. Various
    concrete examples and exercises scattered throughout the text
    enrich this introduction to hyper-K\"ahlerian geometry in a very
    effective way.\par
Apart from being an excellent introduction
    to the subject, these notes also represent a profound source
    for active researchers in complex geometry.}
\RV{Werner Kleinert (Berlin)}
\CC{53C26
53D05
53C28}
%% END ITEM
%% BEGIN ITEM
\AN{02058042}{2003}
\AU{Hyry, Eero; Smith, Karen E.}
\TI{On a non-vanishing conjecture of Kawamata and the core of an ideal.}
\LA{English}
\DT{J}
\SO{Am. J. Math. 125, No.6, 1349-1410 (2003).  ISSN 1080-6377]
\par http://muse.jhu.edu/journals/american\_journal\_of\_mathematics}
\ISSN{002-932}
\AB{Let $(A,M)$ be a local ring essentially of finite type over a
    field of characterisitc zero with maximal ideal $M$. For any
    ideal $I$ we have the notions of integral closure of $I$, reductions
    of $I$ and the core of $I$. The last one was introduced by {\it
    D.~Rees} and {\it J.~Sally} in ["General elements and joint reductions".
    Mich. Math. J. 35, No.2, 241-254 (1988; Zbl 0666.13004)]. These
    notions are of course intimately related. More recently, yet
    another notion has become prominent, the adjoint ideal or the
    multiplier ideal of $I$. I believe this fairly natural notion
    appeared in a more geometric setting in the works of Demailly.
    For a recent exposition by {\it J.-P. Demailly} see ["Multiplier
    ideal sheaves and analytic methods in algebraic geometry". School
    on Vanishing Theorems and Effective Results in Algebraic Geometry
    (Trieste, 2000), 1-148, Abdus Salam Int. Cent. Theoret. Phys.,
    ICTP Lect. Notes 6, 1-148 (2001; Zbl pre01816813)]. The paper
    under review connects these concepts in a very elegant way. The
    authors also prove that if their results can be modified to give
    an appropriate graded version, then a well known conjecture of
    {\it Y. Kawamata} ["On effective non-vanishing and base-point-freeness.
    Kodaira's issue". Asian J. Math. 4, No.1, 173-181 (2000; Zbl
    1060.14505)] will follow. For the review, let me quote just one
    of their results. So, let $(A,M)$ be as above and let $I$ be
    $M$-primary. Assume that the irrelevant ideal of the Rees algebra
    $A[It]$ is a Cohen-Macaulay module over $A[It]$ and $Y=\text{Proj}
    A[It]$ has rational singularities. Let $\omega_A$ denote a canonical
    module of $A$. Then $$\text{core} (I\omega_A)=\text{adj} (I^d\omega_A):=
    \Gamma(Y,I^d\omega_Y).$$ More results in this direction can also
    be found in the authors' article ["Core versus graded core, and
    global sections of line bundles". Trans. Amer. Math. Soc. 356,
    No.8, 3143-3166 (2004; Zbl pre02091194)].}
\RV{N.Mohan Kumar (St.Louis)}
\CC{13A30
13D45
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{1031.14014 (01969550)}{2002}
\AU{Brunella, Marco}
\TI{Courbes enti\`eres dans les surfaces alg\'ebriques complexes. (Entire curves in complex algebraic surfaces).}
\LA{French}
\DT{CA}
\SO{S\'eminaire Bourbaki. Volume 2000/2001. Expos\'es 880-893. Paris: Soci\'et\'e Math\'ematique de France. Ast\'erisque. 282, 39-61 (2002). [ISBN 2-85629-130-9/pbk;}
\ISSN{303-117}
\AB{In this survev article, the author discusses some spectacular
    recent developments in the theory of complex algebraic surfaces.
    The topic explained here is closely related to some elder conjectures
    concerning the configuration of entire holomorphic curves in
    complex varieties, which have regained much attention in the
    last ten years. By definition, an entire (holomorphic) curve
    in a complex variety $X$ is the image of a non-constant holomorphic
    map $f:\bbfC\to X$. If $X$ does not contain any entire curve,
    then $X$ is called a hyperbolic variety (in the sense of Brody-Kobayashi).
    Classical examples of hyperbolic varieties are given by those
    compact complex varieties which admit a K\"ahler metric with
    negative holomorphic sectional curvature. On the other hand,
    projective complex varieties of general type may contain entire
    curves, and so they are natural candidates for the deeper study
    of entire curves and their location. In this context, and almost
    25 years ago, M. Green and Ph. A. Griffiths proposed the following
    conjecture:\par
Let $X$ be a projective complex variety, and
    let $f:\bbfC\to X$ be an entire curve in $X$. Then $f(\bbfC)$
    is contained in an algebraic hypersurface $Y\subset X$, and therefore
    not Zariski-dense in $X$.\par
A slightly stronger conjecture,
    motivated from arithmetic geometry, is due to S. Lang, who predicted
    the existence of a proper subvariety $Z\subset X$ containing
    every entire curve in $X$. Also, there are some conjectures by
    S. Kobayashi, dating back to the 1970s, which predict the hyperbolicity
    of certain generic hypersurfaces in $\bbfP^n_\bbfC$ and their
    complements.\par
Even in the simplest non-trivial case of $\dim
    X=2$ (surfaces of general type), these conjectures turned out
    to be rather complicated, but some substantial progress has recently
    been achieved, in the case of surfaces of general type, by the
    work of {\it M. McQuillan} [Publ. Math., Inst. Hautes \'Etud.
    Sci. 87, 121-174 (1998; Zbl 1006.32020)] and others.\par
The
    main purpose of the survey article under review is to thoroughly
    explain the methods and results of M. McQuillan's work, with
    special emphasis on their significance with respect to the above-mentioned
    (semi-classical) conjectures. Along this line, with a view towards
    the Kobayashi conjectures, the author also discusses the recent
    results by {\it J.-P. Demailly} and {\it J. El Goul} [Am. J.
    Math. 122, 515-546 (2000; Zbl 0966.32014)], {\it Y. Siu} and
    {\it S.-K. Yeung} [Invent. Math. 124, 573-618 (1996; Zbl 0856.32017)],
    {\it P. Vojta} [Am. J. Math. 122, 843-872 (2000; Zbl pre01507615)],
    and himself {\it M. Brunella} [Enseign. Math., II. S\'er. 45,
    195-216 (1999; Zbl 1004.32011)], together with their numerous
    forerunners.\par
All together, this survey article provides
    a masterly summary of the present state of art in the theory
    of entire holomorphic curves, in great methodical detail and
    topicality. In particular, the relations to the classification
    theory and geography of complex surfaces, as well as the crucial
    framework of entire holomorphic foliations, are elaborated at
    great length, which makes this treatise extremely enlightening,
    instructive and valuable.}
\RV{Werner Kleinert (Berlin)}
\CC{14H50
14J29
37F75
32Q45}
%% END ITEM
%% BEGIN ITEM
\AN{1023.53029 (01880705)}{2002}
\AU{Boyer, Charles P.; Galicki, Krzysztof; Nakamaye, Michael}
\TI{Einstein metrics on rational homology 7-spheres.}
\LA{English}
\DT{J}
\SO{Ann. Inst. Fourier 52, No.5, 1569-1584 (2002). 
\par http://annalif.ujf-grenoble.fr/Vol52/E525\_7/E525\_7.html}
\ISSN{373-095}
\AB{In former work, the authors studied Sasakian-Einstein manifolds
    and provided a lot of new examples of compact regular and non-regular
    5-dimensional Sasakian-Einstein spaces. This work is based on
    and extends some work of Demailly-Kol\'a\v{r} and Johnson-Kol\'a\v{r}
    on the existence of K\"ahler-Einstein metrics on compact Fano
    orbifolds.\par
In the present paper they continue their construction
    of examples of compact Einstein spaces along the same lines and
    focus on 7-dimensional Sasakian-Einstein spaces. In particular
    they are interested in the case of rational homology 7-spheres.
    Based on the same work, they provide 1936 distinct Sasakian-Einstein
    structures on certain 2-connected 7-manifolds $M^7$ realized
    as links of weighted homogeneous polynomials in $\bbfC^5$. Among
    them they discovered 184 2-connected rational homology spheres
    and discuss some of these examples in more detail. Here it should
    be noted that before, $S^{2n+1}$, the Stiefel manifold $V_2(\bbfR^{2n+1})$
    of 2-frames in $\bbfR^{2n+1}$ and the 3-Sasakian homogeneous
    11-manifold $G_2/\text{Sp}(1)$ were the only examples, known
    to the authors, of simply-connected rational homology spheres
    that admit Sasakian-Einstein structures.\par
Furthermore, the
    authors discuss 5- and 7-dimensional homology spheres which can
    be equipped with a regular positive Sasakian structure and finally,
    determine all rational homology spheres $M^{2n+1}$ admitting
    a homogeneous Sasakian-Einstein structure or in particular, a
    3-Sasakian homogeneous structure.}
\RV{L.Vanhecke (Leuven)}
\CC{53C25
53C12}
%% END ITEM
%% BEGIN ITEM
\AN{1021.32007 (01827825)}{2002}
\AU{Siu, Yum-Tong}
\TI{A new bound for the effective Matsusaka big theorem.}
\LA{English}
\DT{J}
\SO{Houston J. Math. 28, No.2, 389-409 (2002). 
\par http://www.math.uh.edu/~hjm/Electronic-Editions.html}
\ISSN{362-158}
\AB{Summary: Matsusaka's Big Theorem gives the very ampleness of
    $mL$ for an ample line bundle $L$ over an $n$-dimensional compact
    complex manifold $X$ when $m$ is no less than a number $m_0$
    depending on $L^n$ and $L^{n-1} K_X$. The dependence of $m_0$
    on $L^n$ and $L^{n-1}K_X$ is not effective. An earlier result
    of the author gives an effective $m_0$ containing the factor
    $(L^{n-1}((n + 2)L + K_X))^q$ with $q$ of order $4^n$. Demailly
    improved the order of $q$ to $3^n$ by reducing the twisting required
    for the existence of nontrivial global holomorphic sections for
    anticanonical sheaves of subvarieties which occur in the verification
    of the numerical effectiveness of $pL- K_X$ for some effective
    $p$. Twisted sections of anticanonical sheaves are needed to
    offset the addition of the canonical sheaf in vanishing theorems.\par
We
    introduce here a technique to get a new bound with $q$ of order
    $2^n$. The technique avoids the use of sections of twisted anticanonical
    sheaves of subvarieties by transferring the use of vanishing
    theorems on subvarieties to $X$, and is more in line with techniques
    for Fujita conjecture type results.}
\CC{32L10
14F05
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{1019.14018 (01811889)}{2002}
\AU{Araujo, Carolina}
\TI{K\"ahler-Einstein metrics for some quasi-smooth log del Pezzo surfaces.}
\LA{English}
\DT{J}
\SO{Trans. Am. Math. Soc. 354, No.11, 4303-4312 (2002).  ISSN 1088-6850]
\par http://www.ams.org/journal-getitem?pii=S0002-9947-02-03081-7}
\ISSN{002-994}
\AB{Summary: Recently {\it J. M. Johnson} and {\it J. Koll\'ar} [Ann.
    Inst. Fourier 51, 69-79 (2001; Zbl 0974.14023)] determined the
    complete list of anticanonically embedded quasi-smooth log del
    Pezzo surfaces in weighted projective $3$-spaces. They also proved
    that many of those surfaces admit a K\"ahler-Einstein metric,
    and that some of them do not have tigers. \par
The aim of this
    paper is to settle the question of the existence of K\"ahler-Einstein
    metrics and tigers for those surfaces for which the question
    was left open. In order to do so, we will use techniques developed
    earlier by {\it A. M. Nadel} [Ann. Math. 132, 549-596 (1990;
    Zbl 0731.53063)]and by {\it J.-P. Demailly} and {\it J. Koll\'ar}
    [Ann. Sci. \'Ec. Norm. Sup\'er. 34, 525-556 (2001; Zbl 0994.32021)].}
\CC{14J10
32Q20}
%% END ITEM
%% BEGIN ITEM
\AN{1015.14003 (01663193)}{2002}
\AU{Mustata, Mircea}
\TI{The multiplier ideals of a sum of ideals.}
\LA{English}
\DT{J}
\SO{Trans. Am. Math. Soc. 354, No.1, 205-217 (2002).  ISSN 1088-6850]
\par http://www.ams.org/journal-getitem?pii=S0002-9947-01-02867-7}
\ISSN{002-994}
\AB{From the paper: Let $X$ be a smooth complex variety. To a non-zero
    quasi-coherent sheaf of ideals $\underline{\bold a}$ on $X$ one
    can associate a sequence of ideals called the multiplier ideals
    of $\underline{\bold a}$, which depend on a rational parameter.
    The behaviour of these ideals encodes in a subtle way the properties
    of the singularities of $V(\underline{\bold a})$. Introduced
    first in the analytic context in the work of Demailly, Nadel,
    Siu and others, multiplier ideals have recently found surprising
    applications in algebraic geometry.\par
Here is the definition.
    Suppose that $f:X'\to X$ is a log resolution of $(X,V(\overline
    {\bold a}))$, i.e., $f$ is proper and birational, $X'$ is smooth,
    and $f^{-1}V (\underline{\bold a}) =D$ is a divisor with simple
    normal crossings. If $K_{X'/X}$ is the relative canonical divisor
    of $f$, the multiplier ideal of $\underline {\bold a}$ with coefficient
    $\alpha\in \bbfQ_+$ is $${\cal I}(X,\alpha \cdot\underline {\bold
    a})=f_*{\cal O}_{X'} \bigl( K_{X'/X} -[\alpha D]\bigr).$$ Here
    $[\cdot]$ denotes the integral part function.\par
We prove that
    if $\underline{\bold a},\underline{\bold b}\subseteq {\cal O}_X$
    are non-zero sheaves of ideals on a complex smooth variety $X$,
    then for every $\gamma\in \bbfQ_+$ we have the following relation
    between the multiplier ideals of $\underline{\bold a}$, $\underline{\bold
    b}$ and $\underline {\bold a}+\underline {\bold b}$: $${\cal
    I}(X,\gamma \cdot(\underline {\bold a}+ \underline {\bold b}))\subset
    \sum_{\alpha+ \beta=\gamma} {\cal I}(X,\gamma \cdot \underline
    {\bold a}) \cdot{\cal I}(X,\beta\cdot \underline{\bold b}).$$
    A similar formula holds for the asymptotic multiplier ideals
    of the sum of two graded systems of ideals. We use this result
    to approximate at a given point arbitrary multiplier ideals by
    multiplier ideals associated to zero dimensional ideals. This
    is applied to compare the multiplier ideals associated to a scheme
    in different embeddings.}
\CC{14B05
14E15
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{1010.32017 (01801546)}{2002}
\AU{Phong, D.H.; Sturm, Jacob}
\TI{On the algebraic constructibility of varieties of integrable rational functions on $\Bbb C^n$.}
\LA{English}
\DT{J}
\SO{Math. Ann. 323, No.3, 453-484 (2002).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s002080100310}
\ISSN{025-583}
\AB{The main results of this paper can be roughly described as follows:
    Consider the set of polynomials $f(b;z)$ in $n$ complex variables
    $Z_1,\dots, Z_n$, of degrees bounded by a fixed integer $N$ and
    such that the coefficients of the polynomials are themselves
    polynomials in parameters $b_1,\dots, b_p$. For a fixed real
    number $\delta> 0$, consider $\int_{\bbfC^n}|f|^\delta dv$, where
    $dv$ is the usual Euclidean measure.\par
The authors prove that
    the set of points $(b_1,\dots, b_p)\in \bbfC^p$ such that this
    integral converges is a constructible set. Further, there is
    a stratification of this constructible set such that this integral
    is a continuous function of the parameter on each stratum. The
    proof uses some earlier estimates due to the authors and some
    knowledge from the proof of resolution of singularities of varieties
    due to E. Bierstone and P. Milman.\par
These results are global
    analogous of local results for germs of analytic functions proved
    earlier by Siu, Demailly-Kollar and the authors.}
\RV{R.Gurjar (Mumbai)}
\CC{32S70}
%% END ITEM
%% BEGIN ITEM
\AN{1008.32009 (01786836)}{2002}
\AU{Marinescu, George; Todor, Radu; Ionu\c{t}, Chiose}
\TI{$L^2$ holomorphic sections of bundles over weakly pseudoconvex coverings.}
\LA{English}
\DT{J}
\SO{Geom. Dedicata 91, 23-43 (2002). 
\par http://www.springerlink.com/openurl.asp?genre=journal\&issn=0046-5755}
\ISSN{046-575}
\AB{The Demailly's asymptotic formula gives an estimate for the dimension
    of the cohomology of a compact manifold with coefficients in
    a high tensor power of a line bundle.\par
In this paper the
    authors give an analogue of this formula for the $L^2$-cohomology
    on more general domains. Precisely, let $X$ be a complex manifold
    with a free holomorphic action of a discrete group $\Gamma,M$
    a pseudoconvex $\Gamma$-invariant domain of $X$ such that $\overline
    M/ \Gamma$ is compact and $E$ a Hermitian holomorphic $\Gamma$-invariant
    line bundle on $X$ which is positive on a neighborghood of the
    boundary of $M$. Then, for $k\to +\infty$, one has $$\dim_\Gamma
    H^0_{(2)} (M,E^k)\ge{k^n\over n!} \int_{(M/\Gamma) (\ge 1)}\left({\iota
    \over 2\pi}{\bold c}(E)\right)^n+ o(k^n),$$ where $E^k$ is the
    $k$th tensor power of $E$, $H^0_{(2)}$ is the space of $L^2$-sections
    of $E^k$ (with respect to a $\Gamma$-invariant metric on $X$
    and along the fibers of $E)$ and $\dim_\Gamma$ is the von Neumann
    dimension.\par
As a consequence this formula proves the existence
    of $L^2$-sections of $E^k$ for $k$ sufficiently large provided
    that the $E$ is sufficiently positive.}
\RV{Pierre Mazet (Paris)}
\CC{32L10
58E35}
%% END ITEM
%% BEGIN ITEM
\AN{0998.14009 (01739910)}{2002}
\AU{Musta\c{t}\u{a}, Mircea}
\TI{Singularities of pairs via jet schemes.}
\LA{English}
\DT{J}
\SO{J. Am. Math. Soc. 15, No.3, 599-615 (2002).  ISSN 1088-6834]
\par http://www.ams.org/journal-getitem?pii=S0894-0347-02-00391-0}
\ISSN{894-034}
\AB{From the paper: For an arbitrary scheme $W$, the $m$-th jet scheme
    $W_m$ parametrizes morphisms $\text{Spec} \bbfC [t]/(t^{m+1})\to
    W$. Main result:\par
If $X$ is a smooth variety, $Y\subset X$
    a closed subscheme, and $q>0$ a rational number, then:\par
(1)
    The pair $(X,q\cdot Y)$ is log canonical if and only if $\dim
    Y_m\le (m+1) (\dim X-q)$, for all $m$.\par
(2) The pair $(X,q\cdot
    Y)$ is Kawamata log terminal if and only if $\dim Y_m< (m+1)(\dim
    X-q)$, for all $m$.\par
The main technique we use in the proof
    of this result is motivic integration, a technique due to Kontsevich,
    Batyrev, and Denef and Loeser. As a consequence of the above
    result, we obtain a formula for the log canonical threshold:
    \par
Corollary. If $X$ is a smooth variety and $Y\subset X$
    is a closed subscheme, then the log canonical threshold of the
    pair $(X,Y)$ is given by $c(X,Y)= \dim X-\sup_{m\ge 0} {\dim
    Y_m\over m+1}$.\par
We apply this corollary to give simpler
    proofs of some results on the log canonical threshold proved
    by {\it J.-P. Demailly} and {\it J. Koll\'ar} [Ann. Sci. \'Ec.
    Norm. Sup\'er., IV. S\'er. 34, No. 4, 525-556 (2001; Zbl 0994.32021)]
    using analytic techniques.}
\CC{14E15
14B05
14E30}
%% END ITEM
%% BEGIN ITEM
\AN{01911923}{2002}
\AU{Boucksom, S\'ebastien}
\TI{On the volume of a line bundle.}
\LA{English}
\DT{J}
\SO{Int. J. Math. 13, No.10, 1043-1063 (2002). 
\par http://dx.doi.org/10.1142/S0129167X02001575}
\ISSN{129-167}
\AB{Let $L$ be a holomorphic line bundle on a compact $n$-dimensional
    K\"ahler manifold $X$. The volume $v(L)$ of $L$ is defined as
    $$ v(L):=\limsup_{k\to \infty} \frac{n!}{k^n}h^0(X,kL).$$ The
    bundle $L$ has maximal Kodaira-Iitaka dimension, i.e. $L$ is
    big, if and only if $v(L)>0$. This implies that $v(mF)=m^nv(F)$
    for every $m\in\Bbb Z$ and allows the definition $v(F)=m^{-n}v(mF)$
    for a $\Bbb Q$-line bundle $F$ when $mF$ is a holomorphic line
    bundle. The volume of $L$ depends only on the Chern class $c_1(L)$
    if $L$ is numerically effective. In this case it follows from
    Demailly's Morse inequalities and the asymptotic Riemann-Roch
    formula that $v(L)=\int_Xc_1(L)^n$. \par
The author shows that
    it is possible in general to express $v(L)$ in terms of $c_1(L)$.
    By a theorem of {\it T. Fujita} [Kodai Math. J. 17, No. 1, 1-3
    (1994; Zbl 0814.14006)] the volume of the pull back of a big
    line bundle $L$ on a projective manifold with respect to a suitable
    modification can be arbitrarily close approximated by the volume
    of ample $Q$-line bundles $A$. The author generalizes this theorem
    to K\"ahler manifolds (Theorem 1.4). Using the Calabi-Yau methods
    to solve Monge-Amp\`ere equations he shows that one can find
    a Hermitian metric on $A$ such that the product of the curvature
    eigenvalues equals $v(A)$. Moreover this yields metrics on $L$
    converging to some singular metric whose product of the curvature
    eigenvalues equals $v(L)$. \par
Combining this result with the
    holomorphic Morse equalities the author proves the following
    equality for $v(L)$ when $c_1(L)$ is a pseudoeffective class,
    i.e. it contains closed positive currents $T$: $v(L)=\max_T\int_X
    T^n_{ac}=:v(c_1(L))$, where $T_{ac}$ is the absolutely continuous
    part of $T$. The function $v:H^{(1,1)}(X,\Bbb R)\to\Bbb R$, $v(\alpha):=0$
    if $\alpha\in H^{(1,1)}(X,\Bbb R)$ does not contain a closed
    positive current, is continuous. This is a corollary of Theorem
    4.7, which says that $\alpha\in H^{(1,1)}(X,\Bbb R)$ is the Chern
    class of a big line bundle if and only if $v(\alpha)>0$. The
    proof of this theorem uses methods developed by {\it J. P. Demailly}
    and {\it M. Paun} [math.AG/0105176 (2001)].}
\RV{Eberhard Oeljeklaus (Bremen)}
\CC{32J25
32J27
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{1018.32022 (01690809)}{2001}
\AU{Todor, Radu; Chiose, Ionu\c{t}; Marinescu, George}
\TI{Morse inequalities for covering manifolds.}
\LA{English}
\DT{J}
\SO{Nagoya Math. J. 163, 145-165 (2001). }
\ISSN{027-763}
\AB{Let $\widetilde M$ be a complex manifold of dimension $n$ and
    $\Gamma$ a discrete group acting freely and properly discontinuously
    on $\widetilde M$. Let $E$ be a holomorphic line bundle on $M=\widetilde
    M/\Gamma$ and $\widetilde E$ the pullback of $E$ to $\widetilde
    M$.\par
The authors consider the problem of finding non-trivial
    $L^2$ holomorphic sections of $\widetilde E^k$ (with respect
    to a $\Gamma$-invariant metric) under reasonable conditions on
    $E$ in terms of curvature positivity. They show in fact that,
    if $M$ is complete Hermitian and $(E,h)$ is a holomorphic Hermitian
    line bundle, the von Neumann $\Gamma$-dimension of the space
    of such sections is bounded below by a function of the form $ak^n+o
    (k^n)$, where $a$ is given in terms of the curvature of $(E,h)$.
    This generalises an inequality of {\it Y. T. Siu} [J. Differ.
    Geom. 19, 431-452 (1984; Zbl 0577.32031)] and {\it J. P. Demailly}
    [Ann. Inst. Fourier 35, 189-229 (1985; Zbl 0565.58017)] for the
    case in which $\widetilde M=M$ is compact and $E$ is semi-positive
    and positive at one point.\par
The authors also give criteria
    for some classes of 1-concave manifolds to be Moishezon. They
    prove further the stability of certain Moishezon strongly pseudoconcave
    manifolds under perturbation of complex structures.\par
In the
    final section, the main theorem is used to prove some weak Lefschetz
    theorems for the fundamental group.}
\RV{P.E.Newstead (Liverpool)}
\CC{32L10
32F10
32Q55
57S30}
%% END ITEM
%% BEGIN ITEM
\AN{1014.32024 (01652377)}{2001}
\AU{Rashkovskii, Alexander}
\TI{Lelong numbers with respect to regular plurisubharmonic weights.}
\LA{English}
\DT{J}
\SO{Result. Math. 39, No.3-4, 320-332 (2001). }
\ISSN{422-638}
\AB{Let $\varphi$ be a continuous semi-exhaustive plurisubharmonic
    function in a domain $\Omega\subset\Bbb C^n$. For a plurisubharmonic
    function $u$ the generalized Lelong number of $u$ with respect
    to $\varphi$ is defined by the formula (introduced by Demailly)
    $\nu(u,\varphi):=\lim_{r\to-\infty}\int_{B_r^\varphi} dd^cu\wedge(dd^c\varphi)^{n-1}$,
    where $B_r^\varphi:=\{z: \varphi(z)<r\}$. In the case where $\varphi(z):=\log\|z-x\|$,
    we get the classical Lelong number $\nu(u,x)$. In the case where
    $\varphi(z):=\max_{k=1,\dots,n}(1/a_k)\log|z_k-x_k|$, we get
    $\frac 1{a_1\dots a_n}\nu(u,x,a)$, where $\nu(u,x,a)$ is the
    directional Lelong number (introduced by Kiselman) ($a_1,\dots,a_n>0$).
    \par
The author characterizes two large classes of weight functions
    $\varphi$ which seem to be very useful in applications: so called
    almost homogeneous and almost multicircled functions.}
\RV{Marek Jarnicki (Krak\'ow)}
\CC{32U05
32U25
32W20}
%% END ITEM
%% BEGIN ITEM
\AN{0983.53031 (01653372)}{2001}
\AU{Kebekus, Stefan}
\TI{Lines on contact manifolds.}
\LA{English}
\DT{J}
\SO{J. Reine Angew. Math. 539, 167-177 (2001).  ISSN 1435-5345]
\par http://www.degruyter.de/journals/crelle/2001/539\_167.html}
\ISSN{075-410}
\AB{Complex manifolds $X$ which carry a complex contact structure,
    i.e., a non-degenerate subbundle $F\subset T_X$ of the tangent
    bundle of corank one, appear naturally as twistor spaces over
    Riemannian manifolds with quaternionic-K\"ahlerian holonomy group.
