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MR2077766 (2005k:14014) 14C25 14F43 19E15 55U10 55U35 Peters, Chris ^{(}F-GREN-M2^{)}; Kosarew, Siegmund ^{(}F-GREN-M2^{)} Introduction to Lawson homology. (English. English summary) Transcendental aspects of algebraic cycles, 44–71, London Math. Soc. Lecture Note Ser., 313, Cambridge Univ. Press, Cambridge, 2004. This article is a summary of the definition and basic properties of Lawson homology. In the first part, for a complex projective variety X the definition of Lawson homology is given as a homotopy group of the space Z_{m}(X) of algebraic cycles of dimension m. Explicit calculation of Lawson homology for projective spaces and algebraic curves is given.

In the second part, using the language of simplicial sets and results from homotopy theory, several equivalent definitions of Lawson homology are given. In Theorem 32 it is shown that the space Z_{m}(X) is homotopy equivalent to the loop space ΩBC_{m}(X) associated to the abelian monoid C_{m}(X) of effective cycles of dimension m on X. It is also shown that there are natural homotopy equivalences

\underrightarrow{lim}_{b} \overrightarrow{S_{•}C_{m}X_{b}} → (S_{•}(C_{m}X))^{+} ← S_{•}(Z_{m}(X)).

Here S_{•}(∗) denotes singular simplices of ∗. The inductive structure on the left hand side is given by base system construction and (S_{•}(C_{m}X))^{+} is the naive group completion of the simplicial abelian monoid S_{•}(C_{m}X).

Kenichiro Kimura (J-TSUKS-IM)

MR1962536 (2004c:32026) 32G13 14B12

Kosarew, Siegmund ^{(}F-GREN-F^{)}

Local moduli spaces and Kuranishi maps. (English. English summary) Manuscripta Math. 110 (2003), no. 2, 237–249. In various deformation theories, constructions of local moduli spaces are formulated in terms of differential graded Lie algebras. In such a construction, the Kuranishi map provides a description of the tangent space of the local moduli space so that the latter space is realized as a subspace of the tangent space defined by quadratic equations. In the paper under review, the author proves that the construction of the local moduli space relying on the Kuranishi map is possible in a more general setting than usual. Criteria of universality and smoothness of local moduli spaces are also given in this context.

Kimio Miyajima (Kagoshima)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. Bingener, J.: Offenheit der Versalität in der analytischen Geometrie. Math. Z. 173, 241–281 (1980) MR0592373 (82h:32025)

2. Bingener, J., Kosarew, S.: Lokale Modulräume in der analytischen Geometrie. Aspekte der Mathematik D2, D3, Vieweg Verlag, Braunschweig 1987 MR0912098 (89h:32040)

3. Buchweitz, R.O., Millson, J.J.: CR-geometry and deformations of isolated singularities. Memoirs of the Amer. Math. Soc. Nr. 597 (1997) MR1355712 (97g:32045)

4. Fischer, G.: Complex Analytic Geometry. Springer LNM 538 (1976) MR0430286 (55 #3291)

5. Flenner, H.: Ein Kriterium für Offenheit der Versalität. Math. Z. 178, 449–473 (1981) MR0638811 (83e:32028)

6. Fuchs, D., Lang, L.: Massey products and deformations. J. Pure Appl. Algebra 156, 215–229 (2001) MR1808824 (2002d:17019)

7. Goldman, W.M., Millson, J.J.: The homotopy invariance of the Kuranishi space. Ill. J. Math. 34, 337–367 (1989) MR1046568 (91d:32025)

8. Kleiman, S.: Relativ duality for quasi - coherent sheaves. Comp. Math. 41, 39–60 (1980) MR0578050 (81m:14017)

9. Kontsevich, M.: Deformation quantization of poisson manifolds. I. Preprint, math. q-alg/9709040 MR2062626 (2005i:53122)

10. Kosarew, S., Okonek, C.: Global moduli spaces and simple holomorphic bundles. Publ. RIMS, Kyoto Univ. 25, 1–19 (1989) MR0999347 (90k:32069)

11. Kuranishi, M.: Deformations of compact complex manifolds. Les Presses de l’Université de Montréal, Dépt. de Math., vol. 39 (1971) MR0355111 (50 #7588)

12. Merkulov, S.A.: An L_{∞}-algebra of an unobstructed deformation functor. Int. Math. Res. Not. 2000, 147–164 (2000) MR1741612 (2001f:32019)