    These manifolds have recently gained considerable interest. The
    reader is referred to {\it A. Beauvilles}'s excellent survey
    [``Riemannian holonomy and algebraic geometry'', preprint (1999)]
    for a thorough introduction. Previous results of {\it J. P. Demailly}
    [``On the Frobenius integrability of certain holomorphic $p$-forms'',
    preprint (2000)] and {\it T. Peternell}, {\it A. Sommese}, {\it
    J. A. Wisniewski} and the author [Invent. Math. 142, 1-15 (2000;
    Zbl pre01533359)] practically reduce the study to the case where
    the contact manifold $X$ is Fano and where the second Betti-number
    $b_2(X)$ is one. Since these manifolds can always be covered
    by lines, i.e., by rational curves which intersect the ample
    generator of the Picard-group with multiplicity one, the present
    paper considers the geometry of these lines in greater detail.\par
It
    is shown that if $x$ is a general point on a contact manifold
    $X$, then all lines through $x$ are smooth. Furthermore, if $X$
    is not the projective space, then the tangent spaces to lines
    generate the contact distribution $F$ at $x$. As a consequence
    we obtain that the contact structure on $X$ is unique, a result
    which was previously conjectured by C. LeBrun.}
\RV{S.Kebekus (Bayreuth)}
\CC{53C28
53D10
53C29}
%% END ITEM
%% BEGIN ITEM
\AN{0978.32008 (01586357)}{2001}
\AU{Zaffran, Dan}
\TI{Serre problem and Inoue-Hirzebruch surfaces.}
\LA{English}
\DT{J}
\SO{Math. Ann. 319, No.2, 395-420 (2001).  ISSN 1432-1807]
\par http://dx.doi.org/10.1007/s002080000159}
\ISSN{025-583}
\AB{This paper treats the so-called Serre problem of Stein theory
    [{\it J.-P. Serre}, Colloque fonctions plusieurs variables, Brussels,
    1953, 57-68 (1953; Zbl 0053.05302)]: if $E\to B$ is a (locally
    trivial) holomorphic fiber bundle with fiber type $F$, where
    $B$, $F$ are Stein manifolds, is $E$ also a Stein manifold?\par
There
    are strong positive results on this: {\it K. Stein} [Arch. Math.
    7, 354-361 (1956; Zbl 0072.08002)] proved that $E$ is Stein if
    $F$ is zero-dimensional. {\it Y. Matsushima} and {\it A. Morimoto}
    [Bull. Soc. Math. Fr. 88, 137-155 (1960; Zbl 0094.28104)] showed
    that $E$ is Stein if the structure group of $E$ is a connected
    complex Lie group. {\it J.-L. Stehl\'e} [C. R. Acad. Sci., Paris,
    S\'er. A 279, 235-238 (1974; Zbl 0287.32013); Lect. Notes Math.
    474, 155-179 (1975; Zbl 0309.32011)] showed that $E$ is Stein
    if $F$ is so-called hyperconvex. {\it J. E. Forn{\ae}ss} and
    {\it K. Diederich} [Proc. Natl. Acad. Sci. USA 72, 3279-3280
    (1975; Zbl 0309.32012); Invent. Math. 39, 129-141 (1977; Zbl
    0353.32025)] showed that $E$ is Stein if $F$ is a bounded Stein
    open subset of some ${\Bbb C}^n$ with $C^2$-smooth boundary by
    proving that such an $F$ is hyperconvex. {\it Y.-T. Siu} [Math.
    Ann. 219, 171-192 (1976; Zbl 0318.32010)] proved that $E$ is
    Stein if $F$ is a bounded Stein open subset of some ${\Bbb C}^n$
    and the first Betti number of $F$ is zero. {\it N. Mok} [Math.
    Ann. 258, 145-168 (1981; Zbl 0497.32013)] showed that $E$ is
    Stein if $F$ is one-dimensional.\par
There are, however, counterexamples
    also. The first one is by {\it H. Skoda} [C. R. Acad. Sci., Paris,
    S\'er. A 284, 1199-1202 (1977; Zbl 0353.32032); Invent. Math.
    43, 97-107 (1977; Zbl 0365.32018)], who gave an $E$ not Stein
    with $B={\Bbb C}$, $F={\Bbb C}^2$. Skoda's example was modified
    by {\it J.-P. Demailly} [Invent. Math. 48, 293-302 (1978; Zbl
    0372.32012)], who had for $B$ a disc or ${\Bbb C}$, $F={\Bbb
    C}^2$. In the influential survey article [Bull. Am. Math. Soc.
    84, 481-512 (1978; Zbl 0423.32008)], {\it Y.-T. Siu} conjectured
    that if in addition $F$ is a bounded pseudoconvex domain in some
    ${\Bbb C}^n$, then $E$ will necessarily be Stein. The counterexample
    of {\it G. C{\oe}ur\'e} and {\it J.-J. Loeb} in [Ann. Math. (2)
    122, 329-334 (1985; Zbl 0585.32030)] shows that this is not the
    case: they produced an $E$ not Stein with $B={\Bbb C}\setminus\{0\}$,
    and $F$ a pseudoconvex bounded Reinhardt domain in ${\Bbb C}^2$.\par
The
    goal of the article under review is to illuminate the counterexample
    of C{\oe}ur\'e and Loeb in the light of geometrical and combinatorial
    properties of the so-called Inoue-Hirzebruch surfaces. These
    surfaces relate to cusps and Reinhardt domains in ${\Bbb C}^2$
    of the form $\{(u,v):0<|u|^\alpha<|v|<|u|^\beta\}$ (such as the
    $F$ of C{\oe}ur\'e and Loeb).\par
Part 1 introduces the Inoue--Hirzebruch
    surfaces as surfaces defined in terms of blow-ups governed by
    a so-called Dloussky-matrix. A Dloussky-matrix is one that can
    be written as $A={ 0 1\choose 1 k_1} \ldots { 0 1\choose 1 k_N}$
    with integers $k_i\ge 1$, $1\le i\le N$ -- such a representation
    of $A$ is unique. \S 3 defines an involution (called transposition)
    on the set of all Inoue-Hirzebruch surfaces, and makes use of
    it to provide a counterexample to `conjecture 3' of {\it I. Nakamura}
    [T\^ohoku Math. J. 42, 475-516 (1990; Zbl 0732.14019)] that small
    neighborhoods of the maximal divisor determine the surface. (The
    author credits G. Dloussky and K. Oeljeklaus for an earlier (unpublished)
    counterexample to the same conjecture of Nakamura.)\par
Part
    2 treats the Serre problem proper. The main result is that if
    $A$ is a Dloussky-matrix with $N$ even, then the construction
    and the proof of C{\oe}ur\'e and Loeb can be adapted and carried
    out, and the resulting fiber bundle $E\to B={\Bbb C}\setminus\{0\}$
    is not Stein, and has fiber type $F$ a pseudoconvex bounded Reinhardt
    domain (related to Inoue-Hirzebruch surfaces). The technique
    of proof is an analysis of the form of holomorphic functions
    on $E$ in terms of Laurent series. It turns out (Proposition
    6.1) that the potential theory method from C{\oe}ur\'e and Loeb
    implies that certain monomials are absent from the Laurent series
    of holomorphic functions on $E$. This fact is then applied to
    show that $E$ is not Stein. A consequence of the analysis is
    the curious fact (Corollary 6.4) that if $g$ holomorphic on ${\Bbb
    C}$ is such that $g(z+j)e^{e^{j^2}}\to 0$ uniformly on compacts
    as $j\to\pm\infty$, then $g\equiv 0$. The last point (Proposition
    8.3) is that the manifold $E$ has no Stein envelope of holomorphy.\par
The
    paper is very readable and its arguments are mostly geometric
    and combinatorial in nature.}
\RV{Imre Patyi (Irvine)}
\CC{32E10
32L05
32J15}
%% END ITEM
%% BEGIN ITEM
\AN{0974.14023 (01579351)}{2001}
\AU{Johnson, Jennifer M.; Koll\'ar, J\'anos}
\TI{K\"ahler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces.}
\LA{English}
\DT{J}
\SO{Ann. Inst. Fourier 51, No.1, 69-79 (2001). 
\par http://annalif.ujf-grenoble.fr/aif\_2001.html}
\ISSN{373-095}
\AB{The authors present a list of log del Pezzo surfaces which are
    anticanonically embedded in a weighted projective space as quasi-smooth
    hypersurfaces. There are 22 different sporadic types and one
    infinite series. They show that for all types (except four where
    the answer is unknown), the del Pezzo surface admits a K\"ahler-Einstein
    orbifold metric. The crucial tool here is a result of A. Nadel,
    J.-P. Demailly and J. Koll\'ar which asserts that an $n$-dimensional
    Fano variety $X$ with at most quotient singularities admits a
    K\"ahler-Einstein orbifold metric if there is an $\varepsilon>0$
    such that the pair $(X,{n+\varepsilon\over n+1}D)$ is Kawamata
    log-terminal (klt) for every effective $\bbfQ$-divisor $D\equiv-K_X$.
    A pair $(X,Q)$ as above which is not klt is called a tiger [see
    {\it S. Keel} and {\it J. McKernan}, ``Rational curves on quasi-projective
    surfaces'', Mem. Am. Math. Soc. 669 (1999; Zbl 0955.14031)].
    For each of the types (except four), the authors check whether
    the del Pezzo surface admits a tiger.}
\RV{Igor V.Dolgachev (Ann Arbor)}
\CC{14J26
14J45
32Q20
14Q10}
%% END ITEM
%% BEGIN ITEM
\AN{01972760}{2001}
\AU{Demailly, Jean Pierre}
\TI{Refoundation of mathematics in France.}
\LA{Italian}
\DT{J}
\SO{Lett. Mat. Pristem 42, 10-14 (2001).  ISSN 1123-7198]}
\ISSN{593-588}
\AB{The young and (even more) promising author of the presente study,
    Jean-Piere Demailly, approaches the problem of mathematics refoundation
    from very (highly) French positions, by proundly asserting that
    the ``nouvelles maths'' have been initiated in Frace and nowhere
    else. \par
The role played in France by the ``commission for
    the long-sum reflexion on the teaching of mathematics'' is emphasized,
    and the name of ist president -- Jean-Piere Kahane -- is mentioned.
    \par
Actually, the study is Demailly's contribution to the round
    table organized in Paris on ``Mathematics and the Teaching of
    Sciences''. It is quite a revolutionary text, expressing the
    author's daring opinions, his diraporal of here???? didacticism
    and old-fashioned techniques still in force. \par
Extremely
    ``represhing'' for the reader is the idea of having illustrated
    the text with coloured, challenging reproductions of Picasso.
    The choice of unordinity is explicit.}
\RV{Cristina Irimia (Ia\c{s}i)}
\CC{01-XX}
%% END ITEM
%% BEGIN ITEM
\AN{0999.32015 (01701577)}{2000}
\AU{Do Due Thai; Pham Viet Duc}
\TI{On some classes of hyperbolic complex spaces.}
\LA{English}
\DT{J}
\SO{Acta Math. Vietnam. 25, No.3, 263-272 (2000). }
\ISSN{251-418}
\AB{The authors give some characterization of hyperbolicity in the
    sense of Kobayashi for complex spaces. After studies of Hahn-Kim
    and Thai, they first prove that a complex space $X$ is Kobayashi-hyperbolic
    if and only if $X$ has the Landau property for any Finsler metric
    $H$ on $X$. Next they give the characterization of hyperbolicity
    by Gateaux holomorphic map. Let $X$ be a compact complex space.
    Then they prove that $X$ is hyperbolic if and only if every Gateaux
    holomorphic map from any open subset $\Omega$ of any Banach space
    $B$ into $X$ is holomorphic. Furthermore, they construct the
    Kobayashi algebraic hyperbolic distance on an open subset of
    a Moishezon space and prove that this pseudodistance coincides
    with the Kobayashi pseudodistance. This result is an analogous
    assertion in the case of quasi-projective varieties which was
    proved by {\it J.-P. Demailly, L. Lempert} and {\it B. Shiffman}
    [Duke Math. J. 76, No.~2, 333-364 (1994; Zbl 0861.32006)].}
\RV{Yoshihiro Aihara (Shizuoka)}
\CC{32Q45}
%% END ITEM
%% BEGIN ITEM
\AN{0978.32004 (01585221)}{2000}
\AU{Phong, D.H.; Sturm, Jacob}
\TI{On a conjecture of Demailly and Koll\'ar.}
\LA{English}
\DT{J}
\SO{Asian J. Math. 4, No.1, 221-226 (2000). 
\par http://www.ims.cuhk.edu.hk/~ajm/}
\ISSN{093-610}
\AB{For a holomorphic function $f$ in a neighborhood of a compact
    set $K$ in a complex manifold $M$ of dimension $n$, we denote
    by $c_K(f)$ the supremum of all real numbers $c$ such that $|f|^{-2c}$
    is integrable on some neighborhood of $K$.\par
Demailly and
    Koll\'ar make the following conjecture A: For every compact set
    $L$ containing $K$ in its interior, there is a number $\alpha=
    \alpha(f,K,L)> 0$ such that $\sup_L|g-f|< \alpha$ for a holomorphic
    function $g$ implies $c_K(g)\ge c_K(f)$.\par
The authors prove
    that this conjecture A is true when $n=2$. The authors also give
    a proof for conjecture B, which asserts a property for the set
    $C(n)$ of all $c_0(f)$ for every holomorphic function $f$ in
    a neighborhood of the origin $0$ of $\bbfC^n$, when $n= 2$.}
\RV{A.Morimoto (Nagoya)}
\CC{32A40}
%% END ITEM
%% BEGIN ITEM
\AN{0963.14009 (01461822)}{2000}
\AU{Maillot, Vincent}
\TI{G\'eom\'etrie d'Arakelov des vari\'et\'es toriques et fibr\'es en droites int\'egrables. (Arakelov geometry of toric varieties and integrable line bundles).}
\LA{French}
\DT{J}
\SO{M\'em. Soc. Math. Fr., Nouv. S\'er. 80, 129 p. (2000). 
\par http://www.numdam.org:80/numdam-bin/feuilleter?id=MSMF\_2000\_2\_80\_}
\ISSN{249-633}
\AB{The main object of the monograph under review is an arithmetic
    variety $X$, that is a scheme over $S= \text{Spec}(O_K)$ ($K$
    is a number field) which is assumed flat, projective, integral,
    and regular. One of general goals is to extend the intersection
    theory of {\it H. Gillet} and {\it C. Soul\'e} [Publ. Math.,
    Inst. Hautes \'Etud. Sci. Publ. Math. 72, 93-174 (1990; Zbl 0741.14012)]
    so that in the Arakelov geometry one could consider linear bundles
    equipped with a metric which is not necessarily $C^{\infty }$.
    To be more precise, following {\it S. Zhang} [J. Algebr. Geom.
    4, No. 2, 281-300 (1995; Zbl 0861.14019)], the author introduces
    the notion of an admissible line bundle (generated by its global
    sections and equipped with a positive metric which can be uniformly
    approximated by positive $C^{\infty }$ metrics), and, more generally,
    of an integrable line bundle (the difference of two admissible
    line bundles). Further, the author introduces a new notion of
    a generalized differential form, and for an integrable line bundle
    defines the first arithmetic Chern class which turns out to be
    a generalized differential form in the sense of this new definition.
    The author defines a product of such forms using a product theory
    for positive currents developed by {\it E. Bedford} and {\it
    B. A. Taylor} [Acta Math. 149, 1-40 (1982; Zbl 0547.32012)] and
    {\it J.-P. Demailly} [in: Complex Analysis and Geometry, Univ.
    Ser. Math. 115-193 (1993; Zbl 0792.32006)]. As a result, the
    author constructs a generalized Chow ring containing the usual
    one, with pairing degree functions continuing those of Gillet--Soul\'e
    and with height function compatible with that of Zhang.\par
The
    author's next goal consists in applying the developed theory
    to a particular class of arithmetic varieties, namely, to smooth
    projective toric varieties which are among favorite proving grounds
    in many problems of algebraic geometry. First of all recall that
    the Chow ring of such a variety is classically known to be generated
    by the first Chern classes of linear bundles. As to the arithmetic
    Chow ring, one immediately encounters a problem: Any ``canonical''
    (i.e. functorial in $L$) metric on a linear bundle $L$ is, in
    general, not $C^{\infty }$. This is in fact one of motivations
    for developing a general intersection theory described above.
    \par
Thus, according to this general approach, given a toric
    variety over the integers equipped with a line bundle $\overline
    L$ and its canonical metric, the author defines a current $c_1(\overline
    L)$ replacing the usual first Chern class. Moreover, an explicit
    formula for the generalized product of such currents is given,
    and a vanishing theorem is proved. These results lead not only
    to a correct arithmetic analogue of the above mentioned classical
    result on generating the Chow ring by the first Chern classes,
    but also to a simple algorithm for computing the mixed volume
    of convex polytopes. \par
Furthermore, a more detailed study
    of heights on arithmetic toric varieties yields some more interesting
    consequences, namely a relationship between the canonical height
    of a hypersurface in a toric variety and the Mahler measure of
    the corresponding polynomial, and an arithmetic analogue of the
    Bernstein-Koushnirenko theorem on estimating the number of common
    zeroes of Laurent polynomials in terms of the mixed volumes of
    Newton polytopes.}
\RV{B.Kunyavskii (Ramat Gan)}
\CC{14G40
14M25
11D75
11G50}
%% END ITEM
%% BEGIN ITEM
\AN{0962.32007 (01560912)}{2000}
\AU{Zhan, Huirong; Yao, Zongyuan}
\TI{A generalization of the Koppelman formula for differential forms of type $(p,q)$ on Stein manifolds.}
\LA{Chinese. English summary}
\DT{J}
\SO{J. Xiamen Univ., Nat. Sci. 39, No.2, 147-151 (2000). }
\ISSN{438-047}
\AB{Summary: $(p,q)$-type differential forms on a Stein manifold
    can't adopt the Euclidean metric as they can in $\bbfC^n$ because
    the Euclidean metric on the Stein manifold isn't invariant under
    holomorphic transformations. This article adopts J. P. Demailly
    and Ch. Laurent-Thiebaut's methods to solve the problem of the
    invariant metric by using the Hermitean metric and Chern connection.
    A generalization of the Koppelman formula of differential forms
    of $(p,q)$-type on Stein manifolds is obtained by introducing
    a chosen parameter $m$, a natural number which is more than or
    equal to 2. When $m$ is 2, the formula is just the original Koppelman
    formula of differential forms of $(p,q)$-type on Stein manifolds.
    When $m$ is respectively equal to $3,4, \dots, N(N<+ \infty)$,
    a series of Koppelman formulas with different forms can be given.}
\CC{32A26
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0961.01016 (01500723)}{2000}
\AU{Skoda, Henri}
\TI{Pr\'esence de l'{\oe}uvre de Pierre Lelong dans les grands th\`emes de recherches d'aujourd'hui. (The presence of the work of Pierre Lelong in the main research themes of today).}
\LA{French}
\DT{CA}
\SO{Dolbeault, P. (ed.) et al., Complex analysis and geometry. Proceedings of the international conference in honor of Pierre Lelong on the occasion of his 85th birthday, Paris, France, September 22-26, 1997. Basel: Birkh\"auser. Prog. Math. 188, 1-30 (2000). [ISBN 3-7643-6352-5/hbk;}
\ISSN{743-164}
\AB{This survey covers the years 1935-1995. It concentrates on several
    basic ideas of Pierre Lelong and their development. The essential
    idea is to replace holomorphic functions by plurisubharmonic
    functions, so that computations behind theorems could be done
    using limits. The properties of holomorphic functions and analytic
    sets are very strong; they exclude all but finite partitions
    of unity, they exclude many discontinuous functions completely,
    they demand derivatives higher than the second. Pierre Lelong
    aimed to replace these ``hard'' objects by more ``supple'' or
    pliable objects such as the plurisubharmonic functions in such
    a way that the hard objects could be regenerated when desired
    -- hence the limit theorems. \par
Over the years this approach
    has led to developments in the various problems: Cousin's problem
    on the construction of a holomorphic function which is zero exactly
    on a given analytic set; Levi's problem on the relation between
    analytic and pseudoconvex sets; the existence of certain types
    of holomorphic and plurisubharmonic functions on analytic sets.
    These ideas were developed between 1955 and 1975 by Lars H\"ormander,
    Enrico Bombieri and Henri Skoda, and in the period 1990-1998
    by Jean-Pierre Demailly. There are about 60 items in the bibliography
    at the end. Definitions and statements of theorems are quite
    detailed, and the general thrust of the paper is easy to follow:
    Lelong started the process and it still continues.}
\RV{James J.Cross (Parkville)}
\CC{01A60
32-03}
%% END ITEM
%% BEGIN ITEM
\AN{0957.32001 (01500742)}{2000}
\TI{List of problems.}
\LA{English}
\DT{CA}
\SO{Dolbeault, P. (ed.) et al., Complex analysis and geometry. Proceedings of the international conference in honor of Pierre Lelong on the occasion of his 85th birthday, Paris, France, September 22-26, 1997. Basel: Birkh\"auser. Prog. Math. 188, 237-241 (2000). [ISBN 3-7643-6352-5/hbk;}
\ISSN{743-164}
\AB{A list of open problems is given. They have been suggested by
    Paolo de Bartolomeis, Ingo Lieb, J.-P. Demailly, L. Lempert,
    M. Passare, J. M. Tr\'epreau, and L. Aizenberg.}
\CC{32-02}
%% END ITEM
%% BEGIN ITEM
\AN{0956.32024 (01470860)}{2000}
\AU{Hwang, J.-M.; To, W.-K.}
\TI{On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains.}
\LA{English}
\DT{J}
\SO{J. Reine Angew. Math. 523, 173-197 (2000).  ISSN 1435-5345]
\par http://www.deGruyter.de/journals/crelle/2000/523\_173.html}
\ISSN{075-410}
\AB{Let $L$ be an ample line bundle over a projective algebraic manifold
    $X$ and let $f:Y\to X$ be the blow-up of $X$ at $x\in X$ with
    exceptional divisor $E$. The Seshadri number of $L$ at $x$ is
    defined as $\varepsilon(L,x):=\sup\{\varepsilon\in\Bbb Q\mid
    f^{*}L-\varepsilon E$ is nef over $Y\}$. It is known that the
    knowledge of lower or upper bounds for $\varepsilon(L,x)$ can
    yield important information about $L$ or $K_{X}\oplus L$, where
    $K_{X}$ is the canonical line bundle of $L$ [see {\it J.-P. Demailly},
    Lect. Notes Math. 1507, 87-104 (1992; Zbl 0784.32024)] where
    also the numbers $\varepsilon(L,x)$ were introduced. During the
    last decade several authors obtained bounds for $\varepsilon(L,x)$
    for various types of line bundles [see {\it Th. Bauer}, Math.
    Ann. 313, No. 3, 547-583 (1999; Zbl 0955.14005) and {\it R. Lazarsfeld},
    Math. Res. Lett. 3, No. 4, 439-447 (1996; Zbl 0890.14025) for
    references]. \par
In the paper under review the authors consider
    the canonical bundle of a compact Hermitian local symmetric manifold
    $X$ uniformized by a bounded symmetric domain $\Omega$. Their
    main results are lower and upper bounds for $\varepsilon(K_{X},x)$
    which are independent of $x$ and depend only on geometric properties
    of the irreducible components $\Omega_{1},\ldots,\Omega_{\ell}$
    of $\Omega$ with their Bergman metrics. Let $\kappa_{i}$ be the
    rank and let $-\frac{2}{c_{i}}$ be the minimal sectional curvature
    of $\Omega_{i}$, $1\le i\le \ell$. Assume that $X=\Gamma\backslash\Omega$
    with a torsion-free discrete subgroup of Aut$(\Omega)$ and let
    $\rho_{x}$ denote the injectivity radius with respect to $\Gamma$
    and $D_{x}$ the diameter of $X$ at $x$. Generalizing a former
    result [Compos. Math. 118, No. 2, 203-215 (1999; Zbl 0946.32010)]
    the authors prove: $$\left(\min_{1\le i\le\ell}c_{i}\right)\sinh^{2}
    \left(\frac{\rho_{x}}{\sqrt{\sum^{\ell}_{i=1}\kappa_{i}c_{i}}}\right)
    \le\varepsilon(K_{X},x) \le\left(\min_{1\le i\le\ell}c_{i}\right)\sinh^{2}\left(\frac{D_{x}}
    {\sqrt{\displaystyle\min_{1\le i\le\ell}c_{i}}}\right).$$ The
    proof is based on a subtile combination of a detailed analysis
    of the Lie group structure of Aut$(\Omega)$ and differential
    geometric methods including the direct construction of singular
    Hermitian metrics on $K_{X}$ with prescribed pole orders.\par
The
    paper contains a section with general and special background
    results on bounded symmetric domains which are needed for the
    proof.}
\RV{E.Oeljeklaus (Bremen)}
\CC{32M15
32J27
14C20
14J60
32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0978.14005 (01341965)}{1999}
\AU{K\"uchle, Oliver; Steffens, Andreas}
\TI{Bounds for Seshadri constants.}
\LA{English}
\DT{CA}
\SO{Hulek, Klaus (ed.) et al., New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick, UK, July 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 264, 235-254 (1999). [ISBN 0-521-64659-6/pbk;}
\ISSN{076-055}
\AB{From the introduction: This paper develops a new approach to
    bounding Seshadri constants of nef and big line bundles at a
    general point of a complex projective variety. A modification
    of this approach even allows us to give bounds valid at arbitrary
    points.\par
The Seshadri constant $\varepsilon (L,x)$, introduced
    by {\it J.-P. Demailly} [[in: Complex algebraic varieties, Proc.
    Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-104 (1992; Zbl
    0784.32024)], measures the local positivity of a nef line bundle
    $L$ at a point $x\in X$ of a complex projective variety $X$;
    it can be defined as $$\varepsilon (L,x):= \inf_{C\ni x}\left\{
    {L\cdot C\over \text{mult}_x (C)} \right\},$$ where the infimum
    is taken over all reduced irreducible curves $C \subset X$ passing
    through $x$.\par
{\it L. Ein} and {\it R. Lazarsfeld} [Journ\'ees
    de g\'eom\'etrie alg\'ebrique, Orsay Ast\'erisque 218, 177-186
    (1993; Zbl 0812.14027)] proved that the Seshadri constant of
    an ample line bundle on a smooth surface is $\ge 1$ for all except
    possibly countably many points. -- Examples by Miranda show that
    for any integer $n\ge 2$ and any real number $\delta>0$ there
    is a smooth $n$-fold $X$, an ample line bundle $L$ on $X$ and
    a point $x\in X$ with $\varepsilon (L,x)<\delta$; in other words,
    there does not exist a universal lower bound for Seshadri constants
    valid for all $X$ and ample $L$ at every point $x\in X$. Then
    it was proved by {\it L. Ein}, {\it O. K\"uchle} and {\it R.
    Lazarsfeld} [J. Differ. Geom. 42, 193-219 (1995; Zbl 0866.14004)]
    that, for a nef and big line bundle $L$ on a projective $n$-fold,
    the Seshadri constant at very general points (that is, outside
    a countable union of proper subvarieties) is bounded below by
    $1/n$, which implies the existence of a lower bound at general
    points depending only on $n$.\par
We apply the strategy of differentiation
    in parameter directions in the spirit of Ein, K\"uchle and Lazarsfeld
    (loc. cit.). The result of this method is the following theorem
    on linear systems:\par
Theorem 1. Let $X$ be a smooth projective
    $n$-fold, $L$ a nef and big line bundle on $X$ and $\alpha>0$
    a rational number such that $L^n> \alpha^n$. Let $0=\beta_1 <\beta_2
    <\cdots <\beta_n <\beta_{n+1} =\alpha$ be any sequence of rational
    numbers and $x\in X$ a very general point. Then one of the following
    holds:\par
(a) there exist $k\gg 0$ and a divisor $E\in|kL|$
    having an isolated singularity with multiplicity at least $k(\beta_{n+1}-\beta_n)$
    at $x$, or\par
(b) there exists a proper subvariety $V\subset
    X$ through $x$ of codimension $c<n$ such that $$\eqalign{\deg_LV=
    L^{n-c}\cdot V & \le {1 \over (\beta_{c+1} -\beta_c)^c} \left(1-
    \root n\of {\left(1-{\alpha^n \over L^n}\right)^c} \right)L^n\cr
    & <{\alpha^n \over(\beta_{c+1}-\beta_c)^c}.\cr}$$ To pass
    from theorem 1 to actually bounding the Seshadri constant, we
    use a rescaling trick.\par
Theorem 2. Let $L$ be a nef line
    bundle on an irreducible projective $n$-fold $X$, and $x\in X$
    a very general point. Suppose given positive rational numbers
    $\alpha_1, \dots,\alpha_n$, and set $\gamma= 1+ \sum^{n-1}_{i=1}
    \alpha_i$. For some real number $\varepsilon>0$, suppose that
    every $d$-dimensional subvariety $V\subset X$ containing $x$
    (for any $d$ with $1\le d\le n)$ satisfies $\deg_LV= L^d\cdot
    V\ge\varepsilon^d \gamma^n \alpha_d^{d-n}$. Then $\varepsilon(L,x)
    \ge\varepsilon$.\par
The rescaling argument mentioned above
    can also be applied in another context, leading to bounds for
    Seshadri constants valid at arbitrary points.\par
Corollary
    3. Let $X$ be a smooth projective $n$-fold, $x\in X$ any point,
    $A$ an ample line bundle and $\delta\ge 0$ a real number such
    that $T_X(\delta A)$ is nef. Then $$\varepsilon (A,x)\ge \min\left\{
    {1\over(n-1)^{n-1}(2n-1)^n}, {1\over\delta} \right\}.$$}
\CC{14C20
14C17}
%% END ITEM
%% BEGIN ITEM
\AN{0965.32022 (01341554)}{1999}
\AU{Tsuji, Hajime}
\TI{Existence and applications of analytic Zariski decompositions.}
\LA{English}
\DT{CA}
\SO{Komatsu, Gen (ed.) et al., Analysis and geometry in several complex variables. Proceedings of the 40th Taniguchi symposium, Katata, Japan, June 23-28, 1997. Boston, MA: Birkh\"auser. Trends in Mathematics. 253-271 (1999). [ISBN 0-8176-4067-3/hbk]}
\AB{Let $M$ be a compact complex manifold and let $L$ be a line bundle
    on $M$. A singular Hermitian metric $h$ on $L$ is said to be
    an analytic Zariski decomposition (AZD), if the following hold.\par
1.
    the curvature $\Theta_h$ is a closed positive current,\par
2.
    for every $m\ge 0$, the natural inclusion $$H^0\bigl( M,{\cal
    O}_M(mL)\otimes {\cal I}(h^m)\bigr)\to H^0\bigl(M, {\cal O}_M(mL)\bigr)$$
    is an isomorphim, where ${\cal I}(h^m)$ denotes the multiplier
    ideal sheaf of $h^m$.\par
Firstly, the author proves that every
    big bundle $L$ on a smooth projective variety $M$ has an AZD.
    Secondly, he gives various applications: effective birationality
    of pluricanonical maps, volume of subvarieties, stratification
    of varieties of general type etc. $\dots$\par
A feature of the
    proof of the existence of an AZD:\par
Let $g$ be a $C^\infty$
    K\"ahler metric on $M$, $h_L$ a $C^\infty$ Hermitian metric on
    $L$; let\break $\{f_{0,m},\dots,f_{N(m),m}\}$ be an orthonormal basis
    of $H^0(M,{\cal O}_M (mL))$ with respect to the inner product
    associated to $h^m_L$ and $g$. We put $$K_m(z)= \sum^{N(m)}_{j=0}
    \bigl|f_{j,m} (z)\bigr|^2.$$ The author proves that the function
    $$h(z)=\bigl[ (\lim\sup K_m^{1/m})^* (z)\bigr]^{-1}$$ is an AZD.
    Here Demailly's results of 1992 play an important r\^ole.}
\RV{Thanh Van Nguyen (Toulouse)}
\CC{32L10
32J25
32C30
32U25}
%% END ITEM
%% BEGIN ITEM
\AN{0946.32010 (01347319)}{1999}
\AU{Hwang, Jun-Muk; To, Wing-Keung}
\TI{On Seshadri constants of canonical bundles of compact complex hyperbolic spaces.}
\LA{English}
\DT{J}
\SO{Compos. Math. 118, No.2, 203-215 (1999). 
\par http://www.springerlink.com/openurl.asp?genre=journal\&issn=0010-437X}
\ISSN{010-437}
\AB{Let $X$ be an $n$-dimensional projective algebraic manifold and
    $L$ an ample line bundle over $X$. In 1992, J. P. Demailly has
    introduced the Seshadri number $\varepsilon(L,x)$ at $x\in X$
    as a way to measure the local positivity of $L$ at $x$. Let $f:Y\to
    X$ be the blowing up of $X$ at $x$. Then $\varepsilon(L,x)$ is
    defined as the supremum of all positive number $\varepsilon$
    such that the $\bbfR$-divisor class $f^*L-\varepsilon L$ is nef
    on $Y$.\par
In this paper, the authors give an estimate for
    $\varepsilon(L,x)$ in the case where $X$ is a compact quotient
    of the unit ball in $\bbfC^n$. Let $\bbfB^n$ be the unit ball
    in $\bbfC^n$ equipped with the Poincar\'e metric. We denote by
    $\omega$ its K\"ahler form. For any two points $z$, $z'\in\bbfB^n$,
    we denote by $d(z,z')$ the Poincar\'e distance between them.
    Let $\Gamma \subset {\bold PU}(1,n)$ be a torsion free discrete
    subgroup such that $X=\bbfB^n/ \Gamma$ is compact. For a given
    point $x\in X$, choose an inverse image $x_0\in \bbfB^n$. Then
    we define the injectivity radius of $X$ at $x$ to be $\rho_X=
    (1/2) \min_{\gamma\in \Gamma,\ \gamma x_0\ne x_0} d(x_0,\gamma
    x_0)$. Denote also by $d(\cdot,\cdot)$ the distance function
    on $X$ with respect to $\omega$. We define the diameter $D_x$
    of $X$ at $x$ by $D_x=\max_{y\in X}d(x,y)$. Then we define the
    injective radius $\rho_X$ and the diameter $D_X$ of $(X,\omega)$
    by $\rho_X= \min_{x\in X}\rho_x$ and $D_X= \max_{x\in X}D_x$
    respectively. We also set $\varepsilon(K_X)=\min_{x\in X}\varepsilon(K_X,x)$.\par
The
    following theorem is the main result in this paper:\par
Theorem.
    Let $(X,\omega)$ be as above. Then for any points $x\in X$, the
    following inequality holds. $$(n+1) \sinh^2 (\rho_x) \le\varepsilon
    (K_X,x)\le (n+1)^2 \sinh^2(D_x).$$ In particular, $$(n+1)\sinh^2
    (\rho_X) \le\varepsilon (K_X)\le (n+1)^2 \sinh^2 (D_X).$$ Combining
    the above theorem and {\it J. P. Demailly}'s result [Lect. Notes
    Math. 1507, 87-104 (1992; Zbl 0784.32024)], the authors prove
    the following:\par
Corollary. If $\rho_X> \sinh^{-1} \sqrt{2n/(n+1)}$,
    then $2K_X$ is very ample. In particular there exists a finite
    etale cover $X'$ such that $2K_X$ is very ample.}
\RV{Yoshihiro Aihara (Shizuoka)}
\CC{32J25
32M15}
%% END ITEM
%% BEGIN ITEM
\AN{0944.32018 (01369604)}{1999}
\AU{Alessandrini, Lucia; Bassanelli, Giovanni}
\TI{Compact complex threefolds which are K\"ahler outside a smooth rational curve.}
\LA{English}
\DT{J}
\SO{Math. Nachr. 207, 21-59 (1999).  ISSN 1522-2616]
\par http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=60500208}
\ISSN{025-584}
\AB{Let $M$ be a compact complex manifold with dim $M\ge 3$ and $C$
    an irreducible curve in $M$ such that $M\backslash C$ carries
    a K\"ahler metric with K\"ahler form $\omega$. In general it
    does not follow that $M$ carries a K\"ahler metric, as the classical
    example of Hironaka shows. \par
The authors study the relations
    between cohomological properties of $C$ with respect to $M$ and
    obstructions for $M$ to be K\"ahler. They describe the obstructions
    in terms of plurisubharmonic positive currents by using a result
    of {\it J. P. Demailly} [J. Algebr. Geom. 1, No. 3, 361-409 (1992;
    Zbl 0777.32016)] which connects the regularization of a closed
    positive $(1,1)$-current with its Lelong numbers. The curve $C$,
    as a current, is said to be the $(1,1)$-component of a boundary
    if it is $(\partial+\overline{\partial})$-exact. $C$ is said
    to be part of the $(1,1)$-component of a boundary iff there exists
    a positive $(1,1)$-current $S\not=0$ with $\partial\overline{\partial}S=0$
    and $\chi_C\cdot S=0$, where $\chi_{C}$ denotes the characteristic
    functions of $C$. \par
For the general situation the main results
    are as follows: For a given pair $(M,C)$ exactly one of the following
    three cases will occur: $M$ is K\"ahler, or $C$ is the $(1,1)$-component
    of a boundary, or $C$ is part of the $(1,1)$-component of a boundary.
    If $\omega\cdot C>0$, then $M$ is K\"ahler. If $\omega\cdot C=0$,
    then $C$ is not part of the $(1,1)$-component of a boundary,
    and if $\omega\cdot C<0$ then $C$ is not the $(1,1)$-component
    of a boundary. From this it follows in particular that $M$ is
    K\"ahler if $C$ can be moved in $M$, i.e. if there exists an
    irreducible curve $C'\not=C$ which is homologous to $C$. \par
Finally
    the authors study the situation where dim $M=3$ and $C$ is a
    smooth rational curve in $M$. They define inductively a sequence
    of blow-ups $$\to M_{n+1}\to M_n\to\dots M_1\to M_0:=M$$ where
    $C$ is the center of $M_1\to M_0$, and $E_n$ is the exceptional
    divisor of the blow up $M_n\to M_{n-1}$. For $E_n\not=\Bbb P_1\times\Bbb
    P_1$ there is exactly one curve $C^\infty_n$ in $E_n$ with negative
    self-intersection, and $M_{n+1}\to M_n$ is defined as the blow
    up with center $C^\infty_n$. The sequence terminates at $M_n$
    if $E_n=\Bbb P_1\times\Bbb P_1$. Under the assumption that the
    sequence is infinite, the authors show the equivalence of the
    following three statements: $M$ is K\"ahler. $M$ is bimeromorphic
    to a compact K\"ahler manifold. $C$ is not the $(1,1)$-component
    of a boundary. If the sequence terminates at $M_{n}$, then $M_{n}$
    is K\"ahler.}
\RV{E.Oeljeklaus (Bremen)}
\CC{32J17
32J27
53C55
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0941.14020 (01336848)}{1999}
\AU{Di Rocco, Sandra}
\TI{Generation of $k$-jets on toric varieties.}
\LA{English}
\DT{J}
\SO{Math. Z. 231, No.1, 169-188 (1999). 
\par http://link.springer.de/cgi/linkref?issn=0025-5874\&year=1999\&volume=231\&page=169\kern-9pt}
\ISSN{025-587}
\AB{Let $X$ be a nonsingular complete algebraic variety over the
    field of complex numbers. Generalizing a notion introduced by
    {\it J. P. Demailly} [in: Complex Algebraic Varieties, Proc.
    Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-104 (1992; Zbl
    0784.32024)], {\it M. C. Beltrametti} and {\it A. J. Sommese}
    [in: Complex Analysis and Geometry, The Univ. Ser. Math., 355-376
    (1993; Zbl 0806.14014)] introduced the notion of $k$-jet ampleness:
    \par
A line bundle $L$ on X is said to be $k$-jet ample if for
    any finite collection of points $\{x_1,\ldots,x_r\}\subset X$
    and any $r$-tuple of positive integers $(k_1,\ldots,k_r)$ with
    $\sum k_i=k+1$, the natural map $$ H^0(X,L)\rightarrow\bigoplus_{1\leq
    j\leq r} L\otimes_{{\cal O}_{X,x_i}}{\cal O}_{X,x_i}/{\frak m}_{x_i}^{k_i}
    $$ is surjective, where ${\frak m}_{x_i}$ is the maximal ideal
    of the local ring ${\cal O}_{X,x_i}$ at $x_i$. \par
For $k'\leq
    k$, we easily see that $k$-jet ampleness implies $k'$-jet ampleness.
    $L$ is easily seen to be $0$-jet ample (resp.\ $1$-jet ample)
    if and only if $L$ is generated by global sections (resp.\ very
    ample). \par
When $X$ is a nonsingular complete toric variety
    corresponding to a fan $\Delta$, a line bundle $L$ can be described
    in terms of a $\Delta$-linear support function $\psi_L$. Generalizing
    the ampleness criterion in terms of the convexity of $\psi_L$
    and the toric Nakai criterion, found, for instance, in the reviewer's
    book [{\it T. Oda}, ``Convex bodies and algebraic geometry. An
    introduction to the theory of toric varieties'' (1988; Zbl 0628.52002)],
    the author shows the equivalence of the following: \par
(1)
    $L$ is $k$-jet ample; \par
(2) $L\cdot C\geq k$ for any $T$-invariant
    curve $C$, where $T$ is the algebraic torus for the toric variety
    $X$; \par
(3) $\psi_L$ is $k$-convex; \par
(4) the Seshadri
    constant $\varepsilon(L,x)\geq k$ for each $x\in X$.}
\RV{T.Oda (Sendai)}
\CC{14M25
14C20
14J60
14C25
14E25}
%% END ITEM
%% BEGIN ITEM
\AN{0939.32019 (01366179)}{1999}
\AU{Mortensen, Karen Pinney}
\TI{Vanishing of cohomology for $L$-convex domains.}
\LA{English}
\DT{J}
\SO{Indiana Univ. Math. J. 48, No.2, 557-566 (1999). 
\par http://www.iumj.indiana.edu/TOC/992.htm}
\ISSN{022-251}
\AB{Here the author proves the following result; the subscripts in
    the cohomology group refer to weighted $L^2$ bounds whose definition
    is given in the paper.\par
Theorem. Let $M$ be a K\"ahler manifold,
    $U\Subset M$ a domain with $C^2$ pseudoconvex boundary and $F$
    a coherent analytic sheaf on $M$. Assume the existence of a positive
    line bundle $L$ on $M$. Fix a real number $r>0$ and a sufficiently
    fine admissible open covering $\{U\}$ of $U$. Then there exists
    an integer $k_0$ such that for all integers $k\ge k_0$, $p \ge
    1$ and reals $r'\ge r$, $H^p_{r,r'} (\{U\}, F\otimes L^{\otimes
    k})=0$ and $F \otimes L^{\otimes k}$ is generated by $H^0(U,F
    \otimes L^{\otimes k})$. \par
The proof uses Demailly's vanishing
    theorem for cohomology with values in a vector bundle.}
\RV{E.Ballico (Povo)}
\CC{32L10
32Q15
32L20
32T99}
%% END ITEM
%% BEGIN ITEM
\AN{0933.32031 (01345967)}{1999}
\AU{Brunella, Marco}
\TI{On the automorphism group of a complex sphere.}
\LA{English}
\DT{J}
\SO{Doc. Math., J. DMV 4, 451-462 (1999).  ISSN 1431-0635]
\par http://www.emis.de/journals/DMJDMV/vol-04/13.html}
\ISSN{431-064}
\AB{Let $X$ be a compact complex threefold with the integral homology
    of ${\bold S}^6$ and let $\Aut(X)$ be its holomorphic automorphism
    group. By {\it A. Huckleberry, S. Kebekus} and {\it T. Peternell}
    [Group actions on ${\bold S}^6$ and complex structure on ${\bold
    P}^3$, preprint math. AG/9812076 (1998)] and {\it F. Campana,
    J.-P. Demailly} and {\it T. Peternell} [Compos. Math. 112, No.
    1, 77-91 (1998; Zbl 0910.32032)] the dimension of $\Aut(X)$ is
    at most 2. We prove that $\Aut(X)$ cannot be isomorphic to the
    complex affine group.}
\CC{32M05
32J17}
%% END ITEM
%% BEGIN ITEM
\AN{0926.32025 (01287521)}{1999}
\AU{Buchdahl, Nicholas}
\TI{On compact K\"ahler surfaces.}
\LA{English}
\DT{J}
\SO{Ann. Inst. Fourier 49, No.1, 287-302 (1999). 
\par http://annalif.ujf-grenoble.fr/aif\_1999.html}
\ISSN{373-095}
\AB{The conjecture of Kodaira that every compact complex surface
    $X$ with first Betti number $b_1(X) \equiv 0\pmod 2$ admits a
    K\"ahler metric follows partially from the Enriques-Kodaira classification.
    The remaining cases of elliptic and $K3$ surfaces were affirmatively
    answered by {\it Y. Miyaoka} [Proc. Jap. Acad. 50, 533-536 (1974;
    Zbl 0354.32011)] and {\it Y.-T. Siu} [Invent. Math. 73, 139-150
    (1983; Zbl 0557.32004)] respectively.\par
The paper under review
    contains a general proof of the conjecture which does not make
    any use of the Enriques-Kodaira classification. The author also
    gets criteria analogous to those of Nakai-Moishezon for elements
    in $H_\bbfR^{1,1} (X)$ to be representable by positive closed
    (1,1)-forms. The proofs are based on Hodge theory and $L_2$-cohomology.
    Main ingredients are results of P. Gauduchon on the existence
    of $\partial \overline \partial$-closed positive (1,1)-forms
    on compact complex surfaces, the theorem of Siu on the analyticity
    of the sets associated with the Lelong numbers of closed positive
    currents and results of Demailly about the smoothing of positive
    closed (1,1)-currents.}
\RV{E.Oeljeklaus (Bremen)}
\CC{32J15
14J15
32J27
53C55
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0926.32026 (01287520)}{1999}
\AU{Lamari, Ahc\`ene}
\TI{Courants k\"ahl\'eriens et surfaces compactes. (K\"ahler currents and compact surfaces).}
\LA{French}
\DT{J}
\SO{Ann. Inst. Fourier 49, No.1, 263-285 (1999). 
\par http://annalif.ujf-grenoble.fr/aif\_1999.html}
\ISSN{373-095}
\AB{The unified proof of the existence of K\"ahler metric on a surface
    of even Betti first number is given. The problem was set up by
    Kodaira [{\it K. Kodaira} and {\it J. Morrow}, `Complex manifolds',
    New York: Holt, Rinehart and Winston (1971)], and the solution
    was received by {\it Y.-T.~Siu} [Invent. Math. 73, 139-150 (1983;
    Zbl 0557.32004)]. \par
The author uses the Demailly regularization
    theorem and the Siu decomposition theorem to prove that the compact
    complex manifold $X$ having a K\"ahler current is the K\"ahler
    manifold outside of some analytic subset of codimension at least
    two. Using the Hodge symmetry, which follows from the hypothesis
    $b_1=2 h^{0,1},$ and the criterion similar to that of Harvey-Lawson,
    the author proves elementary the existence of a K\"ahler current
    on $X$, that, due to above-stated result, is enough to ensure
    the existence of the K\"ahler metric on $X.$}
\RV{A.V.Chernecky (Odessa)}
\CC{32J15
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0980.14028 (01383031)}{1998}
\AU{Peternell, Thomas; Serrano, Fernando}
\TI{Threefolds with nef anticanonical bundles.}
\LA{English}
\DT{J}
\SO{Collect. Math. 49, No.2-3, 465-517 (1998). 
\par http://www.mat.ub.es/CM}
\ISSN{010-075}
\AB{From the introduction: In this paper we study the global structure
    of projective threefolds $X$ whose anticanonical bundle $-K_X$
    is nef. In differential geometric terms this means that we can
    find metrics on $-K_X=\det T_X$ (where $T_X$ denotes the tangent
    bundle of $X)$ such that the negative part of the curvature is
    as small as we want. In algebraic terms nefness means that the
    intersection number $-K_X\cdot C\ge 0$ for every irreducible
    curve $C\subset X$. The notion of nefness is weaker than the
    requirement of a metric of semipositive curvature and is the
    appropriate notion in the context of algebraic geometry.\par
{\it
    J.-P. Demailly}, {\it T. Peternell} and {\it M. Schneider} [Compos.
    Math. 89, No. 2, 217-240 (1993; Zbl 0884.32023)] proved that
    the Albanese map $\alpha:X \to\text {Alb}(X)$ is a surjective
    submersion if $-K_X$ carries a metric of semi-positive curvature,
    or, equivalently, if $X$ carries a K\"ahler metric with semipositive
    Ricci curvature. It was conjectured that the same holds if $-K_X$
    is only nef, but there are very serious difficulties with the
    old proof, because the metric of semi-positive curvature has
    to be substituted by a sequence of metrics whose negative parts
    in the curvature converge to 0. The conjecture splits naturally
    into two parts: surjectivity of $\alpha$ and smoothness. Surjectivity
    was proved in dimension 3 already in the paper cited above and
    in general by {\it Qi Zhang} [J. Reine Angew. Math. 478, 57-60
    (1996; Zbl 0855.14007)], using char $p$. Our main result now
    proves smoothness in dimension 3:\par
Let $X$ be a smooth projective
    threefold with $-K_X$ nef. Then the Albanese map is a surjective
    submersion.}
\CC{14J30
14E30
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0966.32011 (01463783)}{1998}
\AU{Bonavero, Laurent}
\TI{In\'egalit\'es de Morse holomorphes singuli\`eres. (Singular holomorphic Morse inequalities).}
\LA{French}
\DT{J}
\SO{J. Geom. Anal. 8, No.3, 409-425 (1998). }
\ISSN{050-692}
\AB{The main result of this paper is the following theorem. Let $X$
    be a compact complex manifold of dimension $n$, $E$ a holomorphic
    line bundle over $X$, equipped with a metric $h$ (which might
    be singular). It is known that to the pair $(E,h)$ and an integer
    $k>0$ one may associate a sheaf of ideals ${\cal I}_k(h)\subset{\cal
    O}_X$, the so-called ``Nadel's multiplier sheaf''. Let $\Theta(E)$
    be the curvature current of $E$, $X(q,E)$ the analytic open subset
    of $X$ of points where the metric is smooth, with index $\le
    q$, where $0\le q\le n$. Finally, let $F$ be a holomorphic vector
    bundle on $X$, of rank $r$. Then, the author proves the following
    ``Morse's inequality'': $$\sum^q_{j=0} \dim H^j(X,{\cal O}(E^k\otimes
    F)\otimes{\cal I}_k(h))\le r(k^n/n!) \int_{X(q,E)} (-1)^q \Theta(E)^n+
    o(k^n).\tag{$*$}$$ The main motivation for this formula is to
    investigate the conjecture of Grauert and Riemenschneider about
    the characterization of Moishezon manifolds (i.e., the compact
    ones having $n$ algebraically independent meromorphic functions,
    where $n$ is the dimension) as those admitting a suitable line
    bundle with semi-positive, generically positive curvature. This
    would be an analog of a celebrated theorem of Kodaira characterizing
    projective manifolds as those admitting a positive line bundle.
    Y. T. Siu and (independently) J. P. Demailly gave sufficient
    conditions for a manifold to be Moishezon, in terms of the existence
    of an appropriate line bundle with a smooth metric. In his work,
    Demailly used a certain type of Morse inequalities (which are
    generalized in the present paper). With formula $(*)$, the author
    gives a characterization of Moishezon manifolds, in terms of
    the existence of a current $T$ on $X$ satisfying suitable conditions.
    He also expands and simplifies previous work by J. Kollar, showing
    that the conditions of Siu-Demailly are not necessary.\par
The
    proof of formula $(*)$ is rather technical. First, one ``simplifies''
    the situation: by means of monoidal transforms with smooth centers,
    one obtains a manifold $X'$, where the induced Nadel ideal defines
    a normal crossings divisor. Then one applies known ``Morse inequalities''
    in this new situation. Finally, one compares the cohomologies
    of $X$ and $X'$, which requires several lemmas and the use of
    a Leray spectral sequence.\par
In the final section of the paper
    (independent of the rest) a conjecture is proposed: if $X$ is
    a compact analytic manifold, $E$ and $F$ are vector bundles over
    $X$ of ranks 1 and $r$ respectively, then $\dim H^q(X, E^k\otimes
    F)= r\dim H^q(X, E^k)+ o(k^n)$. The author proves the conjecture
    in certain cases (if $X$ is Moishezon or a surface, or if $X$
    is arbitrary and $q=0$ or $q= n$).\par
In spite of its technical
    nature, the paper is well written and pleasant to read.}
\RV{Augusto Nobile (Baton Rouge)}
\CC{32J18
58J99
32J99
32L99}
%% END ITEM
%% BEGIN ITEM
\AN{0945.14020 (01216691)}{1998}
\AU{Peternell, Thomas}
\TI{Moishezon manifolds and rigidity theorems.}
\LA{English}
\DT{J}
\SO{Bayreuther Mathematische Schriften 54, 1-108 (1998). }
\ISSN{172-106}
\AB{The article under review is a survey of the theory of Moishezon
    manifolds subsuming, in particular, seven previous papers by
    the same author. A Moishezon variety $X$ is a compact complex
    analytic variety for which the transcendence degree of the field
    of meromorphic functions attains the maximal possible value (being
    equal to the dimension of $X$). Analogs of Moishezon varieties
    exist in abstract algebraic geometry (minischemes of Moishezon
    and algebraic spaces of M.~Artin), but the author considers only
    complex Moishezon manifolds, i.e. nonsingular complex Moishezon
    varieties. Moishezon varieties usually arise from projective
    algebraic varieties by blowing them up, contacting an exceptional
    subvariety or deforming them. The paper is divided into three
    chapters. In the first chapter the author collects preliminaries
    from complex analysis and algebraic geometry necessary for what
    follows. \par
Chapter 2 is devoted to basic theory of Moishezon
    manifolds. The proofs are usually either omitted (substituted
    by references) or sketched. Among the topics covered are the
    proof of the Grauert-Riemenschneider conjecture on characterization
    of Moishezon manifolds by the existence of an almost positive
    torsion-free sheaf of rank one due to Demailly and Siu and Moishezon's
    projectivity criterion (a Moishezon manifold is projective iff
    it is K\"ahler). In this chapter the author also recalls necessary
    facts from Mori theory, obtains a condition sufficient for projectivity
    of Moishezon threefolds in terms of the cone of numerically effective
    curves, introduces ``bad curves'' of two kinds which are obstructions
    for a Moishezon threefold to be projective, and gives some (well
    known) examples of Moishezon manifolds. \par
In chapter 3 the
    author gives a new proof of the theorem to the effect that a
    Moishezon threefold homeomorphic to $\Bbb P^3$ is actually isomorphic
    to $\Bbb P^3$. He also uses Mori theory to study Moishezon threefolds
    whose Picard number is equal to one and shows that every non-projective
    Moishezon threefold contains a rational curve without appealing
    to Mori's theorem on minimal models.}
\RV{Fyodor Zak (Moskva)}
\CC{14J30
32J17
32J25}
%% END ITEM
%% BEGIN ITEM
\AN{0942.32021 (01379350)}{1998}
\AU{Takayama, Shigeharu}
\TI{The Levi problem and the structure theorem for non-negatively curved complete K\"ahler manifolds.}
\LA{English}
\DT{J}
\SO{RIMS Kokyuroku 1058, 105-113 (1998).}
\AB{Recent developments of complex geometry enable us to get global
    sections of an adjoint bundle on a projective manifold, under
    a reasonable numerical condition. The theory which is mainly
    developed by Demailly and Siu, is very concrete and constructive.
    The author applies their method to construct holomorphic functions
    on certain pseudoconvex manifolds. \par
The main result of the
    paper is the following theorem: Let $X$ be a complex manifold
    with a negative canonical bundle $K_X.$ Then $X$ is holomorphically
    convex iff it is pseudoconvex. \par
The proof depends on effective
    construction of singular Hermitian metric and related vanishing
    theorem. \par
This theorem can be considered as a natural generalization
    of a result of Ohsawa for higher dimensional cases [{\it T.~Ohsawa},
    Publ. Res. Inst. Math. Sci. 17, 153-164 (1981; Zbl 0465.32011);
    supplement ibid. 17, 981-982 (1981; Zbl 0504.32012)].}
\RV{A.V.Chernecky (Odessa)}
\CC{32L20}
%% END ITEM
%% BEGIN ITEM
\AN{0934.14002 (01227214)}{1998}
\AU{de Cataldo, Mark Andrea A.}
\TI{Effective nonvanishing, effective global generation.}
\LA{English}
\DT{J}
\SO{Ann. Inst. Fourier 48, No.5, 1359-1378 (1998). 
\par http://annalif.ujf-grenoble.fr/aif\_1998.html}
\ISSN{373-095}
\AB{From the introduction: Let $g:X\to S$ be a surjective morphism
    of proper varieties, where $X$ is nonsingular and complete, $M$
    be a nef and $g$-big line bundle on $X$, $L$ be a nef and big
    line bundle on $S$ and $N=K_X+ M+mg^*L$ be a line bundle varying
    with the positive integer $m$. {\it J. Koll\'ar} [Math. Ann.
    296, No. 4, 595-605 (1993; Zbl 0818.14002)]; theorem 3.2 proved,
    under the necessary assumption that the first direct image sheaf
    $g_*N\ne 0$, that $h^0(X,N)= h^0(S,g_*N)>0$ and the sections
    of $g_*N$ generate this sheaf at a general point of $S$ for every
    $m>(1/2) (\dim S^2+\dim S)$ (this is what makes the result ``effective'').
    The purpose of this note is to observe that more precise statements
    are possible if one considers the local Seshadri constants of
    $L$ on $S$. The main result is the effective nonvanishing theorem
    2.2, a ``multiple-points higher-jets'' version of theorem 3.2
    of J. Koll\'ar's paper cited above.\par
As a first application,
    some generalizations to the case of nef vector bundles of the
    results concerning line bundles by {\it L. Ein}, {\it O. K\"uchle}
    and {\it R. Lazarsfeld} in J. Differ. Geom. 42, No. 2, 193-219
    (1995; Zbl 0866.14004) and by {\it J. Koll\'ar} in Invent. Math.
    113, No. 1, 177-215 (1993; Zbl 0819.14006) are given: effective
    construction of rational and birational maps, and nonvanishing
    on varieties with big enough algebraic fundamental group.\par
As
    another application, it is shown that the global generation results
    for line bundles of Anghern-Siu, Demailly, Tsuji and Siu generalize
    to vector bundles of the form $K_X^{\otimes a}\otimes E\otimes\text{det}
    E\otimes L^{\otimes m}$, where $a$ and $m$ are appropriate positive
    integers, $E$ is a nef vector bundle and $L$ is an ample line
    bundle. Explicit upper bounds on $m$ are given and they depend
    only on the dimension of the variety, and not also on the Chern
    classes of the variety and the bundles in question.\par
The
    paper by {\it M. A. A. de Cataldo} in J. Reine Angew. Math. 502,
    53-122 (1998; Zbl 0902.32012) provides upper bounds as above
    for vector bundles $E$ subject to curvature conditions. The methods
    employed there are analytic. A global generation result for nef
    vector bundles is proved in the final section by the use of algebraic
    Nadel ideals.}
\CC{14C20
14J60
14Q20
14E05
14F17}
%% END ITEM
%% BEGIN ITEM
\AN{0933.14025 (01336960)}{1998}
\AU{Bauer, Thomas}
\TI{Seshadri constants and periods of polarized abelian varieties.}
\LA{English}
\DT{J}
\SO{Math. Ann. 312, No.4, 607-623 (1998). 
\par http://link.springer.de/cgi/linkref?issn=0025-5831\&year=1998\&volume=312\&page=607\kern-9pt}
\ISSN{025-583}
\AB{Consider a polarized abelian variety $(A,L)$ over the field of
    complex numbers. Following J.-P. Demailly, one can associate
    to $(A,L)$ a real number $\varepsilon(A,L)$, its Seshadri constant,
    which in effect measures how much of the positivity of $L$ can
    be concentrated at any given point of $A$. There has been recent
    interest in finding bounds on the Seshadri constants of abelian
    varieties and on smooth projective varieties in general.\par
First,
    we generalize an approach of {\it P. Buser} and {\it P. Sarnak}
    [Invent. Math. 117, No. 1, 27-56 (1994; Zbl 0814.14033)] and
    give a lower bound on the minimal period length of $(A,L)$ in
    terms of the type of the polarization. Combined with a recent
    result of {\it R. Lazarsfeld} [Math. Res. Lett. 3, No. 4, 439-447
    (1996; Zbl 0890.14025)] this leads to a new lower bound on the
    Seshadri constant of $(A,L)$. Secondly, we show that Prym varieties
    have small Seshadri constants and therefore unusually small periods
    -- a result that parallels a result of {\it P. Lazarsfeld} (loc.
    cit.) on Jacobians. Finally, we obtain refined results for the
    case of abelian surfaces, which imply in particular the surprising
    fact that Seshadri constants are rational in this case.}
\RV{T.Bauer (Los Angeles)}
\CC{14K05
14C20
32G20}
%% END ITEM
%% BEGIN ITEM
\AN{0927.14002 (01142994)}{1998}
\AU{Steffens, A.}
\TI{Remarks on Seshadri constants.}
\LA{English}
\DT{J}
\SO{Math. Z. 227, No.3, 505-510 (1998). 
\par http://link.springer.de/cgi/linkref?issn=0025-5874\&year=1998\&volume=227\&page=505\kern-9pt}
\ISSN{025-587}
\AB{From the introduction: Let $X$ be a smooth projective variety
    and let $L$ be a line bundle on $X$. Fix a point $x\in X$. {\it
    J.-P. Demailly} [in: Complex algebraic varieties, Proc. Conf.,
    Bayreuth 1990, Lect. Notes Math. 1507, 87-104 (1992; Zbl 0784.32024)]
    introduced the real number $\varepsilon (L,x)= \inf_{C\ni x}
    {L.C \over m_x(C)}$, which is called the Seshadri constant of
    $L$ at $x$. Here the infimum is taken over all irreducible curves
    $C$ passing through $x$ and $m_x(C)$ is the multiplicity of $C$
    at $x$. For example, if $L$ is very ample then $\varepsilon (L,x)\ge
    1$.\par
Proposition 1. Let $X$ be a surface with $\rho(X)= \text{rk(NS}(X))=1$
    and let $L$ be an ample generator of $\text{NS} (X)$. Let $\alpha$
    be an integer with $\alpha^2\le L^2$. If $x\in X$ is a very general
    point, then $\varepsilon (L,x)\ge \alpha$. In particular if $\sqrt{L^2}$
    is an integer, then $\varepsilon (L,x)= \sqrt{L^2}$.\par
Proposition
    2. Let $X$ be the Jacobian of a hyperelliptic curve of genus
    $g\ge 2$ with $\rho(X)=1$. And let $\Theta$ be the theta divisor
    on $X$. Then $\varepsilon (\Theta)= \varepsilon (\Theta,x) \le{2g
    \over g+1}< \root g\of{g!} =\root g \of {\Theta^g}$. In particular
    if $X$ is an irreducible principal polarized abelian surface,
    then $\varepsilon (\Theta)= \varepsilon (\Theta,x)= {4\over 3}<
    \sqrt 2= \sqrt{\Theta^2}$.\par
Proposition 3. Let $X$ be a general
    principal polarized abelian variety of dimension $g$ with theta
    divisor $\Theta$. Then $$\varepsilon (\Theta)= \varepsilon (\Theta,x)
    \le{\root g-1\of {{g!\over 2^{g-3} (2^g-1)}}} <\root g\of{g!}
    =\root g\of {\Theta^g}.$$ Proposition 4. In dimension $n$, if
    the Seshadri constant is non-maximal, then it is a $d$-th root
    of a rational number for some $1\le d\le n-1$.}
\CC{14C20
14H40
14H42
14K25}
%% END ITEM
%% BEGIN ITEM
\AN{0941.32022 (01405230)}{1997}
\AU{Yotov, M.}
\TI{Nadel's subschemes of Fano manifolds $X$ with a Picard group Pic$(X)$ isomorphic to $Z$.}
\LA{English}
\DT{J}
\SO{Serdica Math. J. 23, No.2, 187-202 (1997). 
\par http://alef.math.acad.bg/~serdica/}
\ISSN{204-411}
\AB{A subscheme $Y$ of a Fano manifold $X$ is called a Nadel subscheme
    if for any Nakano semi-positive holomorphic vector bundle $E$
    on $X$ the cohomology groups $H^q(X, E \otimes I_Y)$, $q \ge
    1$ vanish. As was shown by {\it A. Nadel} [Ann. Math. 132, 549-596
    (1990; Zbl 0731.53063)] and by the author [Izv. Acad. Nauk SSSR,
    Ser. Mat. 58, No. 5, 53-67 (1994; Zbl 0848.32023) in Russian]
    that the non-existence of certain types of Nadel subschemes of
    $X$ implies an existence of a K\"ahler-Einstein metric on $X$.
    The main results of this paper, Theorem 3.1 and Corollary 2.6,
    give global sufficient conditions for suitable subschemes $Y$
    of $X$ to be Nadel subschemes. This generalizes the sufficient
    conditions known by Nadel and Demailly. In Section 4 are given
    examples to which the results of the paper are applicable. In
    particular, Proposition 4.2 describes all the cases when a reduced
    curve of degree 3 is a Nadel subscheme of {\bf P}${}^n$, $n \ge
    3$.}
\RV{A.Iliev (Sofia)}
\CC{32L10
32L20
32Q20}
%% END ITEM
%% BEGIN ITEM
\AN{0932.00086 (01403899)}{1997}
\AU{Hirzebruch, Friedrich; Demailly, Jean-Pierre; Lannes, Jean}
\TI{Journ\'ee en l'honneur de Henri Cartan. (Conference in honor of 
Henri Cartan)\kern-2pt}
\LA{French}
\DT{B}
\SO{SMF Journ\'ee Annuelle. 1997. Paris: Soci\'et\'e Math\'ematique de France, iv, 27 p. (1997).}
\AB{The articles of this volume will be reviewed individually. Contents:
    F. Hirzebruch, Learning complex analysis in M\"unster--Paris,
    Z\"urich and Princeton from 1945 to 1953 (1-2), Jean-Pierre Demailly,
    Vari\'et\'es projectives hyperboliques et \'equations diff\'erentielles
    alg\'ebriques (French) (Hyperbolic projective varieties and algebraic
    differential equations) (3-17), Jan Lannes, Divers aspects des
    op\'erations Steenrod (French) (Diverse aspects of the Steenrod
    operations) (18-27).}
\CC{00B30
00B15
32-06}
%% END ITEM
%% BEGIN ITEM
\AN{0916.14004 (01018557)}{1997}
\AU{Tsai, I-Hsun}
\TI{Dominant morphisms on surfaces of general type modulo biholomorphic equivalence.}
\LA{English}
\DT{J}
\SO{Int. Math. Res. Not. 1997, No.3, 101-111 (1997). }
\ISSN{073-792}
\AB{The Iitaka-Severi conjecture states that the number of smooth
    complex projective varieties of general type that arise as a
    dominant rational image of a fixed variety $X$ is finite up to
    birational equivalence. This article is one of many of the author's
    that are related to this topic. The central goal of the present
    article is giving effective estimates in the case when the target
    is assumed to be a surface.\par
The estimates heavily depend
    on effective very ampleness of adjoint bundles (as in Fujita's
    conjecture) of J.-P. Demailly, J. Koll\'ar, L. Ein and R. Lazarsfeld,
    and Y. T. Siu. Therefore, although these estimates are probably
    far from being sharp, as soon as there are better estimates for
    very ampleness they should yield better ones in this case.\par
Besides
    using effective very ampleness results, the paper studies Chow
    varieties and uses results of {\it F. Catanese} [J. Algebr. Geom.
    1, No. 4, 561-595 (1992; Zbl 0807.14006)].}
\RV{Sandor J.Kovacs (MR 99f:14017)}
\CC{14E05
14J29}
%% END ITEM
%% BEGIN ITEM
\AN{0914.32003 (01172114)}{1997}
\AU{Coman, Dan}
\TI{Integration by parts for currents and applications to the relative capacity and Lelong numbers.}
\LA{English}
\DT{J}
\SO{Mathematica 39, No.1, 45-57 (1997). }
\ISSN{222-901}
\AB{Some formulas for integration by parts of plurisubharmonic functions
    are proved. Let us give one statement.\par
Theorem 3.1. Let
    $\Omega$ be a bounded domain in $\bbfC^n$ and let $T$ be a closed
    positive current of bidimension (1.1) in $\Omega$. Assume that
    $u$ and $v$ are negative locally bounded plurisubharmonic functions
    in $\Omega$, such that $u=v$ outside a compact set of $\Omega$.
    Then $$\int_\Omega vdd^cu\wedge T=\int_\Omega udd^c v\wedge^T.$$
    This and relations of such type are used to obtain upper and
    lower bounds for the relative capacity $C(K,\Omega)$ of a compact
    subset $K$ of a hyperconvex domain $\Omega$. Also they are used
    to prove a generalization of a formula of Demailly regarding
    Lelong numbers with weights.}
\RV{S.M.Ivashkovich (Villeneuve d'Ascq)}
\CC{32U05
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0889.32019 (01057911)}{1997}
\AU{V\^aj\^aitu, V.}
\TI{Runge theorem in higher dimensions.}
\LA{English}
\DT{CA}
\SO{Ancona, V. (ed.) et al., Complex analysis and geometry. Proceedings of the CIRM conferences on complex analysis and geometry XII, Trento, Italy, June 5--9, 1995 and vector bundles on Fano threefolds II, Trento, Italy, December 2--5, 1995. Bonn: Longman. Pitman Res. Notes Math. Ser. 366, 165-181 (1997). [ISBN 0-582-29276-X;}
\ISSN{269-367}
\AB{Let $X$ be a complex space of dimension $n$ without compact irreducible
    components of dimension $n$. Let $K\subset X$ be a compact set.
    Then $K$ is said to be $n$-convex if for every open neighborhood
    $D$ of $K$ in $X$, there exists an exhaustion strongly $n$-convex
    function $\varphi\in C_R^\infty (X)$ such that $$K\subset \bigl\{x\in
    X\mid \varphi(x) < 0\bigr\} \Subset D.$$ The main result of this
    paper provided various characterizations of $K$: algebraic cohomology
    groups, sheaf-theoretic cohomology groups and set-theoretic topology.\par
When
    $X$ is nonsingular, this was established by {\it G. Lupacciolu}
    [Trans. Am. Math. Soc. 337, No. 1, 427-435 (1993; Zbl 0776.32013)]
    by constructing a family of subharmonic (resp. strongly subharmonic)
    functions $\{h_i\}$ with respect to certain Laplacian $\Delta_0$;
    the existence of those functions were established by {\it R.
    E. Greene} and {\it H. Wu} [Ann. Inst. Fourier 25, No. 1, 215-235
    (1975; Zbl 0307.31003)] which worked only when $X$ is a manifold.
    \par
Now for arbitrary $X$, a device of {\it J.-P. Demailly}
    [Math. Z. 204, No. 2, 283-295 (1990; Zbl 0702.32013)] shows how
    to handle the $n$-convexity of $\varphi$ in a neighborhood of
    the singular set of $X$. The main job of this expository article
    transplants Demailly's device into the construction of Lupacciolu;
    certainly this has a flavor of an elegant term paper by an undergraduate
    student. Also fairly complete references are provided.}
\RV{Vo Van Tan (Boston)}
\CC{32F10}
%% END ITEM
%% BEGIN ITEM
\AN{0877.14013}{1997}
\AU{Biswas, Indranil}
\TI{Parabolic ample bundles.}
\LA{English}
\DT{J}
\SO{Math. Ann. 307, No.3, 511-529 (1997). 
\par http://link.springer.de/cgi/linkref?issn=0025-5831\&year=1997\&volume=307\&page=511\kern-9pt}
\ISSN{025-583}
\AB{In this paper the notion of ample parabolic bundle is introduced
    and ample parabolic bundles over a Riemann surface are characterized.
    Parabolic bundles with respect to an effective divisor were defined
    by {\it V. B. Mehta} and {\it C. S. Seshadri} for Riemann surfaces
    [Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)] and by {\it
    M. Maruyama} and {\it K. Yokogawa} for higher dimensional varieties
    [Math. Ann. 293, No. 1, 77-100 (1992; Zbl 0735.14008)]. A parabolic
    structure on a torsion free sheaf $E$ is equivalent to the existence
    of certain decreasing filtration $\{E_t\}_{t\in \bbfR}$ of $E$.
    For a parabolic bundle $E_\ast$, one can define a parabolic $n$-fold
    symmetric product $S^n(E_\ast)$; it is a parabolic sheaf so that
    one can consider the associated filtration and in particular
    the subsheaf $S^n(E_\ast)_0$. The author then defines a parabolic
    ample sheaf as a parabolic sheaf $E_\ast$ such that $S^n(E_\ast)_0\otimes
    V$ is generated by its global sections for any vector bundle
    $V$. When the divisor $D$ is empty, an ample parabolic sheaf
    is nothing but a usual ample sheaf. Parabolic sheaves on compact
    Riemann surfaces are characterized: a parabolic vector bundle
    $E_\ast$ over a compact Riemann surface is ample parabolic if
    and only if the parabolic degree of $E_\ast$ and that of all
    its quotient bundles with the induced parabolic structure are
    all strictly positive numbers. In particular a parabolic semistable
    bundle over a compact Riemann surface is parabolic ample if and
    only if its parabolic degree is strictly positive. The paper
    gives also a vanishing theorem for the cohomology of the sheaves
    of vector bundle valued differential forms: \par
If the divisor
    $D$ has normal crossings and $E_\ast$ satisfies certain conditions,
    then $$H^q(X,\Omega^p(\log D)\otimes E)=0\text{ for }p+q\ge r+d,$$
    $r$ being the rank of $E$ and $d$ the dimension of $X$. \par
This
    generalizes a celebrated theorem of {\it J. Le Potier} [Math.
    Ann. 218, 35-53 (1975; Zbl 0313.32037)]. The paper also gives
    parabolic analogues of vanishing theorems for tensor powers of
    ample bundles due to {\it J.-P. Demailly} [Invent. Math. 91,
    No. 1, 203-220 (1988; Zbl 0647.14005)] and {\it L. Manivel} [J.
    Reine Angew. Math. 422, 91-116 (1991; Zbl 0728.14011)]. The paper
    is very clearly written despite of the fact that certain proofs
    are rather technical.}
\RV{D.Hern\'andez Ruip\'erez (Salamanca)}
\CC{14F05
14F17
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0926.32027 (01154108)}{1996}
\AU{Ohsawa, Takeo}
\TI{Some applications of $L^2$ estimates to complex geometry.}
\LA{Japanese, English}
\DT{J}
\SO{Sugaku Expo. 12, No.2, 127-150 (1999); translation from Sugaku 48, No.2, 142-161 (1996). }
\ISSN{039-470}
\AB{This paper surveys applications of the $L^{2}$ estimate method
    to complex algebraic geometry, which consists of 6 sections:
    (1) Plurisubharmonic functions (PSH) and Siu's theorem; (2) Vanishing
    theorems and extension theorems; (3) Fujita conjecture and Demailly-Siu's
    theory; (4) $L^{2}-$estimates; (5) On deformation of open manifolds;
    (6) $L^{2}$-cohomology of singularities. \par
The first 3 sections
    just review some materials related to this topic, such as Siu's
    theorem on the analyticity of a set determined by PSH, Nadel's
    vanishing theorem on analytic coherent sheaves (a generalization
    of Kodaira's vanishing theorem), extension theorem on weighted
    $L^{2}$-holomorphic functions, Fujita's conjecture on ample bundles,
    Demailly's Morse inequality, etc. Section 4 describes a basis
    of the $L^{2}$ estimate method for the following application.
    Section 5 concerns application to the deformation of complex
    manifolds. Section 6 is to the topology of algebraic varieties,
    in particular, to solving the Cheeger-Goresky-MacPherson conjecture
    on the isomorphism of intersection cohomology group and $L^{2}$-cohomology
    group in the case of isolated singularities.\par
At last a bibliography
    of 71 papers is given.}
\RV{Na Jisheng (Beijing)}
\CC{32J25
32V05
32G15
14F43
32L20
32C35}
%% END ITEM
%% BEGIN ITEM
\AN{0893.14014 (00915377)}{1996}
\AU{Nakamaye, Michael}
\TI{Seshadri constants on abelian varieties.}
\LA{English}
\DT{J}
\SO{Am. J. Math. 118, No.3, 621-635 (1996). 
\par http://muse.jhu.edu/journals/american\_journal\_of\_mathematics/toc/ajm118.3.html}
\ISSN{002-932}
\AB{For an ample line bundle $L$ over a smooth projective variety
    $X$, the Seshadri constant of $L$ at a point $x\in X$ is defined
    following {\it J.-P. Demailly} [in: Complex algebraic varieties,
    Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-104 (1992;
    Zbl 0784.32024)] as $\varepsilon(x,L)=\inf_{x\in C} L.C/\text{mult}_x(C)$,
    where the infimum is taken over all curves $C$ passing through
    $x$. It is known that $\varepsilon (\eta ,L)\geq (\dim X)^{-1}$
    for any line bundle and a very general point $\eta\in X$ [{\it
    L. Ein, O.K\"uchle} and {\it R. Lazarsfeld}, J. Differ. Geom.
    42, No. 2, 193-219 (1995; Zbl 0866.14004)]. \par
Note that if
    $X=A$ is a complex abelian variety, since translation preserves
    numerical equivalence the Seshadri constant of a line bundle
    $L$ is independent of the point; it will be denoted by $\varepsilon
    (L)$. The author proved that $\varepsilon(L)\geq 1$ for any ample
    line bundle $L$ on $A$; in this paper he shows that for dim$A\geq
    2$ there exists an ample line bundle $L$ with $\varepsilon(L)=1$
    if and only if $A$ is isomorphic to a product $E\times B$, with
    $E$ is an elliptic curve and $B$ an abelian variety. \par
The
    global Seshadri constant is defined as $\varepsilon(X)=\inf_{x,L}\varepsilon(x,L)$,
    where the infimum is taken over all $x\in X$ and all ample line
    bundles $L$ on $X$. Regarding this invariant, this paper shows
    that $\varepsilon(A)>1$ if $A$ is a simple abelian variety of
    dimension $\geq 2$. In the case when $A$ is an abelian surface
    which is not a product of elliptic curves, the author shows that
    $\varepsilon(A)\geq {4\over 3}$. For a general abelian surface
    $A$ equality holds, i.e. $\varepsilon(A)={4\over 3}$ [see {\it
    A. Steffens}, ``Remarks on Seshadri constants'', Math. Z. 227,
    No. 3, 505-510 (1998)].\par
The proofs use differential operators
    on the abelian variety to analyze certain Gauss maps and graded
    linear series.}
\RV{E.Previato (Boston)}
\CC{14K10
14C20
14H52}
%% END ITEM
%% BEGIN ITEM
\AN{0892.14007 (01054916)}{1996}
\AU{Ein, Lawrence; Lazarsfeld, Robert; Nakamaye, Michael}
\TI{Zero-estimates, intersection theory, and a theorem of Demailly.}
\LA{English}
\DT{CA}
\SO{Andreatta, Marco (ed.) et al., Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, June 15--24, 1994. Berlin: Walter de Gruyter. 183-207 (1996). [ISBN 3-11-014503-0/hbk]}
\AB{These notes are an excellent presentation from the point of view
    of algebraic geometry of the deep analytic results of Demailly
    in his work on Fujita's conjectures about adjoint line bundles
    on projective varieties.}
\RV{A.Dimca (Bordeaux)}
\CC{14C20
14F05
14C17}
%% END ITEM
%% BEGIN ITEM
\AN{0883.14005}{1996}
\AU{Demailly, Jean-Pierre}
\TI{$L\sp 2$ vanishing theorems for positive line bundles and adjunction theory.}
\LA{English}
\DT{CA}
\SO{Catanese, F. (ed.) et al., Transcendental methods in algebraic geometry. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, July 4--12, 1994. Cetraro: Springer. Lect. Notes Math. 1646, 1-97 (1996). [ISBN 3-540-62038-9/pbk;}
\ISSN{075-843}
\AB{Main goal of the paper is to describe a few analytic tools which
    are useful to study questions such as linear series and vanishing
    theorems for algebraic vector bundles. Also, algebraic and analytic
    proofs of some results are compared. One of the first applications
    of the analytic method in algebraic geometry is Kodaira's use
    of the Bochner technique (1950-60) to relate cohomology and curvature
    via harmonic forms. Well known is the Akizuki-Kodaira-Nakano
    theorem (1954): If $X$ is a nonsingular projective algebraic
    variety and $L$ is a holomorphic line bundle on $X$ with positive
    curvature, then $H^q(X,\Omega^p_X\otimes L)=0$ for $p+q>$ dim$X$.
    H\"ormander (1965) used a refinement of this technique to obtain
    a fundamental $L^2$ estimate, concerning solutions of the Cauchy-Riemann
    operator. Except vanishing theorems, more precise quantative
    information about solutions of $\bar{\partial}$-equations was
    obtained. Main tools to relate analytic and algebraic geometry
    are the multiplier ideal sheaf $I(\phi)$ and positive currents.
    $I(\phi)$ is defined as a sheaf of germs of holomorphic functions
    $f$ such that $|f|^2e^{-2\phi}$ is locally summable, where $\phi$
    is a (locally defined) plurisubharmonic function. Since $I(\phi)$
    is a coherent algebraic sheaf over $X$, we have a direct correspondence
    between analytic and algebraic objects which takes into account
    singularities efficiently. Currents, introduced by Lelong (1957),
    play the role of algebraic cycles, and many classical results
    of intersection theory apply to currents. Also an analytic interpretation
    of the Seshadri constant of a line bundle is given and it represents
    a measure of local positivity. One of the motivations for this
    work was the conjecture of Fujita: If $L$ is an ample (i.e. positive)
    line bundle on a projective $n$-dimensional algebraic variety
    $X$ then $K_X+(n+2)L$ is very ample. Reider (1988) gave a proof
    of the Fujita conjecture in the case of surfaces. \par
Using
    an analytic approach, in the paper under review it is shown that
    $2K_X+L$ is very ample under suitable numerical conditions for
    $L$. The first part of the proof is to choose an appropriate
    metric using a complex Monge-Amp\`ere equation and the Aubin-Calabi-Yau
    theorem. Solution $\phi$ of the equation assumes logarithmic
    poles and they are controlled using the intersection theory of
    currents. Detailed relations to the existing algebraic proofs
    of similar results are given (Ein-Lazarsfeld, Fujita, Siu). In
    the last section, a proof of the effective Matsusaka big theorem
    obtained by {\it Y.-T. Siu} [Ann. Inst. Fourier 43, No. 5, 1387-1405;
    Zbl 0803.32017)] is presented. Siu's proof is based on the very
    ampleness of $2K_X+mL$ together with the theory of holomorphic
    Morse inequalities [{\it J.-P. Demailly}, Ann. Inst. Fourier
    35, No. 4, 189-229 (1985; Zbl 0565.58017)]. Long and detailed
    preliminary sections dedicated to the basic facts of complex
    differential geometry are included which make the main ideas
    of the paper easier to understand.}
\RV{N.Bla\v{z}i\'c (Beograd)}
\CC{14F17
32L05
14F43
14F05
32L20
32C30
32W20}
%% END ITEM
%% BEGIN ITEM
\AN{0871.14006}{1996}
\AU{Fern\'andez del Busto, Guillermo}
\TI{A Matsusaka-type theorem on surfaces.}
\LA{English}
\DT{J}
\SO{J. Algebr. Geom. 5, No.3, 513-520 (1996). }
\ISSN{056-391}
\AB{Let $X$ be a smooth complex projective variety of dimension $n\ge
    1$ polarized by an ample line bundle $A$. Matsusaka's theorem
    states that there exists a constant $t_0$, depending only on
    the coefficients of the Hilbert polynomial of $A$, such that
    $tA$ is very ample for every $t\ge t_0$. Recently, after the
    work of Demailly and Nadel, some interest grew-up about effective
    versions of this theorem, but the lower bounds obtained for $t$
    are far from being optimal. In the paper under review, focusing
    on the case of surfaces, the author proves the following theorem:\par
Let
    $(X,A)$ be as above with $n=2$; if $t> [(A(K_X +4A)+ 1)^2/A^2+3]/2$,
    then $tA$ is very ample. This follows from a similar but more
    general result proved in the paper for the $k$-jet ampleness
    of $tA$, a notion of higher order embedding introduced by {\it
    M. C. Beltrametti} and {\it A. J. Sommese} [in: Complex analysis
    and geometry, Univ. Ser. Math., 355-376 (1993; Zbl 0806.14014)].
    More recently, Beltrametti and Sommese [``Sharp Matsusaka-type
    theorems on surfaces'', Preprint (1997)] obtained sharper lower
    bounds than the author and also Matsusaka-type theorems for the
    $k$-spannedness and the $k$-very ampleness of $tA$, by putting
    the nef-value of the polarized surface $(X,A)$ into the picture.}
\RV{A.Lanteri (Milano)}
\CC{14C20
14J60}
%% END ITEM
%% BEGIN ITEM
\AN{0866.32014}{1996}
\AU{Bonavero, Laurent}
\TI{Sur des vari\'et\'es de Moishezon dont le groupe de Picard est de rang un. (On Moishezon manifolds with Picard group of rang one).}
\LA{French}
\DT{J}
\SO{Bull. Soc. Math. Fr. 124, No.3, 503-521 (1996). 
\par http://www.numdam.org:80/numdam-bin/feuilleter?id=BSMF\_1996$\_\_$124\_3 \par http://smf.emath.fr/Publications/Bulletin/124/html/smf\_bull\_124\_503-521.html\kern-3pt}
\ISSN{037-948}
\AB{The well known Demailly-Siu criterion [{\it Y. T. Siu}, J. Differ.
    Geom. 19, 431-452 (1984; Zbl 0577.32031); Lect. Notes Math. 1111,
    169-192 (1985; Zbl 0577.32032), and {\it J. P. Demailly}, Ann.
    Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)] gives
    a sufficient condition in order that a compact complex analytic
    manifold $X$ is Moishezon. The condition is not necessary as
    shown by J. Koll\'ar and K. Oguiso by giving examples of Moishezon
    manifolds $X$ with $\text {Pic} (X)=\bbfZ \cdot {\cal O}_X(1)$,
    $m_X\le 0$, where $m_X$ is defined by $K_X={\cal O}_X(m_X)$,
    without any big and nef bundle on them.\par
The paper deals
    with the problem of the existence of such manifolds with $m_X>0$.
    From a result of {\it J. Koll\'ar} [Surv. Differ. Geom. 1, 113-199
    (1991; Zbl 0755.14003)] such manifolds do not exist in dimension
    3 since if the canonical bundle is big then it is nef. The author
    proves that in dimension 4 or larger than 4 there exist such
    Moishezon manifolds where $K_X$ is big but it is not nef and,
    using Mori theory, he produces explicitly some interesting examples,
    which give also new manifolds not satisfying the Demailly-Siu
    criterion.}
\RV{A.Tancredi (Perugia)}
\CC{32J99
32Q99
14J40
14C22}
%% END ITEM
%% BEGIN ITEM
\AN{0860.14019}{1996}
\AU{Angelini, Flavio}
\TI{An algebraic version of Demailly's asymptotic Morse inequalities.}
\LA{English}
\DT{J}
\SO{Proc. Am. Math. Soc. 124, No.11, 3265-3269 (1996).  ISSN 1088-6826]
\par http://www.ams.org/journal-getitem?pii=S0002-9939-96-03829-4}
\ISSN{002-993}
\AB{{\it J.-P. Demailly} deduces, from his Morse inequalities [Ann.
    Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)], some
    asymptotic estimates for the dimensions of cohomology groups
    of the difference of two numerically effective line bundles in
    which only algebraic invariants are involved. The purpose of
    this note is to give an elementary algebraic proof of these estimates
    in the case of a smooth projective variety. -- The theorem is
    the following:\par
Let $L=F-G$ be a line bundle on a smooth
    complex projective variety $X$ of dimension $n$, where $F$ and
    $G$ are ample line bundles. Then, for every $q =0$, $1,\dots,n$
    and $k$ sufficiently large, we have: $$h^q (X,kL) \le k^n {F^{n-q}
    \cdot G^q \over (n-q)!q!} + o(k^n), \tag 1$$ $$\sum^q_{i=0} (-1)^{q-i}
    h^i(X,kL) \le {k^n \over n!} \sum^q_{i = 0} (-1)^{q-i} {n\choose
    i} F^{n-i} \cdot G^i+o (k^n).\tag 2$$.}
\CC{14F25
14F05
32J99}
%% END ITEM
%% BEGIN ITEM
\AN{0855.14007}{1996}
\AU{Zhang, Qi}
\TI{On projective manifolds with nef anticanonical bundles.}
\LA{English}
\DT{J}
\SO{J. Reine Angew. Math. 478, 57-60 (1996). 
\par http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN243919689}
\ISSN{075-410}
\AB{We prove that if $X$ is a projective manifold with nef anti-canonical
    bundle $- K_X$, then the Albanese map $\alpha : X \to A(X)$ is
    surjective and has connected fibers which gives an affirmative
    answer to a conjecture of Demailly, Peternell and Schneider.
    We also study morphisms between manifolds with nef anticanonical
    bundles and the fundamental groups of such manifolds.}
\RV{Qi Zhang (Columbia)}
\CC{14J60
14F35
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0851.32018}{1996}
\AU{Kaliman, Sh.; Zaidenberg, M.}
\TI{A transversality theorem for holomorphic mappings and stability of Eisenman-Kobayashi measures.}
\LA{English}
\DT{J}
\SO{Trans. Am. Math. Soc. 348, No.2, 661-672 (1996).  ISSN 1088-6850]
\par http://www.ams.org/journal-getitem?pii=S0002-9947-96-01482-1}
\ISSN{002-994}
\AB{The paper treats a version of Thom's transversality theorem in
    complex analytic category. Let $X$ and $M$ be connected complex
    manifolds. Denote by $\text{Hol}(X,M)$ the space of holomorphic
    mappings $X \to M$ and by $J^k(X,M)$ the space of $k$-jets of
    holomorphic mappings $f : X \to M$. To state a main result suppose
    that $X$ is Stein and $M$ is endowed with a Hermitian metric
    $h$. Let $f \in \text{Hol}(X,M)$, $Y$ a closed analytic subset
    of $X$ and $A_i \subset J^{k_i} (X,M)$, for $i = 1,2,\dots$ and
    $k_i$ non-negative integers, be a sequence of stratified analytic
    subsets. Here a subset $A$ of $J^k (X,M)$ is said to be a stratified
    analytic subset if $A$ is a closed analytic subset of $J^k(X,M)$
    and satisfies Whitney's condition (a) [{\it H. Whitney}, Complex
    analytic varieties, Addison-Wesley, (1972; Zbl 0265.32008)].
    Suppose that the restriction $j^{k_i} (f)|_Y : Y \to J^{k_i}
    (X,M)$ is transversal to $A_i$ for each $i$. Fix any compact
    set $K \subset X$, any $k \in \bbfN$ and any $\varepsilon > 0$.
    Then there exists an open neighbourhood $\Omega$ of $K$ in $X$
    and a holomorphic mapping $\widehat{f} = f_{\varepsilon,K,k}
    : \Omega \to M$ such that the following conditions are true:
    (i) $j^{k_i} (\widetilde{f})$ is transversal to $A_i$ for each
    $i = 1,2,\dots;$ (ii) $j^k(\widetilde{f}) |_{Y \cap \Omega} =
    j^k(f) |_{Y \cap \Omega};$ (iii) $\text {dist}_h (f|_\Omega,
    \widetilde{f}) < \varepsilon$ uniformly in $\Omega$. In the algebraic
    setting the theorem can be strengthened, as a corollary of the
    main result above using an approximation theorem in [{\it J.
    P. Demailly, L. Lempert} and {\it B. Shiftman}, Duke Math. J.
    76, No. 2, 333-363 (1994); see also {\it L. Lempert}, Invent.
    Math. 121, No. 2, 335-354 (1995; Zbl 0837.32008)] so that under
    the appropriate algebraic conditions ($X$ being affine algebraic,
    $M$ smooth quasi-projective, and $Y$ a closed algebraic subvariety
    of $X$ \dots etc.) with $j^k(f)|_Y : Y \to J^k(X,M)$ a regular
    mapping and with each $n \in \bbfN$, given $K \subset X$, $\varepsilon
    > 0$ as before, there exists $\Omega$ as before (depending on
    $n$) such that the map $\widetilde{f} : \Omega \to M$ can be
    chosen to be a Nash algebraic mapping satisfying the preceding
    conclusions (i) for $i = 1,2,\dots,n$, (ii) and (iii). Here $\widetilde{f}$
    being Nash algebraic means that the graph of $\widetilde{f}$
    is contained in an algebraic subvariety $N \subset X \times M$
    with the dimension of $N$ equal to that of $X$. As an application
    of the present transversality theorem it is shown that the Eisenmann
    $k$-measure does not change under deleting analytic subsets of
    codimension at least $k + 1$.}
\RV{I-Hsun Tsai (Taipei)}
\CC{32E10
32H02
58A35
58A07
58C10}
%% END ITEM
%% BEGIN ITEM
\AN{0849.14002}{1996}
\AU{Bertin, Jos\'e; Demailly, Jean-Pierre; Illusie, Luc; Peters, Chris}
\TI{Introduction \`a la th\'eorie de Hodge. (Introduction to Hodge theory).}
\LA{French. English summary}
\DT{B}
\SO{Panoramas et Synth\`eses. 3. Paris: Soci\'et\'e Math\'ematique de France. vi, 272 p. (1996). [ISBN 2-85629-049-3]}
\AB{The origin of what is currently meant by the notion of Hodge
    theory can be traced back to {\it W. V. D. Hodge}'s fundamental
    work accomplished in the 1930s. In modern terminology, Hodge
    prepared the ground for describing the De Rham cohomology algebra
    of a Riemannian manifold in terms of its harmonic differential
    forms. In the following two decades, Hodge's decomposition principle
    has been extended to the (then) new sheaf-theoretic and cohomological
    framework of Hermitean differential geometry, complex-analytic
    geometry, and transcendental algebraic geometry. The names of
    G. De Rham, A. Weil, K. Kodaira, and many others stand for the
    tremendous progress achieved during this period, in particular
    with regard to deformation and classification theory in these
    areas. The special algebraic structures (Hodge structures) arising
    from Hodge decompositions and their generalizations have led
    to a rather independent field of research in geometry, precisely
    to the so-called Hodge theory, which represents a powerful and
    indispensible toolkit for contemporary complex geometry, general
    algebraic geometry, and -- nowadays -- also for mathematical
    physics. The vast activity in Hodge theory and its related areas,
    especially during the recent twenty years, is not reflected in
    the current textbook literature, at least not comprehensively
    or in an updated form compiling the various recent aspects and
    applications, so that a panoramic overview of the present state
    of art must be regarded as a highly welcome (and needed) service
    to the mathematical community. \par
A conference on the present
    state of Hodge theory, serving exactly that purpose, took place
    at the University of Grenoble (France) in November 1994. The
    book under review grew out of the series of lectures which the
    authors delivered at this meeting. The aim of the text is to
    develop a number of fundamental concepts and results of classical
    and modern Hodge theory, and in this the book is prepared for
    students and non-expert researchers in the field, who wish to
    get acquainted in depth with the subject, and obtain a profound
    up-to-date knowledge of its present level of development. --
    The material is divided into three main parts, each of which
    is written by different authors and devoted to various central
    and complementary aspects of the theory.\par
Part I, written
    by {\it J.-P. Demailly}, is entitled ``$L^2$-Hodge theory and
    vanishing theorems''. The author discusses in detail two fundamental
    applications of Hilbert $L^2$-space methods to complex analysis
    and algebraic geometry, respectively. This part adopts basically
    the analytic viewpoint and consists, on its side, of two chapters.
    Chapter 1 provides an introduction to standard complex Hodge
    theory, including the basics on Hermitean and K\"ahler geometry,
    differential operators on vector bundles, Hodge decomposition,
    Hodge degeneration, the spectral sequence of Hodge-Fr\"olicher,
    Gauss-Manin connexion, and the deformation behavior of the Hodge
    groups (after Kodaira). Chapter 2 is devoted to $L^2$-estimates
    for the $\overline \partial $-operator and the resulting vanishing
    theorems for cohomology groups of K\"ahler manifolds and projective
    varieties. The main topics here are the classical methods of
    Oka, Bochner, and H\"ormander in pseudo-convex analysis, their
    consequences for cohomology vanishing, as well as the more recent
    but already well-known fundamental contributions by the author
    himself towards the interpretation of the great vanishing theorems
    of A. Nadel and of Kawamata-Viehweg. -- The concluding two sections
    of this chapter deal with the property of very-ampleness of line
    bundles on projective varieties. The first central result discussed
    here is the author's analytic approach to the famous conjecture
    of Fujita, culminating in an improvement of {\it Y.-T. Siu}'s
    very recent theorem on an effective bound for very-ampleness
    [cf. ``Effective very ampleness'', Invent. Math. 124, No. 1-3,
    563-571 (1996)]. The second central result is an effective version
    of the classical ``Big embedding theorem of Matsusaka'', whose
    surprisingly simple proof is due to the author himself (1996),
    based on some foregoing work of {\it Y.-T. Siu} [Ann. Inst. Fourier
    43, No. 5, 1387-1405 (1993; Zbl 0803.32017)], and methodically
    related to the effective bound for very-ampleness discussed before.
    These two last sections provide a particularly up-to-data account
    on the newest developments in analytical Hodge theory and its
    (algebraic) applications.\par
Part II of the text, written by
    {\it L. Illusie}, is entitled ``Frobenius and Hodge degeneration''.
    These notes aim at introducing non-specialists to those methods
    and techniques of algebraic geometry over a field of characteristic
    $p > 0$, which have been used by P. Deligne and the author to
    give an algebraic proof of the Hodge degeneration and the Kodaira-Akizuki-Nakano
    vanishing theorem for smooth projective varieties in characteristic
    zero. Basically, this part of the book is a careful, detailed
    introduction to the important work ``Rel\`evements modulo $p^2$
    et d\'ecomposition du complexe de De Rham'' [Invent. Math. 89,
    247-270 (1987; Zbl 0632.14017)] by {\it P. Deligne} and {\it
    L. Illusie}. Here the reader is assumed to bring along some basic
    knowledge of the theory of algebraic schemes and of homological
    algebra (in categories). After recalling the basics on schemes,
    differentials and the algebraic De Rham complex in characteristic
    $p>0$, the author discusses the following topics: smoothness
    and coverings, the Frobenius morphism and the Cartier isomorphism,
    derived categories and spectral sequences, decomposition theorems,
    vanishing theorems in characteristic $p$, degeneration theorems,
    the standard techniques for passing from characteristic $p$ to
    characteristic zero, and the proof of the above mentioned degeneration
    and vanishing theorems. The concluding section of this part points
    to some recent developments and open problems concerning Hodge
    theory in characteristic $p$.\par
Also this part is essentially
    self-contained, and most proofs are given in detail. Some proofs
    are -- quite naturally -- at least outlined, assuming the reader
    to follow the precise hints to the related textbook literature
    (mostly EGA) and original papers.\par
Part III of the book,
    written by {\it J. Bertin} and {\it C. Peters}, is entitled ``Variations
    of Hodge structures, Calabi-Yau manifolds, and mirror symmetry''.
    It consists again of two main chapters, whose interrelation is
    beautifully explained in a comprehensive introduction. -- Chapter
    I is devoted to the comparatively elementary part of the theory
    of variation of Hodge structures and its applications in complex
    algebraic geometry. This includes detailed descriptions of the
    Hodge bundles, the Hodge filtrations, the De Rham cohomology
    sheaves, the Gauss-Manin connexion in its general setting (after
    Katz and Oda) and with its transversality property (due to Griffiths),
    variations and infinitesimal variations of Hodge structures,
    the Griffiths period domains for polarized Hodge structures,
    mixed Hodge structures, limits of Hodge structures (after Deligne),
    the Picard-Lefschetz theory and the local monodromy theorem,
    Deligne's degeneration criteria for Hodge spectral sequences,
    and a brief discussion of the method of vanishing cycles. At
    the end, the authors give a sketch of the use of Higgs bundles
    for the construction of variations of Hodge structures, mainly
    by following Simpson's approach [cf.: {\it C. T. Simpson}, Proc.
    Int. Congr. Math., Kyoto 1990, Vol. I, 747-756 (1991; Zbl 0765.14005)],
    as well as some comments on M. Saito's work on Hodge modules,
    intersection cohomology, and ${\cal D}$-modules in algebraic
    analysis. -- Chapter II reflects the fact that Calabi-Yau manifolds,
    their Hodge theory, and their mirror symmetry have recently gained
    enormous significance in both algebraic geometry and theoretical
    physics, particularly in constructing two-dimensional conformal
    quantum field theories. The material presented here covers the
    fundamental facts on Calabi-Yau manifolds, their construction
    and deformation theory, and their mirror properties. After a
    digression on the cohomology of hypersurfaces (after Griffiths
    and Dimca), which is used for the description of the link between
    the Picard-Fuchs equation and the variation of Calabi-Yau structures,
    the variation of Hodge structures for families of Calabi-Yau
    threefolds, their Yukawa couplings, and their mirror symmetries
    are explained in more depth. The interested reader can find a
    very complete and comprehensive account on this subject in the
    recent monography ``Sym\'etrie miroir'' by {\it C. Voisin} [Panoramas
    et Synth\`eses, No. 2 (1996; see the preceding review)]. In a
    concluding section, the authors discuss (following an idea of
    P. Deligne) a possible approach to mirror symmetry via a certain
    duality between variations of Hodge structures for Calabi-Yau
    threefolds. A rich bibliography enhances this very systematic
    and lucid treatise.\par
Altogether, the present book, in all
    its three parts, which consistently refer to each other, may
    be regarded as a masterly introduction to Hodge theory in its
    classical and very recent, analytic and algebraic aspects. Aimed
    to students and non-specialists, it is by far much more than
    only an introduction to the subject. The material leads the reader
    to the forefront of research in many areas related to Hodge theory,
    and that in a detailed and highly self-contained manner. As such,
    this text is also a valuable source for active researchers and
    teachers in the field, in particular due to the utmost carefully
    arranged index at the end of the book.}
\RV{W.Kleinert (Berlin)}
\CC{14C30
14F17
14-02
14D07
13A35
58A14
14D05
81T30
14J32}
%% END ITEM
%% BEGIN ITEM
\AN{0891.32008 (01115812)}{1995}
\AU{Coltoiu, M.; Diederich, K.}
\TI{Convexity properties of analytic complements in Stein spaces.}
\LA{English}
\DT{J}
\SO{J. Fourier Anal. Appl. Spec. Iss., 153-160 (1995). }
\ISSN{069-586}
\AB{From the Introduction: ``It is well known that geometric convexity
    properties of complex manifolds or, more generally, complex spaces,
    imply strong analytic consequences. For the case of geometric
    1-completeness this is the heart of the solution of the Levi
    problem together with Theorem B of Cartan and Serre; more generally,
    for $q$-completeness this follows from the fundamental theory
    of Andreotti and Grauert. Therefore, it is important to have
    criteria that give geometric convexity of suitable classes of
    complex spaces. The most general fact in this respect is probably
    the results from T. Ohsawa (1984) and J. P. Demailly (1990) namely,
    any complex space $X$ of dimension $n$ without any compact irreducible
    $n$-dimensional component is $n$-complete. For further results
    more specific hypotheses have to be made.\par
An important situation
    that appears very often in complex analysis is the following:
    A complex space $X$ is given together with a certain complex
    analytic subvariety $A\subset X$, and one wants to study properties
    of the complement $X':=X\setminus A$. In the light of the aforementioned
    facts, it can then be important to know how the convexity properties
    of $X$ and the nature of $A$ influence the convexity of the complement
    $X'$. Many more specific questions of this sort are still open.
    In this article, we want to consider the case in which $X$ is
    known to be Stein, or, in other words, 1-complete, and $A$ has
    strictly positive dimension at any of its points. In terms of
    $q$-convexity (resp. $q$-completeness) with corners, this question
    has been studied by M. Peternell (1986).\par
The main purpose
    of this article now is to show that in the situation of the corollary
    $X'$ is at least always $(n-1)$-complete if $q<n$. In fact, more
    generally, we show the following theorem.\par
Theorem. Let $X$
    be a Stein space of pure dimension $n\ge 2$ and $A\subset X$
    a (closed) complex subvariety such that for any point $x\in A$
    and any local irreducible component $X_{i,x}$ of $X$ at $x$,
    the point $x$ is not isolated in $A\cap X_{i,x}$. Then $X\subset
    A$ is a union of $(n-1)$ open Stein subspaces. In particular,
    $X\setminus A$ is $(n-1)$-complete''.}
\CC{32F10
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0880.14003}{1995}
\AU{Demailly, Jean-Pierre}
\TI{Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties (joint work with Thomas Peternell and Michael Schneider).}
\LA{English}
\DT{CA}
\SO{Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6--14, 1992. Oxford: Oxford University Press. Stud. Math., Tata Inst. Fundam. Res. 13, 67-86 (1995). [ISBN 0-19-563740-2/hbk]}
\AB{A compact Riemann surface always has a hermitian metric with
    constant curvature, in particular the curvature sign can be taken
    to be constant: the negative sign corresponds to curves of general
    type (genus $\ge 2)$, while the case of zero curvature corresponds
    to elliptic curves (genus 1), positive curvature being obtained
    only for $\bbfP^1$ (genus 0). In higher dimensions the situation
    is much more subtle and it has been a long standing conjecture
    due to Frankel to characterize $\bbfP_n$ as the only compact
    K\"ahler manifold with positive holomorphic bisectional curvature.
    Hartshorne strengthened Frankel's conjecture and asserted that
    $\bbfP_n$ is the only compact complex manifold with ample tangent
    bundle. In his famous paper in Ann. Math., II. Ser. 110, 593-606
    (1979; Zbl 0423.14006), {\it S. Mori} solved Hartshorne's conjecture
    by using characteristic $p$ methods. Around the same time {\it
    Y.-T. Siu} and {\it S.-T. Yau} [Invent. Math. 59, 189-204 (1980;
    Zbl 0442.53056)] gave an analytic proof of the Frankel conjecture.
    Combining algebraic and analytic tools Mok classified all compact
    K\"ahler manifolds with semi-positive holomorphic bisectional
    curvature. -- From the point of view of algebraic geometry, it
    is natural to consider the class of projective manifolds $X$
    whose tangent bundle is numerically effective (nef). This has
    been done by Campana and Peternell and -- in case of dimension
    3 -- by Zheng. In particular, a complete classification is obtained
    for dimension at most three. The main purpose of this work is
    to investigate compact (most often K\"ahler) manifolds with nef
    tangent or anticanonical bundles in arbitrary dimension. We first
    discuss some basic properties of nef vector bundles which will
    be needed in the sequel in the general context of compact complex
    manifolds. We refer to papers by {\it J.-P. Demailly}, {\it T.
    Peternell} and {\it M. Schneider} [Compos. Math. 89, No. 2, 217-240
    (1993) and J. Algebr. Geom. 3, No. 2, 295-345 (1994; Zbl 0827.14027)]
    for detailed proofs. Instead, we put here the emphasis on some
    unsolved questions.}
\CC{14C20
32J27
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{0878.53039}{1995}
\AU{Gauduchon, Paul}
\TI{Topics in Riemannian twistor geometry.}
\LA{English}
\DT{J}
\SO{An. \c{S}tiin\c{t}. Univ. ``Ovidius" Constan\c{t}a, Ser. Mat. 3, No.2, 15-35 (1995). }
\ISSN{223-723}
\AB{In the first part of this paper, the author reviews the definition
    of the twistor space $Z$ of an oriented Riemannian manifold $M$
    and the Atiyah-Hitchin-Singer almost-complex structure $\cal
    J$ on $Z$ as well as the integrability condition for $\cal J$.
    The latter condition is obtained as a consequence of an expression
    for the Nijenhuis tensor of $\cal J$ in terms of the Weyl tensor
    of $M$. The interpretation of an almost complex structure $I$
    on $M$ compatible with the metric and the orientation of $M$
    as a map (section) from $(M,I)$ to $(Z,{\cal J})$ is also treated.\par
In
    the second part of the paper, the author discusses {\it F. Campana's}
    result [J. Differ. Geom. 33, 541-549 (1991; Zbl 0694.32017)]
    that if the twistor space $Z$ of a compact self-dual Riemannian
    four-manifold $M$ is Moishezon, then $M$ is homeomorphic to the
    connected sum $k\bbfC\bbfP^2$ or $k\ge 0$ copies of $\bbfC\bbfP^2$.
    In fact, {\it Y. S. Poon} [J. Differ. Geom. 36, 451-491 (1992;
    Zbl 0742.53024)] showed that ``homeomorphic'' can be replaced
    by ``diffeomorphic''. Using this and applying {\it J. P. Demailly}'s
    criteria [Ann. Inst. Fourier 35, 189-229 (1985; Zbl 0565.58017)]
    for a compact complex manifold to be Moishezon, the author proves
    the following result: Theorem. Let $M$ be a compact connected
    oriented four-manifold admitting a self-dual Riemannian metric
    $g$ of positive scalar curvature. Let $\chi$ and $\tau$ denote
    the Euler characteristic and the signature of $M$, respectively,
    and assume that $2\chi-3\tau>0$. Then the twistor space of $(M,g)$
    is Moishezon and $M$ is diffeomorphic to the connected sum $k\bbfC\bbfP^2$,
    $0\le k\le 3$.}
\RV{J.Davidov (Sofia)}
\CC{53C25
32L25}
%% END ITEM
%% BEGIN ITEM
\AN{0871.14015}{1995}
\AU{Koll\'ar, J\'anos}
\TI{Shafarevich maps and automorphic forms.}
\LA{English}
\DT{B}
\SO{Princeton, NJ: Princeton University Press. ix, 199 p. \$ 37.50; \sterling 29.50 (1995). [ISBN 0-691-04381-7/hbk]}
\AB{The aim of this book is to study the geometric properties of
    a smooth complex projective variety $X$ (usually expressed in
    terms of cohomological invariants obtained starting with the
    canonical class $K_X)$ under the assumption that $X$ has a large
    fundamental group.\par
The author makes a systematic use of
    the Shafarevich maps, a non-abelian generalization of the classical
    Albanese morphisms. Let $X$ be a normal proper variety. Then
    a normal proper variety $\text {Sh} (X)$ and a morphism $\text
    {sh}_X: X\to \text {Sh} (X)$ are called the Shafarevich variety
    of $X$ and, respectively, the Shafarevich morphism of $X$ if
    $\text {sh}_X$ has connected fibers and for any closed subvariety
    $Z\subset X$ either $\text {sh}_X (Z)$ is a point, or $\text
    {image} (\pi_1(Z) \to\pi_1 (X))$ is infinite. -- The existence
    of Shafarevich morphisms in general depends on an open conjecture
    by Shafarevich going back to 1972. What the author shows is that
    the rational version of a Shafarevich morphism (called a Shafarevich
    map) always exists.\par
The following notion describes roughly
    the varieties $X$ for which the Shafarevich map is birational.
    One says that a proper variety $X$ has a generically large fundamental
    group if $\text {image} (\widehat \pi_1 (Z)\to \widehat \pi_1(X))$
    is infinite for ``most'' subvarieties $Z\subset X$, where $\widehat
    \pi_1$ denotes the algebraic fundamental group. -- Here are an
    example of a main result and an example of an open question from
    this book:\par
Theorem. Let $X$ be a smooth projective variety
    of general type, $X$ having a generically large algebraic fundamental
    group. Then $\dim H^0 (X,K_X) \ge 1$.\par
Conjecture. Let $X$
    be as in the theorem above. Then $\chi (X,K_X) \ge 0$.\par
The
    methods used in the book come from very diverse sources, e.g.
    Atiyah's $L^2$-index theorem, Gromov theory of Poincar\'e series
    and recents generalizations of the Kodaira vanishing theorem
    by Demailly, Esnault-Viehweg, Kawamata, Koll\'ar et Nadel.\par
The
    exposition is clear and very attractive, with a continuous care
    to explain what is hiding behind the various formalisms that
    one is forced to introduce along the way.}
\RV{A.Dimca (Bordeaux)}
\CC{14F35
14Exx
11F03
14-02}
%% END ITEM
%% BEGIN ITEM
\AN{0862.14022}{1995}
\AU{Bonavero, Laurent}
\TI{Sur des vari\'et\'es de Moishezon dont le groupe de Picard est de rang un. (On Moishezon manifolds with Picard group of rank one).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 321, No.4, 443-446 (1995). 
\par http://www.sciencedirect.com/science/journal/07644442}
\ISSN{764-444}
\AB{In this note, we use Mori theory to analyse the structure of
    Moishezon manifolds with Picard group equal to $\bbfZ$, with
    big canonical bundle, and which become projective after one blow-up.
    In this context, we study the Mori contraction on the projective
    model, and we show that in general the center of the blow-up
    has ``low'' codimension. In dimension 3, the canonical bundle
    is nef by a result of Koll\'ar. We show that this result is no
    longer true in dimension 4 or larger than 4 by constructing explicitly
    some examples, which give also new Moishezon manifolds not satisfying
    the Demailly-Siu criterion. In dimension 4, we show that the
    center of the blow-up is a surface, and that our construction
    is the only one possible when the canonical bundle is not nef;
    in particular, the center of the blow-up must be $\bbfP^2$ in
    this last case.}
\CC{14C22
32J18
14C22}
%% END ITEM
%% BEGIN ITEM
\AN{0859.32017}{1995}
\AU{Abe, Makoto}
\TI{On the line bundle convexity of a totally pseudoconvex open set.}
\LA{English}
\DT{J}
\SO{Complex Variables, Theory Appl. 26, No.4, 283-290 (1995). }
\ISSN{278-107}
\AB{The author proves the following theorem. Let $X$ be a complex
    manifold and $L$ a positive holomorphic line bundle on $X$. Let
    $D$ be a relatively compact open set in $X$, which is both totally
    pseudoconvex and weakly pseudoconvex. Then $D$ is convex with
    respect to $L^m$ for sufficiently large $m$. The proof uses a
    vanishing theorem of {\it J.-P. Demailly} [Ann. Sci. \'Ec. Norm.
    Super., IV. Ser. 15, 457-511 (1982; Zbl 0507.32021)].}
\RV{P.E.Newstead (Liverpool)}
\CC{32L15
32T99
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{0853.32035}{1995}
\AU{Siu, Yum-Tong}
\TI{Very ampleness criterion of double adjoints of ample line bundles.}
\LA{English}
\DT{CA}
\SO{Bloom, Thomas (ed.) et al., Modern methods in complex analysis. The Princeton conference in honor of Robert C. Gunning and Joseph J. Kohn, Princeton University, Princeton, NJ, USA, Mar. 16-20, 1992. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 137, 291-318 (1995). [ISBN 0-691-04428-7/pbk]}
\AB{Let $X$ be a compact complex manifold of complex dimension $n$,
    and let $L$ be an ample line bundle on $X$. Fujita has conjectured
    in 1987 that $(n+1)L+K_X$ is free and that $(n+2)L+K_X$ is very
    ample. Evidence for this conjecture comes from its validity in
    dimensions 1 and 2 and the proof by Ein-Lazarsfeld of the freeness
    part in dimension 3. A great deal of effort was put in proving
    the freeness and very ampleness of bundles of the form $aL +
    bK_X$, with $a$ and $b$ much larger than in the conjecture [see
    {\it J.-P. Demailly}, J. Differ. Geom. 37, No. 2, 323-374 (1993;
    Zbl 0783.32013) and {\it L. Ein, R. Lazarsfeld} and {\it M. Nakayame},
    preprint (1994)].\par
The paper under review contains a numerical
    criterion for $L + 2K_X$ to be very ample. The proofs use Shokurov's
    technique for a non-vanishing theorem [{\it V. V. Shokurov},
    Math. USSR, Izv. 26, 591-604 (1986); translation from Izv. Akad.
    Nauk SSSR, Ser. Mat. 49, No. 3, 635-651 (1985; Zbl 0605.14006)]
    and Nadel's multiplier ideal sheaves [cf. {\it N. Nadel}, Proc.
    Natl. Acad. Sci. USA 86, No. 19, 7299-7300 (1989; Zbl 0711.53056)
    and {\it J. Koller}, Ann. Math. 296, No. 4, 595-605 (1993; Zbl
    0818.14002)].\par
The author has improved the results of this
    paper in several more recent articles, to mention only the latest
    in Invent. Math. 124, No. 1-3, 563-571 (1996; see the paper above).}
\RV{M.Putinar (Riverside)}
\CC{32L20
14F17
32C30
32J15}
%% END ITEM
%% BEGIN ITEM
\AN{0847.32035}{1995}
\AU{Angehrn, Urban; Siu, Yum-Tong}
\TI{Effective freeness and point separation for adjoint bundles.}
\LA{English}
\DT{J}
\SO{Invent. Math. 122, No.2, 291-308 (1995). 
\par http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356556735}
\ISSN{020-991}
\AB{Let $X$ be a compact complex manifold of dimension $n$ and let
    $K_X$ be its canonical bundle. A conjecture of Fujita raised
    about ten years ago asks whether for an ample line bundle $L$
    on $X$, the bundle $(n + 1) L + K_X$ is free and $(n + 2) L +
    K_X$ is very ample. A great deal of effort was put recently to
    prove these conjectures. So far the answer is known for combinations
    of the form $pL + qK_X$ with $p$ roughly of the order of $n^n$
    and similarly for $q$ [see {\it J.-P. Demailly}, J. Differ. Geom.
    37, No. 2, 323-374 (1993; Zbl 0783.32013)], {\it L. Ein}, {\it
    R. Lazarsfeld} and {\it M. Nakamaye}, Zero-estimates, intersection
    theory, and a theorem of Demailly, preprint (1995), and {\it
    J. Kollar}, Math. Ann. 296, No. 4, 595-605 (1993; Zbl 0818.14002)].\par
Two
    main applications of the results contained in the paper by authors'
    state that $mL + K_X$ is free for $m \ge {1 \over 2} (n^2 + n
    + 2)$ and more generally, the global sections of $mL + K_X$ separate
    any $r$ distinct points if $m \ge {1 \over 2} (n^2 + 2rn - n
    + 2)$. The innovation in the proofs consists in combining the
    algebro-geometric (traditional) methods with some new transcendental
    methods. Quite specifically, starting from fractionar powers
    of $L$, the authors construct a singular metric whose (Nadel)
    multiplier ideal sheaf has a finite and controllable set of zeroes.
    Then a deformation argument, plus the recent $L^2$-estimates
    of Ohsawa and Takegoshi and an induction argument lead to the
    bounds of the necessary fractions of $L$.\par
[See for completeness
    and comparison: the second author, Effective very ampleness,
    Invent. Math. 124, No. 1-3, 563-571 (1996)].}
\RV{M.Putinar (Riverside)}
\CC{32L20
32L10
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0840.14008}{1995}
\AU{Fulton, William}
\TI{Positive polynomials for filtered ample vector bundles.}
\LA{English}
\DT{J}
\SO{Am. J. Math. 117, No.3, 627-633 (1995). }
\ISSN{002-932}
\AB{In this small paper, the author studies the positivity of polynomials
    in the Chern classes of ample or nef vector bundles $E$ with
    a filtration. Let $n= \text {rank } E$, $0= E_k \subset E_{k-1}
    \subset \cdots \subset E_0 =E$ a filtration such that $\text
    {rank} E/E_j= r_j$, $0= r_0< r_1< \cdots< r_k=n$. Fix $n$ and
    $\underline {r}= (r_0, \dots, r_k)$. Choose Chern roots $y_1,
    \dots,y_n$ of $E$ so that $y_{r_j+1}, \dots, y_{r_{j+1}}$ correspond
    to the Chern roots of $E_j/ E_{j+1}$. Let $P(r,d)$ denote the
    vector space of homogeneous polynomials of degree $d$ in $n$
    variables $x_1, \dots, x_n$ with rational coefficients and symmetric
    in the variables $x_i$ and $x_{i+1}$ for all $i$ such that $r_j<
    i< r_{j+1}$ for some $j= 0, \dots, k-1$. For $P\in P(r, d)$ let
    $P(E)= P(y_1, \dots, y_n)$. Then $P(E)\in H^{2d} (X; \bbfQ)$
    if $X$ is a complex variety, $P(E)\in A^d (X; \bbfQ)$ the Chow
    cohomology group for an algebraic scheme $S$ in general. Hence
    if $X$ is complete of dimension $d$, then $P(E)\cap [X]$ has
    a degree. A polynomial $P\in P(r, d)$ is called numerically positive
    for $\underline {r}$-filtered ample (resp. nef) vector bundles
    if degree$(P (E)\cap [X])\geq 0$ for all ample (resp. nef) vector
    bundles with filtration of the above type. The Schubert polynomials
    $S_\omega$, as $\omega$ varies over all permutations of length
    $d$ with $\omega (i)< \omega (i+1)$ if $r_j< i< r_{j+1}$ for
    some $j$, form a basis of $P(r, d)$ over $\bbfR$. The main results
    are: \par
Theorem 1: A nonzero polynomial $P= \sum a_\omega
    S_\omega\in P(r,d)$ is numerically positive for $\underline {r}$-filtered
    ample vector bundles if and only if every coefficient $a_\omega\geq
    0$. \par
Theorem 2: Let $X$ be a $d$-dimensional complex compact
    K\"ahler manifold. Then any $P\in P(r, d)$ with $P= \sum a_\omega
    S_\omega$, $a_\omega\geq 0$ is numerically positive for $\underline
    {r}$-filtered nef vector bundles. \par
In case $k=1$, the Schubert
    polynomials are Schur polynomials. In this case, theorem 1 was
    proved by the author and {\it R. Lazarsfield} [Ann. Math., II.
    Ser. 118, 35-60 (1983; Zbl 0537.14009)] while theorem 2 was proved
    by {\it J.-P. Demailly}, {\it T. Peternell} and {\it M. Schneider}
    [J. Algebr. Geom. 3, No. 2, 295-345 (1994; Zbl 0827.14027)].}
\RV{U.N.Bhosle (Bombay)}
\CC{14F05
14M15}
%% END ITEM
%% BEGIN ITEM
\AN{0828.14002}{1995}
\AU{Trapani, Stefano}
\TI{Numerical criteria for the positivity of the difference of ample divisors.}
\LA{English}
\DT{J}
\SO{Math. Z. 219, No.3, 387-401 (1995). 
\par http://link.springer.de/link/service/journals/00209/}
\ISSN{025-587}
\AB{Let $X$ be a compact complex projective manifold of complex dimension
    $n$; let $L$ be a nef divisor, and $D$ an effective divisor on
    $X$. In the paper it is proved that if $L^n > n (L + b_2 (T)
    \{u\})^{n - 1} D$ then the Kodaira dimension of $L - D$ is maximal.
    (The paper contains is in fact the proof of a slightly more general
    result.) Here $T$ denotes the non-negative curvature current
    of a singular metric of $D$, $b_2(T)$ is a non-negative real
    number which is the largest of the Lelong numbers of $T$ which
    are non-zero on a codimension one analytic subvariety of $X$,
    whereas the $1 - 1$ cohomology class $\{u\}$ measures the ``defect
    of positivity'' of the tangent bundle of $X$. [For precise definition
    see {\it J. P. Demailly}, J. Algebr. Geom. 1, No. 3, 361-409
    (1992; Zbl 0777.32016) and Complex algebraic varieties, Proc.
    Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-103 (1992; Zbl
    0784.32024).]\par
It is also shown by an example that the class
    $\{u\}$ cannot be dropped in general. However, if the complete
    linear system of some power of $D$ has no fixed codimension-1
    components, then the theorem gives conditions, only in terms
    of the Chern classes of $L$, $D$, to insure that $L - D$ has
    maximal Kodaira dimension. By using the theorem one can also
    obtain inequalities between intersection numbers of nef line
    bundles, which can be applied to the study of convex compact
    sets in $\bbfR^n$, as by {\it B. Teissier} [in Semin. Differ.
    Geom., Ann. Math. Stud. 102, 85-105 (1982; Zbl 0494.52009)].
    If the tangent bundle of $X$ is nef (i.e., if we can choose $\{u\}
    = 0)$, then by the second paper by {\it J. P. Demailly} cited
    above every line bundle with maximal Kodaira dimension is ample;
    hence, in this case, the above inequality is a sufficient condition
    for $L - D$ to be ample.}
\RV{S.Trapani (Roma)}
\CC{14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0827.53051}{1995}
\AU{Yeung, Sai Kee}
\TI{Properties of complete non-compact K\"ahler surfaces of negative Ricci curvature.}
\LA{English}
\DT{J}
\SO{J. Reine Angew. Math. 464, 129-141 (1995). 
\par http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN243919689}
\ISSN{075-410}
\AB{This article presents some results which characterize non-compact
    complete K\"ahler surfaces satisfying condition $\cal C$, which
    means that the metric has finite volume and bounded sectional
    curvature, and the Ricci curvature satisfies $\text {Ric} (\omega)
    < -c \omega$ for some constant $c > 0$. The author first gives
    a characterization of such a manifold with K\"ahler-Einstein
    metric. Then he considers analytic Chern numbers of such manifolds.
    He shows that they have to be integers and gives them interpretations.
    He also proves analogues of Matsusaka's Big Theorem and a theorem
    of Demailly on very ampleness in the non-compact surface case.
    Finally, he gives remarks on $L^2$-cohomology corresponding to
    Z\"ucker's conjecture.}
\RV{S.K.Yeung}
\CC{53C55
53C21}
%% END ITEM
%% BEGIN ITEM
\AN{0886.32017 (01058118)}{1994}
\AU{Noguchi, Junjiro}
\TI{Some topics in Nevanlinna theory, hyperbolic manifolds and Diophantine geometry.}
\LA{English}
\DT{CA}
\SO{Mabuchi, T. (ed.) et al., Geometry and analysis on complex manifolds. Festschrift for Professor S. Kobayashi's 60th birthday. Singapore: World Scientific. 140-156 (1994). [ISBN 981-02-2067-7/hbk]}
\AB{The author proposes a large series of open problems and conjectures
    of several complex variables at intersection with algebraic or
    hyperbolic geometry and number theory, with historical surveys,
    comments and rich informations on their present stage. He dedicated
    more than 20 years of researches to the subject and brought important
    contributions to their solution.\par
The paper has three parts,
    each with several sections, as follows:\par
$\underline {1.\text
    { Nevanlinna theory.}}$ 1.1. Transcendental B\'ezout problem:
    What kind of measures can provide a generalization of Stoll's
    average B\'ezout theorem? 1.2. Nevanlinna's inverse problem:
    Does it hold for $f:\bbfC^m \to P^n (\bbfC)$ with respect to
    hyperplanes in general position or for $f: \bbfC^m \to V$, $f$
    a dominant meromorphic map into a projective manifold $V$ with
    respect to a union of hypersurfaces with simple normal crossings?
    1.3. Order of convergence of Nevanlinna's defects: Specify Krutin's
    result for $f$ -- roughly speaking -- as in 1.2. 1.4. Order of
    a meromorphic map into a projective manifold: Problems when the
    order is less than 2. 1.5. Holomorphic curves: Griffiths' conjectures
    about the defect relation and respectively, about a holomorphic
    curve into an Abelian variety $A$ missing an ample divisor of
    $A$; proposals to compute the constant in Noguchi's inequality
    of the second main theorem type, to apply Nevanlinna calculus
    to a moduli problem.\par
$\underline {2.\text { Hyperbolic manifolds.}}$
    2.1. Finiteness and rigidity: Let $X$ and $Y$ be compact complex
    spaces, $Y$-hyperbolic. S. Lang's conjecture is discussed beginning
    with S. Kobayashi and T. Ochiai result till Noguchi's solution
    in 1992 (there are only a finite number of surjective meromorphic
    maps $X\to Y)$ and a non-compact version by Makoto Suzuki. By
    fixing $X$, Noguchi conjectures that Severi set $\text {Sev}
    (X)= \{(f,Y)$; $Y$ hyperbolic, $f:X\to Y$ holomorphic and surjective\}
    is finite. Other problems refer to $X$ and $Y$ complete hyperbolic
    complex spaces with finite hyperbolic volume. 2.2. Hyperbolic
    fiber spaces and extension problems: To remark again the author's
    results and interesting existence problems. 2.3. Hypersurfaces
    of $P^n(\bbfC)$: One deals with Kobayashi's conjectures (a generic
    hypersurface of large degree and its complement are hyperbolic).
    The author presents an existence theorem, which he proved together
    with K. Masuda. Significant examples, many of them new, are given.
    2.4. Does any non-algebraic compact hyperbolic manifold exist?\par
$\underline{3.
    \text{ Algebraic and arithmetic pseudodistances.}}$ 3.1. Starting
    with Lang's conjecture on rational points, Noguchi proposes an
    algebraic hyperbolic pseudodistance on a complex algebraic variety
    $V$. In 3.2. he defines an arithmetic one on an algebraic variety
    $V$ over a number field. In both cases he uses chains of algebraic
    curves joining two points. He presents recent results by Demailly-Lempert-Shiffman,
    Kitayama, Kollar-Miyaoka-Mori and formulates problems.\par
A
    large bibliography completes the work. Proofs are included where
    reference is not available.}
\RV{Cabiria Andreian Cazacu (Bucure\c{s}ti)}
\CC{32H30
32Q45
32F45
32A22
32-02
11J68}
%% END ITEM
%% BEGIN ITEM
\AN{0811.32011}{1994}
\AU{Peternell, Th.}
\TI{Pseudoconvexity, the Levi problem and vanishing theorems.}
\LA{English}
\DT{CA}
\SO{Grauert, H. (ed.) et al., Several complex variables VII. Sheaf- theoretical methods in complex analysis. Berlin: Springer-Verlag. Encycl. Math. Sci. 74, 221-257 (1994). [ISBN 3-540-56259-1/hbk]}
\AB{The author reviews results concerning the topic mentioned in
    the title. The contents of the paper is as follows:\par
\S1.
    Plurisubharmonic functions and pseudoconvexity (1. The notion
    plurisubharmonic functions. 2. Properties of plurisubharmonic
    functions. 3. Pseudoconvex domains).\par
\S2. Convex spaces
    (1. Remmert reduction. 2. The Levi problem for 1- convex spaces.
    3. Maximal compact analytic sets. 4. Positive sheaves and the
    normal bundle. 5. The cohomology of 1-convex spaces).\par
\S3.
    The Levi problem (1. The classical Levi problem. 2. Counterexamples.
    3. Characterizing Stein spaces. 4. The local Stein problem).\par
\S4.
    Positive sheaves and vanishing theorems (1. The projective bundle.
    2. The vanishing theorem for positive sheaves. 3. The embedding
    theorem. 4. Characterization of positivity by cohomology vanishing.
    5. Functorial properties of positive sheaves. 6. Differential-geometric
    positivity notions. 7. Hodge metrics. 8. Relative positive sheaves).\par
\S5.
    More vanishing theorems (1. Demailly's vanishing theorem. 2.
    The notion of $k$-ampleness. 3. Grauert-Riemenschneider vanishing
    theorem and direct images of dualizing sheaves.).}
\RV{L.I.Ronkin (Khar'kov)}
\CC{32T99
32L20
32L10
32E05}
%% END ITEM
%% BEGIN ITEM
\AN{0812.14027}{1993}
\AU{Ein, Lawrence; Lazarsfeld, Robert}
\TI{Seshadri constants on smooth surfaces.}
\LA{English}
\DT{CA}
\SO{Journ\'ees de g\'eom\'etrie alg\'ebrique d'Orsay, France, juillet 20-26, 1992. Paris: Soci\'et\'e Math\'ematique de France, Ast\'erisque. 218, 177-186 (1993). 
\par http://smf.emath.fr/Publications/Asterisque/1993/218/html/smf\_ast\_218\_177-186.html\kern-9pt}
\ISSN{303-117}
\AB{For a complex projective variety $X$, a local measure at a point
    $x$ of the positivity of a nef line bundle $L$ is the so-called
    Seshadri constant (introduced by Demailly), which is the real
    number $\varepsilon (L,x) = \inf \{{L \cdot C \over m\sb x(C)}
    \vert C$ is an irreducible curve passing through $x\}$. It is
    easy to see that $\varepsilon (L,x) \ge 1$ for all points in
    $X$ if $L$ is very ample. The main result in this paper consists
    in proving that, if $X$ is a surface, the set of points for which
    $\varepsilon(L,x)<1$ is countable, and even finite with the extra
    hypothesis $L\sp 2>1$. In fact the authors show that pairs $(C,x)$
    for which $m\sb x (C) > C \cdot L$ are rigid under deformation.
    There is a section in the paper devoted to examples and open
    problems. The authors give for instance an example due to Miranda
    that shows that $\varepsilon (L,x)$ can be arbitrarily small
    for isolated points of the surface.}
\RV{E.Arrondo (Madrid)}
\CC{14J60
14C20
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{0811.32022}{1993}
\AU{Hamada, Hidetaka}
\TI{Cohomology vanishing theorems on a domain exhausted by complete K\"ahler domains.}
\LA{English}
\DT{J}
\SO{Math. Rep. Coll. Gen. Educ., Kyushu Univ. 19, 17-25 (1993). }
\ISSN{287-998}
\AB{The author generalizes results of Nakano, Kazama, Demailly and
    gives cohomology vanishing theorems for complex manifolds exhausted
    by K\"ahler domains.}
\RV{J.Kajiwara (Fukuoka)}
\CC{32L20}
%% END ITEM
%% BEGIN ITEM
\AN{0799.32023}{1993}
\AU{Bonavero, Laurent}
\TI{In\'egalit\'es de Morse holomorphes singuli\`eres. (Singular holomorphic Morse inequalities).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 317, No.12, 1163-1166 (1993). }
\ISSN{764-444}
\AB{The author generalises the holomorphic Morse inequalities [{\it
    J.-P. Demailly} in `Several complex variables and complex geometry,
    Proc. Summer. Res. Inst., Santa Cruz/CA (USA) 1989', Proc. Symp.
    Pure Math. 52, Part 2, 93-114 (1991; Zbl 0755.32008)] to the
    case of a line bundle with singular metric, in the sense of {\it
    J.-P. Demailly}, Singular Harmitian metrics on positive line
    bundles, in `Complex algebraic varieties', Proc. Conf., Bayreuth/Ger.
    1990, Lect. Notes Math. 1507, 87-104 (1992; Zbl 0784.32004).
    These inequalities estimate the dimension of cohomology groups
    with values in high powers of the line bundle, twisted by the
    corresponding sequence of Nadel's multiplier ideal sheaves.}
\RV{J.Stevens (Hamburg)}
\CC{32L10}
%% END ITEM
%% BEGIN ITEM
\AN{0796.14010}{1993}
\AU{Takayama, Shigeharu}
\TI{Ample vector bundles on open algebraic varieties.}
\LA{English}
\DT{J}
\SO{Publ. Res. Inst. Math. Sci. 29, No.6, 885-910 (1993). }
\ISSN{034-531}
\AB{Let $X$ be a smooth complex projective variety, let $D$ be an
    effective, reduced divisor on $X$ with only simple normal crossings.
    A vector bundle $E$ on $X$ is called ample modulo $D$ if for
    every torsion free sheaf ${\cal F}$ on $X$, for large $m$, ${\cal
    F} \otimes S\sp m(E) \vert\sb{X \backslash D}$ is generated by
    $H\sp 0 (X,{\cal F} \otimes S\sp m(E))$. A line bundle $L$ is
    said to be very ample modulo $D$ if the rational map $\Phi\sb{\vert
    L \vert}$ gives an embedding of $X \backslash D$ into a projective
    space. The author generalizes a theorem of {\it J.-D. Demailly}
    on ample line bundles [J. Differ. Geom. 37, No. 2, 323-374 (1993;
    Zbl 0783.32013)]. He proves:\par
Theorem 1. There is a function
    $C(n)$ in $n \in \bbfN$ such that, for $X$ and $D$ as above and
    $L$ a nef line bundle on $X$ being ample modulo $D$, the bundle
    $2(K\sb X + D)+mL$ is very ample modulo $D$ for $m \ge C (\dim
    X)$.\par
A corollary to the proof is theorem 2: Under the notation
    above, let $K\sb X + D$ (resp. $-(K\sb X+D))$ be nef on $X$ and
    ample modulo $D$. Then there is a function $C(n)'$ in $n \in
    \bbfN$ such that $m(K\sb X + D)$ (resp. $-m(K\sb X + D))$ is
    very ample modulo $D$ for $m>C(\dim X)'$.\par
The methods of
    the proofs are similar to the analytic methods of Demailly such
    as H\"ormander's $L\sp 2$ estimates for $\overline \partial$,
    Lelong number theory and Aubin-Yau's solution of the Calabi conjecture.
    The author gives his ideas on how to apply theorem 2 to the construction
    of the moduli space of open algebraic varieties of general type.
    He also collects some elementary properties of vector bundles
    which are ample modulo $D$.}
\RV{K.Dabrowski (Piastow)}
\CC{14F05
14C20
32L10
32J25}
%% END ITEM
%% BEGIN ITEM
\AN{0789.32002}{1993}
\AU{Ji, Shanyu; Ru, Min}
\TI{Global Lojasiewicz inequality, defect relation and applications of holomorphic curve theory.}
\LA{English}
\DT{CA}
\SO{Yang, Chung-Chun (ed.) et al., Several complex variables in China. Providence, RI: American Mathematical Society. Contemp. Math. 142, 49-60 (1993). [ISBN 0-8218-5164-0/pbk]}
\AB{This is a survey article on recently published papers (related
    to the topics mentioned in the title) by Demailly, Koll\'ar,
    Shiffman, Stoll, Wong, and the authors.}
\RV{M.Jarnicki (Krak\'ow)}
\CC{32-02
32A22
32H30}
%% END ITEM
%% BEGIN ITEM
\AN{0784.32009}{1993}
\AU{Ji, Shanyu; Shiffman, Bernard}
\TI{Properties of compact complex manifolds carrying closed positive currents.}
\LA{English}
\DT{J}
\SO{J. Geom. Anal. 3, No.1, 37-61 (1993). }
\ISSN{050-692}
\AB{The authors show that a compact complex manifold is Moishezon
    if and only if it carries a strictly positive integral (1,1)-current.
    Also the problem of intersecting closed positive currents is
    treated. Smoothing and approximation techniques are an essential
    part of the paper. Also several results of J. P. Demailly are
    used who was able to show in the meantime an analytic characterization,
    by singular Hermitian metrics, of big line bundles.}
\RV{S.Kosarew (Saint-Martin-d'H\`eres)}
\CC{32C30
32J99
32J25}
%% END ITEM
%% BEGIN ITEM
\AN{0771.32011}{1993}
\AU{Demailly, Jean-Pierre}
\TI{Holomorphic Morse inequalities on q-convex manifolds.}
\LA{English}
\DT{CA}
\SO{Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 245-257 (1993).}
\AB{[For the entire collection see Zbl 0759.00008.]\par
This paper
    is a nice report (``high level propaganda for a very interesting
    result and for very interesting tools'') of at that time recent
    work [{\it T. Bouche}, Ann. Sci. Ec. Norm. Super., IV. Ser. 22,
    No. 4, 501-513 (1989; Zbl 0693.32016)] which extends previous
    work by Demailly to the case of strongly $q$-convex manifolds.
    The main result of Bouche's paper is the following one.\par
Theorem
    A: Let $X$ be a strongly $q$-convex complex manifold with $n:=\dim(X)$,
    $E$ a rank $r$ vector bundle on $X$ and $L$ a line bundle on
    $X$ with hermitian metric such that the curvature form $ic(L)$
    has at least $n-p+1$ eigenvalues $\ge 0$ outside a compact subset
    of $X$; set $X(m,L):=\{x\in X:ic(L)$ is non degenerate at $x$
    and with exactly $m$ negative eigenvalues\}, $X(\le m,L):=\bigcup\sb{t\le
    m}X(t,L)$ and $X(\ge m,L):=\bigcup\sb{t\ge m}X(t,L)$. Then for
    all $m\ge p+q-1$ the following asymptotic inequalities hold:\par
$(a\sb
    m)$ Weak Morse inequalities $$\dim H\sp m(X,E\otimes L\sp k)\le
    r{k\sp n\over n!}\int\sb{X(m,L)}(-1)\sp m\left({i\over 2\pi}c(L)\right)\sp
    n+o(k\sp n)$$ $(b\sb m)$ Strong Morse inequalities: $$\sum\sb{m\le
    t\le n}(-1)\sp{t-m}\dim H\sp t(X,E\otimes L\sp k)\le r{k\sp n\over
    n!}\int\sb{X(\ge m,L)}(-1)\sp m\left({i\over 2\pi}c(L)\right)\sp
    n+o(k\sp n)$$ This note contains a sketch of the proof of this
    theorem. Here the main recent and very powerful techniques are
    explained and used (e.g. Witten's complex); the main tool for
    the proof of Morse inequalities is a spectral theorem for Schr\"odinger
    operators which describes very precisely the asymptotic distribution
    of eigenvalues for a suitable quadratic form. This report contains
    the statement, the history and the motivation of two important
    applications of Theorem A: a very general a priori estimate for
    Monge-Amp\`ere operator $(id'd'')\sp n$ on $q$-convex manifolds
    and the following stronger form of Grauert-Riemenschneider conjecture:\par
Theorem
    B: Let $X$ be a connected $n$-dimensional compact manifold; if
    $X$ has a hermitian line bundle $L$ such that $\int\sb{X(\le
    1,L)}(ic(L)\sp n>0$, then $X$ is Moishezon.}
\RV{E.Ballico (Povo)}
\CC{32F10
32C35
32J99
32W20}
%% END ITEM
%% BEGIN ITEM
\AN{0785.32006}{1992}
\AU{Blel, Mongi}
\TI{Sur le c\^one tangent \`a un courant positif ferm\'e. (Tangent cone associated to a closed positive current).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 314, No.12, 887-890 (1992). }
\ISSN{764-444}
\AB{The author's summary: ``Let $K\sb p$ be the set of $(p,p)$ closed
    positive and conic currents on $\bbfC\sp n$ such that the Lelong
    number is equal to 1 at the origin $O$, the convex subset $K\sb
    p$ is a weakly compact metrizable subspace of the space of currents
    on $\bbfC\sp n$. For every current $T$ on $\bbfC\sp n$, the set
    of limits of the family $((h\sb r)* T)\sb{r<1}$ is connex and
    weakly compact in $m$. $K\sb p$ where $m$ is the Lelong number
    of $T$ at $O$. Conversely we prove in this note that if $M$ is
    any closed connex subset of $K\sb p$ $(n-1\ge p\ge 2)$; there
    exists a closed positive current $T$ such that $M$ is the limit
    set of the family $((h\sb r)*T)\sb{r<1}$. The case $p=1$ was
    first treated by {\it C. O. Kiselman} [``Tangents of plurisubharmonic
    functions'', preprint], and then refined by the author in collaboration
    with {\it J. P. Demailly} and {\it M. Mouzali} [Ark. Mat. 28,
    No. 2, 231- 248 (1990; Zbl 0724.32005)] to get precise quantitative
    estimates. In the second part we give a sufficient condition
    relative to the trace measure for the existence of the tangent
    cone of a closed positive current of type $(p,p)$ with $p=1$
    or $n-1$. We prove that this condition is not valid for the case
    $2 \le p \le n-2$''.}
\RV{D.Barlet (Vand{\oe}uvre-les-Nancy)}
\CC{32C30
58A25}
%% END ITEM
%% BEGIN ITEM
\AN{0774.14018}{1992}
\AU{Maehara, Kazuhisa}
\TI{(Vanishing theorems in algebraic geometry).}
\LA{JP}
\DT{B}
\SO{Sophia Kokyuroku in Mathematics. 34. Tokyo: Sophia University, Department of Mathematics, 283 p. (1992).}
\AB{This book is a survey over the development of vanishing theorems
    in algebraic geometry. Theorems assuring that a cohomology group
    is zero often play a very important role in algebraic geometry.
    Perhaps the most useful theorem in this style is the well-known
    Kodaira vanishing theorem. Besides Kodaira's one, even in textbooks
    of algebraic geometry, we can find many vanishing theorems such
    as Serre's vanishing theorem, Bott's vanishing theorem and so
    forth. Experts know many other vanishing theorems with the name
    of the contributor. In fact, new development in algebraic geometry
    has often appeared together with a new vanishing theorem.\par
This
    book contains many vanishing theorems. Perhaps they are all that
    the author knows. It contains also theorems on Hodge decomposition,
    because it was discovered that vanishing theorems follow from
    the Hodge decomposition theorem. Enumerating names of contributors
    from this book, we have Kodaira, Akizuki, Nakano, Serre, Bott,
    Andreotti, Grauert, Vesentini, Riemenschneider, Griffiths, Hartshorne,
    Gigante, Girbau, Le Potier, Sommese, Ohsawa, Abdelkader, Siu,
    Demailly, Ramanujam, Mumford, Miyaoka, Norimatsu, Kawamata, Viehweg,
    Bogomolov, Esnault, Tankeev, Koll\'ar, Raynaud, Deligne, Illusie,
    Fontaine, Messing, Kato, Faltings, Saito, Shatz and Ekedahl.
    To each theorem the statement and necessary definitions are given.
    Moreover, the author tries to explain key ideas hidden in the
    background. More than a half of this book is devoted to the recent
    theories related to the theory of positive characteristic and
    the theory of $D$-modules due to Raynaud, Deligne, Illusie, Fontaine,
    Messing, Kazuya, Kato, Faltings, and Morihiko Saito.\par
Perhaps
    students cannot read this book, I think, because many concepts
    appear without any explanations. However, for experts this book
    is useful, because we can understand essential ideas in the recent
    development without reading original papers. It is a matter to
    regret that this book was not written in English. Japanese algebraic
    geometers will be able to use this book as a very good reference.}
\RV{T.Urabe (Tokyo)}
\CC{14F17
14C30
14D07
14-02}
%% END ITEM
%% BEGIN ITEM
\AN{0771.32010}{1992}
\AU{Demailly, Jean-Pierre}
\TI{Courants positifs et th\'eorie de l'intersection. (Positive currents and intersection theory).}
\LA{French}
\DT{J}
\SO{Gaz. Math., Soc. Math. Fr. 53, 131-159 (1992). }
\ISSN{224-899}
\AB{This is a nice tool for spreading mathematical culture and ideas
    among mathematicians. It starts with the notion of current (after
    de Rham) and ends with the use in the subject of top level research
    and new extremely powerful methods of the author (around 1991).
    In the middle it is shown how to use the notion of positive current
    to define and work (via the integration current of the fundamental
    class of a subvariety) in Intersection Theory (key words: Lelong
    numbers and multiplicities). The tools come from Analysis and
    bring (and often solve) with them several interesting problems
    which cannot be formulated in a purely algebraic way inside Algebraic
    Geometry. But these methods are very, very strong competitors
    even on natural very important algebraic problems. Of course,
    in this paper most of the proofs are omitted, but ideas and difficulties
    are not skipped. It is pleasant reading and even specialists
    in not too far fields can find here some ideas/tools useful for
    their job; everybody can find some recent deep idea (mostly from
    Demailly brain).}
\RV{E.Ballico (Povo)}
\CC{32C30
58A25
32J25
14C17}
%% END ITEM
%% BEGIN ITEM
\AN{0768.32020}{1992}
\AU{Bouche, Thierry}
\TI{Noyau de la chaleur et in\'egalit\'es de Morse holomorphes pour un fibr\'e en droites \`a courbure d\'eg\'en\'er\'ee. (Heat kernel and holomorphic Morse inequalities for line bundles of degenerate curvature).}
\LA{French}
\DT{J}
\SO{Bull. Sci. Math., II. S\'er. 116, No.2, 167-183 (1992). 
\par http://www.sciencedirect.com/science/journal/00074497}
\ISSN{007-449}
\AB{The aim of this paper is to generalize Demailly's asymptotic
    Morse inequalities for the $d''$-cohomology of the $k$-fold twist
    of a vector bundle $G$ with some holomorphic Hermitian line bundle
    $E$ on a compact complex manifold $X$ [see {\it J. P. Demailly},
    Ann. Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)
    and {\it J.-M. Bismut}, J. Funct. Anal. 72, 263-278 (1987; Zbl
    0649.58030)].\par
Demailly's bounds involve certain integrals
    of the curvature of $E$ which are vanishing if the curvature
    of $E$ is degenerate at every point of $X$. To be able to obtain
    similar inequalities in the case of degenerate curvature of $E$
    he introduces a second holomorphic Hermitian line bundle $F$
    and investigates the $d''$-cohomology of $E\sp k\otimes F\sp
    l\otimes G$.\par
As in the classical setting [see {\it R. Bott},
    Ann. Math., II. Ser. 60, 248-261 (1954; Zbl 0058.091)] he should
    assume to have sufficiently good structure on the set of critical
    points. His assumption is that the curvature form vanishes along
    a foliation of $X$. Under this hypothesis and if $k$, $l$ and
    $k/l$ tend to infinity he obtains asymptotic Morse inequalities.\par
For
    the proof he computes an equivalent for the kernel of the heat
    operator using some localization principle, which is proved with
    results from {\it P. Li} and {\it S.-T. Yau} [Acta Math. 156,
    154-201 (1986; Zbl 0611.58045)].\par
If the rank of the curvature
    form on $X$ is at most $r$, he obtains as a consequence of his
    inequality an asymptotic estimate for the Betti numbers of the
    $d''$-complex of $E\sp k\otimes G$ by a constant multiple of
    $k\sp r$.}
\RV{B.Kreu{\ss}ler (Kaiserslautern)}
\CC{32L10
35K05
58J35
58E05
57R70}
%% END ITEM
%% BEGIN ITEM
\AN{0751.32010}{1992}
\AU{Lauber, Marianne}
\TI{Nombres de Lelong microlocaux it\'er\'es et nombres de Lelong g\'en\'eralis\'es avec poids. (Iterated microlocal Lelong numbers and weighted generalized Lelong numbers).}
\LA{French. Abridged English Version}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 314, No.7, 517-522 (1992). }
\ISSN{764-444}
\AB{Let $u$ be a plurisubharmonic function on an $n$-dimensional
    holomorphic manifold $X$, $\gamma$ be a smooth curve on $X$,
    and $x\in\gamma$. The author defines a microlocal Lelong number
    of order $p$ of $u$ along $\gamma$ at $x$ by making $p$ iterations
    of pointwise blow ups at the origin points of the strict transforms
    of the curve $\gamma$. This number is related to a generalized
    Lelong number introduced by J.-P. Demailly [{\it J.-P. Demailly},
    Acta Math. 159, 153-169 (1987; Zbl 0629.32011)].}
\RV{A.Yu.Rashkovsky (Khar'kov)}
\CC{32U05
31C10}
%% END ITEM
%% BEGIN ITEM
\AN{0786.58013}{1991}
\AU{Bouche, Thierry}
\TI{Sur les in\'egalit\'es de Morse holomorphes lorsque la courbure du fibr\'e en droites est d\'eg\'en\'er\'ee. (On holomorphic Morse inequalities when the curvature of the line bundle is degenerate.).}
\LA{French}
\DT{J}
\SO{Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No.4, 501-523 (1991).}
\AB{Let $X$ be a compact complex manifold of dimension $n$, $E$ a
    holomorphic line bundle over $X$ with a Hermitian metric, and
    $G$ a holomorphic vector bundle of rank $g$ over $X$. Denote
    by $X(\le q)$ the set of points where the curvature form $c(E)$
    is nondegenerate and the real form $i c(E)$ has at most $q$ negative
    eigenvalues. Then, {\it J.-P. Demailly} proved in Ann. Inst.
    Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017) a holomorphic
    Morse inequality $$\sum\sb{\nu = 0}\sp q(-1)\sp{q-\nu}\dim H\sp
    \nu(X,E\sp k\otimes G)\le g{k\sp n\over n!}\int\sb{X(\le q)}\left({i\over
    2\pi}c(E)\right)\sp n+o(k\sp n).$$ The author in this paper considers
    the case when $c(E)$ is degenerate everywhere. To overcome the
    difficulty in the description of the shape of degeneracy of $c(E)$,
    he assumes certain regularity on $c(E)$: one says that the kernel
    of the form $i c(E)$ is well-foliated if $X$ is foliated so that
    each leaf is of codimension equal to the rank of $i c(E)$ and
    that the tangent space of each leaf at any point is included
    in the kernel of $i c(E)$. With this assumption, he proves that
    there exists a constant $C$ such that $\dim H\sp q(X,E\sp k\otimes
    G)\le Ck\sp r$ holds asymptotically when $k$ tends to $\infty$.\par
In
    order to prove the result, the author introduces a supplementary
    holomorphic Hermitian line bundle $F$ so that the sum of the
    form $i c(E)$ along the normal space of the foliation and the
    form $i c(F)$ along the tangent space of the foliation is nondegenerate
    and proves an inequality similar to the displayed above first
    on $\dim H\sp q(X,E\sp k\otimes F\sp \ell\otimes G)$ and, then,
    on the alternating sum of this estimate; concretely he shows
    $$\sum\sb{\nu = 0}\sp q(-1)\sp{q-\nu}\dim H\sp \nu(X,E\sp k\otimes
    F\sp \ell\otimes G)\le g{k\sp n\over n!}\int\sb{X(\le q)}\left({i\over
    2\pi}c(E)\right)\sp r\wedge\left({i\over 2\pi}c(F)\right)\sp{n-r}+o(k\sp
    r\ell\sp{n-r}).\kern-15pt$$}
\RV{T.Sasaki (Kobe)}
\CC{58E35
32Q20
58E05}
%% END ITEM
%% BEGIN ITEM
\AN{0777.32003}{1991}
\AU{Qiu, Chunhui}
\TI{The Koppelman--Leray--Norguet formula of type $(p,q)$ on Stein manifolds.}
\LA{English}
\DT{J}
\SO{J. Math. Res. Expo. 11, No.4, 611-618 (1991). }
\ISSN{000-341}
\AB{L'Auteur red\'emontre la formule \'etablie par {\it J. P. Demailly}
    et {\it Ch. Laurent-Thi\'ebaut} [Ann. Sci. Ec. Norm. Super.,
    IV. Ser. 20, 579-598 (1987; Zbl 0632.32004)], sous les m\^emes
    hypoth\`eses, \`a savoir: $f$ forme de type $(p,q)$, $f$ et $\overline\partial
    f$ continues sur $\overline D$, $D$ ouvert relativement compact
    dans une vari\'et\'e de Stein, $\partial D$ de classe ${\cal
    C}\sp 1$ par morceaux.}
\RV{M.Herv\'e (Paris)}
\CC{32A25
32E20
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0747.58021}{1991}
\AU{Marinescu, George}
\TI{Asymptotic Morse inequalities.}
\LA{Romanian. English summary}
\DT{J}
\SO{Stud. Cercet. Mat. 43, No.5/6, 243-297 (1991). }
\ISSN{039-406}
\AB{We present some methods and results about asymptotic Morse inequalities
    and some applications. We review the result of Demailly [see
    {\it J.-P. Demailly}, Ann. Inst. Fourier 35, No. 4, 189-229 (1985;
    Zbl 0565.58017)] concerning these topics including a detailed
    spectral analysis of Laplace and Schr\"odinger operators. Also
    a proof of the Grauert-Riemenschneider conjecture is given. Finally
    we extend these inequalities to compact complex manifolds with
    cone-like singularities.}
\CC{58E05
58J50}
%% END ITEM
%% BEGIN ITEM
\AN{0742.32008}{1991}
\AU{Berndtsson, Bo}
\TI{Cauchy-Leray forms and vector bundles.}
\LA{English}
\DT{J}
\SO{Ann. Sci. \'Ec. Norm. Sup\'er., IV. S\'er. 24, No.3, 319-337 (1991). 
\par http://www.numdam.org/numdam-bin/feuilleter?id=ASENS\_1991\_4\_24\_3}
\ISSN{012-959}
\AB{The paper deals with integral formulas for $(p,q)$-forms on complex
    manifolds that generalize the well-known formulas of Cauchy-Leray
    type for domains in $\bbfC\sp n$ and, in the case of Stein manifolds
    and $p=0$, those of Henkin and Leiterer. Following an idea of
    {\it J. P. Demailly} and {\it C. Laurent-Thi\'ebaut} [Ann. Sci.
    Ec. Norm. Super., IV. Ser. 20, 579-598 (1987; Zbl 0632.32004)]
    instead of the exterior differential operator the author uses
    a connection on a suitable bundle. As an illustration of this
    technique the author gives also proofs of classical results in
    differential geometry such as the Hopf index theorem, the Gauss-Bonnet
    theorem and the Bott residue theorem.}
\RV{A.Tancredi (Perugia)}
\CC{32C30
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0703.32008}{1991}
\AU{Foote, Robert L.}
\TI{Homogeneous complex Monge-Amp\`ere equations and algebraic embeddings of parabolic manifolds.}
\LA{English}
\DT{J}
\SO{Indiana Univ. Math. J. (to appear). }
\ISSN{022-251}
\AB{Let $M\sp n$ be a Stein manifold, and let $\tau: M\to [0,R\sp
    2)$, $0<R\le \infty$, be a strictly plurisubharmonic exhaustion.
    The main (M,$\tau$) is said to be strictly parabolic at infinity
    if $u=\log \tau$ satisfies the Monge-Amp\`ere equation ($\partial
    {\bar \partial}u)\sp n\equiv 0$ outside some compact set K. Additionally,
    (M,$\tau$) is of Reinhardt type if $\tau$ can be written locally
    as a Reinhardt function on M-K. The set M-K admits a Monge-Amp\`ere
    foliation in a well-known way. With an assumption on the holomorphic
    twist of this foliation when $R=\infty$ we prove a curvature
    estimate, which, via a theorem due to Demailly, implies that
    M embeds algebraically into ${\bbfC}\sp{2n+1}$. When $R\le \infty$
    and $K=\tau\sp{-1}(0)$ we prove, under a weaker regularity assumption
    than usual, that M is biholomorphic to the ball in ${\bbfC}\sp
    n$ of radius R.}
\RV{R.L.Foote}
\CC{32W20
32E10
32J99
14J99}
%% END ITEM
%% BEGIN ITEM
\AN{0728.14018}{1990}
\AU{Manivel, Laurent}
\TI{Sur la cohomologie des fibr\'es associ\'es au fibr\'e quotient universel sur la grassmannienne. (On the cohomology of fiber bundles associated to the universal quotient fiber bundle on the Grassmannian).}
\LA{French}
\DT{J}
\SO{Bull. Soc. Math. Fr. 118, No.1, 67-84 (1990). 
\par http://www.numdam.org:80/numdam-bin/feuilleter?id=BSMF\_1990$\_\_$118\_1}
\ISSN{037-948}
\AB{The paper is concerned with the calculation of some standard
    cohomology groups on Grassmannians of a complex vector space
    and especially with vanishing theorems for them. The kind of
    results is in the spirit of Demailly's work in this field. The
    author uses a classical theorem of Bott and a filtration of the
    inverse image of the tangent space of the grassmannian to the
    flag variety.}
\RV{S.Kosarew (Saint-Martin-d'Heres)}
\CC{14F17
32L20
14M15}
%% END ITEM
%% BEGIN ITEM
\AN{0688.32022}{1990}
\AU{Azhari, Abdelhak}
\TI{Sur la conjecture de Chudnovsky-Demailly et les singularites des hypersurfaces algebriques. (On the Chudnovsky-Demailly conjecture and singularities of algebraic hypersurfaces).}
\LA{French}
\DT{J}
\SO{Ann. Inst. Fourier 40, No.1, 103-116 (1990). 
\par http://www-fourier.ujf-grenoble.fr/AIF/aif\_1990.html}
\ISSN{373-095}
\AB{Soit S une partie finie de ${\bbfP}\sp n$, t un entier positif
    et $\omega\sb t(S)$ le plus petit degr\'e des hypersurfaces de
    ${\bbfP}\sp n$ ayant en chaque point de S une singularit\'e de
    multiplicit\'e $\ge t$. Un th\'eor\`eme d'existence de J.-P.
    Demailly concernant le prolongement des fonctions analytiques
    d\'efinies au voisinage d'une sous-vari\'et\'e lin\'eaire de
    ${\bbfC}\sp n$ nous permet d'obtenir des minorations fines de
    $\omega\sb t(S)/t$ pour tout t. En particulier, nous montrons
    $$ (\omega\sb{t\sb 1}(S)+n-a-1)/(t\sb 1+n-1)\le \omega\sb t(S)/t
    $$ o\`u a est la dimension de l'ensemble des points singuliers
    non \`a croisements normaux du diviseur de P lorsque $\text{deg}
    P=\omega\sb t(S)$. Ceci redonne le r\'esultat de H. Esnault et
    E. Viehwegpar une m\'ethode analytique assez \'el\'ementaire
    et contribue \`a renforcer la vraisemblance de la conjecture
    de Chudnovsky-Demailly.}
\RV{A.Azhari}
\CC{32J25}
%% END ITEM
%% BEGIN ITEM
\AN{0714.58053}{1989}
\AU{Getzler, Ezra}
\TI{An analogue of Demailly's inequality for strictly pseudoconvex CR manifolds.}
\LA{English}
\DT{J}
\SO{J. Differ. Geom. 29, No.2, 231-244 (1989). }
\ISSN{022-040}
\AB{Let L be a Heisenberg line bundle on a compact $(2n+1)$-dimensional
    Heisenberg manifold M. The author gives an expression involving
    local data for the limit as $m\to \infty$, of $(4\pi t/m)\sp{n+1}\cdot
    trace(\exp (-t\square /m))$ on $\Omega\sp{0,q}(M;L\sp m).$ \par
In
    defining Heisenberg structures, the author states that the most
    important examples of Heisenberg manifolds are strictly pseudo-convex
    hypersurfaces in ${\bbfC}\sp{n+1}$. The restriction of a holomorphic
    vector bundle with Hermitian connection on ${\bbfC}\sp{n+1}$
    to the hypersurface is then an example of a Heisenberg bundle
    on the hypersurface. A Heisenberg manifold has a Cauchy-Riemann
    operator that generalizes the operator ${\bar \partial}\sb b$,
    and this Cauchy-Riemann operator can be defined on $\Omega\sp{0,*}(M;E)$
    when E is a Heisenberg bundle. The Laplacian associated to the
    Cauchy-Riemann operator is denoted $\square.$ \par
As in an
    earlier paper [Commun.\ Math.\ Phys.\ 92, No.2, 163-178 (1983; Zbl
    0543.58026)], the author proves his theorem by developing an
    extended pseudodifferential operator calculus. This time the
    basic calculus is that developed for CR manifolds by {\it R.
    Beals}, {\it P. C. Greiner} and {\it N. K. Standon} [J. Differ.
    Geom. 20, No.2, 343-387 (1984; Zbl 0553.58029)] from the algebra
    of left-invariant operators on the Heisenberg group. \par
A
    preprint by J.-M. Bismut suggested the author's analogue of the
    result of {\it J.-P. Demailly} [Ann. Inst. Fourier 35, No.4,
    189-229 (1985; Zbl 0565.58017)]. For a complex line bundle on
    a compact complex manifold M with $\dim\sb CM=n$, Demailly calculated
    the asymptotic dimension of the ${\bar \partial}$-cohomology
    of $L\sp m$ as $m\to \infty$. Observing that the dimension of
    the ${\bar \partial}$-cohomology was bounded by the trace of
    the ${\bar \partial}$-heat operator, Bismut recovered Demailly's
    result by using methods from probability theory to develop an
    expression for the limit, as $m\to \infty$, of $(2\pi /m)\sp
    n$ times the trace of the appropriate ${\bar \partial}$-heat
    operator.}
\CC{58J35
32V40
58J40
35K05
32W05}
%% END ITEM
%% BEGIN ITEM
\AN{0633.32019}{1989}
\AU{Lelong, Pierre}
\TI{Fonction de Green pluricomplexe et lemmes de Schwarz dans les espaces de Banach. (Pluricomplex Green functions and Schwarzian lemmata in Banach spaces).}
\LA{French}
\DT{J}
\SO{J. Math. Pures Appl., IX. S\'er. 68, No.3, 319-347 (1989). }
\ISSN{021-782}
\AB{Pour f(z) holomorphe dans un domaine $\Omega$ $\subset {\bbfC}$,
    v\'erifiant $\vert f(z)\vert \le M$, s'annulant aux points $a\sb
    i$ avec les multiplicit\'es $\nu \sb i$, le lemme de Schwarz
    donne une majoration $\log \vert f(z)\vert \le \log M +\sum \sb{i}\nu
    \sb ig\sb{\Omega}(z,a\sb i)$ \`a partir de la fonction de Green
    $g\sb{\Omega}$. On \'etablit dans tout espace de Banach complexe
    E, l'existence d'une fonction de Green $g\sb{\Omega}(z,a)$ pour
    les domaines $\Omega$ born\'es et hyperconvexes: PSH($\Omega)$
    notant la classe des fonctions plurisousharmoniques de $\Omega$,
    l'hyperconvexit\'e \'equivaut \`a l'existence de $\phi\le 0$,
    $\phi\le PSH(\Omega)$, $\lim \sb{z\to \partial \Omega}\phi (z)=0,$
    uniform\'ement avec la distance d(z,$\partial \Omega)$ au bord.
    Pour ${\bbfC}\sp n$, cf. {\it M. Klimek}, Bull. Soc. Math. Fr.
    113, 231-240 (1985; Zbl 0584.32037) et {\it J.-P. Demailly},
    Math. Z. 194, 519-564 (1987; Zbl 0595.32006). La construction
    est faite dans l'espace de Banach E directement pour en ensemble
    fini $A=(a\sb i,\nu \sb i)$ de points affect\'es de coefficients
    $\nu \sb i>0$ et proc\`ede sans r\'ef\'erence aux op\'erateurs
    diff\'erentiels, la construction donne $g\sb{\Omega}$ comme enveloppe
    sup\'erieure d'une famille ${\cal F}\subset PSH(\Omega)$ et permet
    l'\'etude de la continuit\'e de $g\sb{\Omega}(z,A)$ par rapport
    au couple (z,A); on notera que m\^eme en dimension finie on a
    en g\'en\'eral $g\sb{\Omega}(z,A)\ge \sum \sb{i}\nu \sb ig\sb{\Omega}(z,a\sb
    i),$ $a\sb i\in A$, au lieu de l'\'egalit\'e due dans ${\bbfC}$
    \`a la lin\'earit\'e du laplacien. \par
De $g\sb{\Omega}$ on
    d\'eduit des lemmes de Schwarz pour $v\in PSH(\Omega)$, ainsi
    que pour les applications holomorphes F dans les trois cas $E\sb
    1\to E\sb 2,\quad E\to {\bbfC}\sp n,\quad {\bbfC}\sp n\to {\bbfC}\sp
    n.$ En dimension infinie on utilise $\nu (a)=\lim (\log r)\sp{-1}
    M\sb v(a,r)$ [cf. l'A., Lect. Notes Math. 919, 167-176 (1982;
    Zbl 0477.32017)] o\`u $M\sb v(a,r)=\sup \sb{\Vert y\Vert \le
    r}\nu (a+y)$ et $v=(2\pi)\sp{-1} \log \Vert F\Vert:$ on dit que
    F a en a un z\'ero de multiplicit\'e r\'eguli\`ere $\nu$ (a)
    si $\nu (a)=\lim \sb{y\to 0}(\log \Vert y\Vert)\sp{-1} v(a+y).$
    Le lemme de Schwarz \'enonc\'e au d\'ebut est alors \'etabli
    pour les applications $F: E\sb 1\to E\sb 2$ quand la fibre $F\sp{-1}(0)$
    a dans $\Omega$ des points isol\'es $(a\sb i,\nu \sb i)$ de multiplicit\'es
    $\nu \sb i$ r\'eguli\`eres, puis \'etendu \`a une classe plus
    g\'en\'erale. Pour les applications ${\bbfC}\sp n\to {\bbfC}\sp
    n$ il suffit que les $a\sb i$ soient isol\'es dans $F\sp{- 1}(0)$
    $\{$ pour le cas ${\bbfC}\sp n\to {\bbfC}$ le r\'esultat est
    \'evanescent pour $n\to \infty$, cf. l'A. et {\it L. Gruman},
    Entire functions of several complex variables (1986; Zbl 0583.32001),
    p. 160]. Pour $F: {\bbfC}\sp n\to {\bbfC}\sp n$, il y a \'egalit\'e
    entre (1) l'indice de Kronecker au z\'ero isol\'e a, (2) la masse
    $\mu$ (a) de la mesure $(2\pi)\sp{-n}(dd\sp a \log \Vert F\Vert)\sp
    n$ en a, (3) la multiplicit\'e dans la repr\'esentation int\'egrale
    de Bochner-Martinelli $\mu (a)f(a)=\int \sb{\partial \Omega}f(\zeta)K\sb{BM}(\zeta,a)$
    pour f holomorphe sur ${\bar \Omega}$ et $F\sp{-1}(0)\cap {\bar
    \Omega}=\{a\}$ et l'on a $\mu (a)\ge [\nu (a)]\sp n$ avec \'egalit\'e
    si la multiplicit\'e est r\'eguli\`ere.}
\CC{32U05
31C10
32K05
30C80
32A30}
%% END ITEM
%% BEGIN ITEM
\AN{0649.58030}{1987}
\AU{Bismut, Jean-Michel}
\TI{Demailly's asymptotic Morse inequalities: a heat equation proof.}
\LA{English}
\DT{J}
\SO{J. Funct. Anal. 72, 263-278 (1987). 
\par http://www.sciencedirect.com/science/journal/00221236}
\ISSN{022-123}
\AB{From the introduction: ``{\it J. P. Demailly} [C. R. Acad. Sci.,
    Paris, Ser. I 301, 119-122 (1985; Zbl 0595.58014); Ann. Inst.
    Fourier 35, 189- 229 (1985; Zbl 0575.58013)] proved remarkable
    asymptotic Morse inequalities for the ${\bar \partial}$ complex
    constructed over the line bundle $E\sp{\otimes k}$ as $k\uparrow
    \uparrow +\infty$, where E is a holomorphic hermitian line bundle.
    The inequalities of Demailly give asymptotic bounds on the Morse
    sums of the Betti numbers of ${\bar \partial}$ on $E\sp{\otimes
    k}$ in terms of certain integrals of the curvature of E. The
    analysis of Demailly is based on the beautiful remark that in
    the formula for (${\bar \partial}+{\bar \partial}\sp*)\sp 2$
    on $E\sp{\otimes k}$, the metric of E plays formally the role
    of the Morse function. In this paper, we give a heat equation
    proof of Demailly's inequalities.''}
\RV{G.Warnecke}
\CC{58J10
58J35
57R70}
%% END ITEM
%% BEGIN ITEM
\AN{0644.32002}{1987}
\AU{Filali, Adib Khalid}
\TI{\'Etude des fonctions m\'eromorphes sur certains espaces fibr\'es. (The study of meromorphic functions on certain fibre spaces).}
\LA{French}
\DT{CA}
\SO{S\'emin. d'analyse P. Lelong - P. Dolbeault - H. Skoda, Paris 1985/86, Lect. Notes Math. 1295, 59-95 (1987).}
\AB{[For the entire collection see Zbl 0623.00006.] \par
{\it H.
    Skoda} [Invent. Math. 43, 97-107 (1977; Zbl 0365.32018)] and
    {\it J.-P. Demailly} [Lect. Notes Math. 694, 15-41 (1978; Zbl
    0418.32011)] have explicitly constructed some fibre space X with
    fibre ${\bbfC}\sp 2$ and with base space an open set in ${\bbfC}$
    such that all plurisubharmonic functions on X are constant; in
    particular, X is not Stein. In this paper, the author shows that
    for such fibre space X all meromorphic functions on it are constant
    along its fibres. In order to do so, the author establishes some
    inequalities by using the defect Nevanlinna theory.}
\RV{Vo Van Tan}
\CC{32A20
32H25
32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0627.46054}{1987}
\AU{Lelong, Pierre}
\TI{Notions capacitaires et fonctions de Green pluricomplexes dans les espaces de Banach. (Notions of capacity and Green pluricomplex functions in Banach spaces).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 305, 71-76 (1987). }
\ISSN{764-444}
\AB{J. P. Demailly has defined recently a pluricomplex Green function
    for hyperconvex domains $\Omega$ in $C\sp n$ by using a certain
    differential operator. Such a notion exists in a complex Banach
    space E with no reference to differential operators or potential
    theory, and is obtained as maximal element of some family of
    plurisubharmonic functions.}
\RV{L.Nachbin}
\CC{46G20
31C10}
%% END ITEM
%% BEGIN ITEM
\AN{0614.32022}{1987}
\AU{Getzler, Ezra}
\TI{In\'egalit\'es asymptotiques de Demailly pour les fibr\'es vectoriels. (A vector bundle version of Demailly's asymptotic inequalities).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 304, 475-478 (1987). }
\ISSN{764-444}
\AB{Recently, {\it J.-P. Demailly} [Ann. Inst. Fourier 35, No.4,
    189-229 (1985; Zbl 0565.58017)] has proved an asymptotic generalization
    of the celebrated Kodaira's vanishing theorem. The purpose of
    the note under review is to extend Demailly's formulae from line
    bundles to certain higher rank vector bundles. \par
Let M be
    a compact Hermitian manifold. Consider a principal bundle P on
    M with fibre a compact Lie group G, and suppose that P carries
    a fixed holomorphic connection. To every complex irreducible
    representation $V\sp{\lambda}$ of G, determined by its highest
    weight $\lambda$, corresponds the analytic vector bundle $E\sp{\lambda}=P\times\sb
    GV\sp{\lambda}$, defined on M. The author's main result computes
    the asymptotic of the heat kernel: $$ \lim\sb{m\to \infty} Tr\sb{\Omega\sp{0,q}(M,E\sp{m\lambda})}\exp
    (-t\Delta\sb m/m),\quad q\ge 0, $$ where $\Delta\sb m$ is the
    Laplacian of the Dolbeault's complex $(\Omega\sp{0,.}(M,E\sp{m\lambda}),{\bar
    \partial})$. The expression of the above limit depends only on
    the curvature of P and on $\lambda$. \par
The proof reduces
    by a clever use of an equivariant index theorem of Bott to Demailly's
    framework.}
\RV{M.Putinar}
\CC{32L20
58J35
32M05}
%% END ITEM
%% BEGIN ITEM
\AN{0613.32014}{1987}
\AU{Siu, Yum-Tong}
\TI{Asymptotic Morse inequalities for analytic sheaf cohomology [according to J.-P. Demailly].}
\LA{English}
\DT{CA}
\SO{S\'emin. Bourbaki, 38\`eme ann\'ee, Vol. 1985/86, Exp. 666, Ast\'erisque 145/146, 283-297 (1987). }
\ISSN{303-117}
\AB{[For the entire collection see Zbl 0601.00002.] \par
The author
    considers the Morse inequalities, their background, Demailly's
    proof [{\it J.-P. Demailly}, C. R. Acad. Sci., Paris, S\'er.
    I 301, 119- 122 (1985; Zbl 0595.58014); Ann. Inst. Fourier 35,
    No.4, 189-229 (1985; Zbl 0565.58017)], a later probabilistic
    proof by Bismut, and some applications.}
\RV{M.S.Marinov}
\CC{32L10
32J99
58E05
14C40}
%% END ITEM
%% BEGIN ITEM
\AN{0629.32010}{1985}
\AU{Nguyen van Khue}
\TI{Stein morphisms and Riemann domains over Stein spaces.}
\LA{English}
\DT{J}
\SO{Acta Math. Vietnam. 10, 75-92 (1985). }
\ISSN{251-418}
\AB{The authors results are closely related to a conjecture of {\it
    J.-P.\ Serre} [``Quelques pro\-bl\`emes globaux relatifs aux vari\'et\'es
    de Stein'', Colloque sur les fonctions de plusieurs variables,
    Bruxelles, Li\`ege, Paris, 57-68 (1953; Zbl 0053.053)] that a
    holomorphic bundle with a Stein base and a Stein fiber is Stein.
    A counterexample for Serre's problem was given in 1977 by {\it
    H. Skoda} [Invent. Math. 43, 97-107 (1977; Zbl 0365.32018)].
    {\it J. P. Demailly} [Lect. Notes Math. 694, 15- 41 (1978; Zbl
    0418.32011)] showed that the first cohomology group of the bundle
    space of the structure sheaf of this counterexample is not Hausdorff
    in the canonical topology. Based on this result the author's
    two main theorems are: \par
Theorem 1. Let X be a complex space
    having a Stein morphism. Then X is Stein iff $H\sp 1(X,{\cal
    O}\sb X)$ is Hausdorffsch. \par
Theorem 2. Let X be a Riemann
    domain over a Stein space. Then X is Stein iff $H\sp q(X,{\cal
    O}\sb X)=0$ for $q=1,2,...,\dim X-1.$}
\CC{32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0597.10033}{1986}
\AU{Azhari, Abdelhak}
\TI{D\'emonstration analytique d'un lemme de multiplicit\'es. (Analytic proof of a multiplicity estimate).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 303, 269-272 (1986). }
\ISSN{764-444}
\AB{L'A. montre le r\'esultat suivant: Soit t et $t\sb 1$ des entiers
    positifs. Pour tout $P\in {\bbfC}[z\sb 1,...,z\sb n]$, $P\ne
    0$, ayant en chaque point d'une partie finie non vide S de ${\bbfC}\sp
    n$ un z\'ero d'ordre $\ge t$, il existe $F\in {\bbfC}[z\sb 1,...,z\sb
    n]$, $F\ne 0$, ayant en tout point de S un z\'ero d'ordre $\ge
    t\sb 1$, et dont les degr\'es partiels v\'erifient $$ \deg\sb{z\sb
    j} (F)\quad \le \quad ((n+t\sb 1-1)/t)\quad \deg\sb{z\sb j} (P).
    $$ Il s'agit d'un analogue du r\'esultat de {\it M. Waldschmidt}
    [Nombres transcendants et groupes alg\'ebriques (Ast\'erisque
    69/70) (1979; Zbl 0428.10017), lemme 7.5.2] qui ne consid\'erait
    que les degr\'es totaux. Le th\'eor\`eme de M. Waldschmidt se
    d\'eduisait des estimations $L\sp 2$ de H\"ormander- Bombieri-Skoda.
    Ici l'A. utilise la g\'en\'eralisation de ces r\'esultats due
    \`a {\it J.-P. Demailly} [Ann. Sci. \'Ec. Norm. Sup\'er., IV.
    S\'er. 15, 457-511 (1982; Zbl 0507.32021)]. Il obtient en fait
    un r\'esultat un peu plus g\'en\'eral en pond\'erant les multiplicit\'es
    des z\'eros suivant diff\'erentes directions, dans l'esprit du
    lemme de Dyson. L'A. en d\'eduit l'encadrement $$ (t/n) \omega\sb
    1(S)\le {\tilde \omega}\sb t(S)\le t {\tilde \omega}\sb 1(S),
    $$ o\`u ${\tilde \omega}{}\sb t(S)$ est la borne inf\'erieure
    du plus grand des degr\'es partiels d'une hypersurface alg\'ebrique
    admettant en tous les points de S une singularit\'e d'ordre au
    moins t.}
\RV{F.Gramain}
\CC{11J81
32L20
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0596.32020}{1986}
\AU{Skoda, H.}
\TI{Extension problems and positive currents in complex analysis.}
\LA{English}
\DT{CA}
\SO{Contributions to several complex variables, Hon. W. Stoll, Proc. Conf. Complex Analysis, Notre Dame/Indiana 1984, Aspects Math. E9, 299-327 (1986).}
\AB{[For the entire collection see Zbl 0585.00003.] \par
This paper
    is a survey of recent developments in the theory of the extension
    of analytic sets and closed positive currents. The following
    points are studied: \par
1) The extension across a closed subset
    A in an open set $\Omega \subset {\bbfC}\sp n$ of a closed positive
    current T which is locally of finite mass near A: if i) A is
    a complete pluripolar set, or ii) A is a complete p-pluripolar
    set $(0$-pluripolar$=pluripolar)$, and T is of bidimension $(p+1,p+1)$,
    then the trivial extension $\tilde T$ of T is a closed positive
    current. \par
2) The extension of pluripositive currents (i.e.
    T is normal, positive or negative and $dd\sp cT$ is positive)
    of bidimension $(p+1,p+1)$ across a closed p-pluripolar set,
    satisfying mass finiteness conditions. \par
3) The extension
    of a positive current of bidimension (p,p) across a Cauchy-Riemann
    submanifold A of CR dimension k: if i) T is closed and $p\ge
    k+1$ or ii) T is pluripositive and $p\ge k+2$, then T has always
    locally finite mass. If moreover $p\ge k+2$, $\tilde T$ is closed.
    4) Local finiteness of the mass of the Monge-Amp\`ere operator
    $dd\sp c$ near a closed set $M\subset \Omega \subset {\bbfC}\sp
    n$ which satisfies a proper condition, for example when M is
    a totally real submanifold. \par
5) The extension of a meromorphic
    map $f: \Omega$ $\setminus M\to N$ to a compact K\"ahler manifold
    N, where for example M is a CR submanifold of CR dimension $k\le
    n-2$ contained in a (n-2)-pluripolar complete subset (it is the
    case when M is an analytic subset of codimension $\ge 2)$. It
    is an open question to know if a meromorphic map from a Hartogs
    open set extends meromorphically to the polydisc. \par
6) The
    extension of a closed positive current on a Runge domain of ${\bbfC}\sp
    n$ to ${\bbfC}\sp n.$ \par
The author gives some proofs and
    some open problems. For sections 1,2,3,4,5, the explained methods
    and some results come from {\it N. Sibony}, Duke Math. J. 52,
    157-197 (1985; Zbl 0578.32023); for section 6 see {\it J.-P.
    Demailly}, Ark. Mat. 23, 35-52 (1985; Zbl 0566.32005). For an
    introduction to flat currents and slicing of Federer see {\it
    R. Harvey} and {\it B. Shiffman}, Ann. Math., II. Ser. 99, 553-587
    (1974; Zbl 0287.32008)].}
\RV{G.Dloussky}
\CC{32D15
32B15
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0555.10016}{1984}
\AU{Wakabayashi, Isao}
\TI{Meilleures estimations possibles pour l'ensemble des points alg\'ebriques de fonctions analytiques.}
\LA{French}
\DT{J}
\SO{S\'emin. Th\'eor. Nombres, Univ. Bordeaux I 1983/1984, Exp. No.19, 8 p. (1984).}
\AB{Classical transcendence criteria, due in particular to Straus,
    Schneider and Lang, bound the number of points in a number field
    K at which the Taylor expansions of a given entire transcendental
    function satisfy certain arithmetic properties. The reviewer
    [Transcend. Theory, Proc. Conf., Cambridge 1976, 187-193 (1977;
    Zbl 0365.10024)] showed that under highly technical assumptions,
    the number field K may be allowed to vary with the points. The
    author here proves that these technical assumptions are, surprisingly,
    best possible. A similar assertion concerning the work of Bombieri,
    Waldschmidt, Demailly, Chudnovsky et al. in the multidimensional
    case is also proven.}
\RV{D.Bertrand}
\CC{11J81
30A10
32A15}
%% END ITEM
%% BEGIN ITEM
\AN{0539.32006}{1984}
\AU{Skoda, H.}
\TI{A survey of the theory of closed, positive currents.}
\LA{English}
\DT{CA}
\SO{Complex analysis of several variables, Proc. Symp., Madison/Wis. 1982, Proc. Symp. Pure Math. 41, 181-190 (1984).}
\AB{[For the entire collection see Zbl 0527.00006.] \par
This paper
    is a survey of some recent progress in the theory of closed,
    positive currents. At first, we recall the fundamental structure
    theorems of R. Harvey, J. R. King and Y. T. Siu which give nice
    characterization of analytic cycles in the set of closed, positive
    currents and which are closely connected with the deep $L\sp
    2$-estimates for ${\bar \partial}$ of L. H\"ormander. Then we
    explain how the classical result of E. Bishop of extension of
    an analytic set of finite mass across another analytic set has
    a nice generalization to the case of currents: closed, positive
    currents of finite mass extend across an analytic set or even
    across a closed complete pluripolar set (El Mir result). The
    main tool is a study of the distribution of the mass of the current
    in the neighborhoods of the exceptional set. - At least, we describe
    an explicit example of J.- P.Demailly of an extremal, positive,
    closed current in ${\bbfP}\sp 2$ or ${\bbfC}\sp 2$ which is not
    a cycle. The proof of the fact that the current is extremal,
    uses two new support theorems for currents. \par
In a recent
    paper (to appear in Duke Math. J.), {\it N. Sibony} gives a new
    proof of extension theorems for currents of H. El Mir and the
    author, using a simple and beautiful Chern-Levine-Nirenberg estimate
    for currents. He also has nice results on extension of pluripositive
    (not necessarily closed) currents and also on extension of positive
    closed currents across a C. R. submanifold.}
\CC{32C30
32C25
49Q15
58A25}
%% END ITEM
%% BEGIN ITEM
\AN{0539.32007}{1983}
\AU{Jarnicki, Marek}
\TI{A method of holomorphic retractions and pseudoinverse matrices in the theory of continuation of $\delta$-tempered functions.}
\LA{English}
\DT{J}
\SO{Prepr., Inst. Math., Pol. Acad. Sci., Warsz. 276, 66 p. (1983).}
\AB{In this paper the author continues his investigations of extensions
    of holomorphic functions with controlled growth. The functions
    considered are holomorphic on certain analytic submanifolds M
    of Stein domains X over ${\bbfC}\sp n$ and are extended to all
    of X with growth controlled by weight functions $\delta:X\to(0,1]$
    for which -log $\delta$ is plurisubharmonic and $\delta\sp{\alpha}$
    is square integrable with $\alpha\ge 0$. The main theorem is
    the following: For ${\cal O}\sp{(k)}(X,\delta)= \{f\in {\cal
    O}(X):\Vert \delta\sp kf\Vert\sb{\infty}<\infty \}$ set ${\cal
    O}(X,\delta)= \cup\sb{k\ge 0}{\cal O}\sp{(k)}(X,\delta).$ If
    there is a map $G\in({\cal O}(X,\delta))\sp m$ with $M\subset
    G\sp{-1}(0)$, $rank(d\sb xG)=co\dim\sb xM=r(x),$ $x\in M$, and
    $\Vert d\sb xG\wedge...\wedge d\sb xG\Vert \ge b\delta\sp{\beta}(x),$
    $x\in M$, $r(x)$ times $d\sb xG$, for $b>0$, $\beta\ge 0$, then
    it exists $\sigma\ge 0$ such that for all $\eta>1$ there is $c\ge
    0$ with the property that for all $k\ge 0$ there is a continuous
    linear map $L\sb k:{\cal O}\sp{(k)}(M,\delta)\to {\cal O}\sp{(k+\sigma)}(X,\delta)$
    with $L\sb x(f)\vert M=f$ and $\Vert L\sb k\Vert \le c\eta\sp
    k.$- The proof of this theorem is based on a construction of
    holomorphic retractions using pseudoinverse matrices. Unfortunately,
    some pages of the paper are missing. This theorem generalizes
    results of I. Cnop, J. P. Demailly, R. Narasimhan and Y. Nishimura
    as well as former results of the author.}
\RV{S.Hayes}
\CC{32D15
32A22}
%% END ITEM
\bye