13. Palamodov, V.: Deformations of complex spaces. Russian Math. Surveys 31, 129–197 (1976) MR0508121 (58 #22671)

14. Rim, D.S.: Formal deformation theory. In: SGA 7, I: Springer LNM 288, 32–132 (1972)

15. Rim, D.S.: Equivariant G-structure on versal deformations. Trans. AMS 257, 217–226 (1980) MR0549162 (80k:14021)

MR1991002 (2004k:14026) 14F05 14D20 14J99 18E30 32C37

32G08

Kosarew, Siegmund ^{(}F-GREN-F^{)}

Geometric and categorical nonabelian duality in complex geometry. (English. English summary) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 4, 769–797. In the paper under review the author explores the problem of what should be a duality theory for varieties and complex spaces. In his approach, a proper notion of duality should satisfy contravariance and biduality. Motivated by some concrete examples such as Tannaka duality for compact topological groups or abstract Spanier-Whitehead duality, the author develops a notion of dual of a variety or a complex space following a geometric and categorical approach. Given a projective variety X over Spec(ℂ), he defines the dual of X as the tensor category of algebraic vector bundles on X, that is, X^{∨} := (τ_{X}, ⊗) and proves a biduality statement. Here the bidual is given by right exact tensor functors Hom_{⊗,e}(τ_{X}, τ_{0}) where τ_{0} := τ_{Spec}(ℂ). Another instance of dual object X^{∨} that he considers is given by the derived category of coherent sheaves X^{∨} = D(X).

A different notion of dual object proposed is given by moduli spaces of simple or (semi)-stable sheaves or vector bundles. The author interprets the moduli space M as a quotient M = Mor(X, P)/G where P is a system of projective varieties parametrized by finite-dimensional nonzero ℂ-vector spaces and he proves the surjectivity (under some suitable conditions) of the biduality map X → Mor(M, P).

Finally the author discusses duality arising in the context of nonabelian group cohomology or Hilbert schemes.

Fabio Pioli (Madrid)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. J. F. Adams, ”Stable homotopy and generalized homology”, The University of Chicago Press, Chicago, 1974. MR0402720 (53 #6534)

2. A. Altman–S. Kleiman, Compactifying the Picard Scheme, Adv. Math. 35 (1980), 50–112. MR0555258 (81f:14025a)

3. V. Balaji–L. Brambila-Paz–P. Newstead, Stability of the Poincaré bundle, Math. Nachr. 188 (1997), 5–15. MR1484665 (99a:14013)

4. A. Bondal–D. Orlov., Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125

(2001), 327–344. MR1818984 (2001m:18014)

5. A. Bondal–D. Orlov, Semiorthogonal decomposition for algebraic varieties, MPI Bonn, Preprint 95–15.

6. T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), 25–34. MR1651025 (99k:18014)

7. I. Ciocan-Fontanine–M. M. Kapranov, Derived Quot schemes, Ann. Sci. Ecole Norm. Sup. (4) 34 (2001), 403–440. MR1839580 (2002k:14003)

8. I. Ciocan-Fontanine–M. M. Kapranov, Derived Hilbert schemes, J. Amer. Math. Soc. 15 (2002), 787–815. MR1915819 (2003h:14007)

9. A. Dold–D. Puppe, Duality, trace and transfer, In: ”Proc. Steklov Inst.” (4); Topology, A collection of papers, P. S. Alexandrov (ed.) (AMS translation), 1984, pp. 85–103. MR0733829 (85i:55007)

10. A. A. Kirillov, ”Elements of the theory of representations”, Grundl. math. Wiss. 220, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR0412321 (54 #447)

11. S. Kosarew, Nonabelian duality on Stein spaces, Amer. J. Math. 120 (1998), 637–648. MR1623408 (99d:32015)

12. S. Kosarew–C. Okonek, Global Moduli Spaces and Simple Holomorphic bundles, Publ. RIMS, Kyoto Univ. 25 (1989), 1–19. MR0999347 (90k:32069)

13. A. Maciocia, Generalized Fourier-Mukai transforms, J. Reine Angew. Math. 480 (1996), 197–211. MR1420564 (97g:14013)

14. S. Mac Lane, ”Categories for the Working Mathematician”, Grad. Texts in Math. 5, Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR1712872 (2001j:18001)

15. S. Mukai, On the moduli space of bundles on a K3-surface, In: ”Vector Bundles on Algebraic Varieties” Bombay Coll., 1984, Tata Inst., Oxford University Press, 1987, pp. 341–413. MR0893604 (88i:14036)

16. M. S. Narasimhan–S. Ramanan, Deformations of the moduli spaces of vector bundles over an algebraic curve, Ann. Math. 101 (1975), 391–417. MR0384797 (52 #5669)

17. A. Rosenberg, ”Noncommutative Algebraic Geometry and Representations of Quantized Algebras”, Kluwer Acad. Publ., Dordrecht-London-Boston, 1995. MR1347919 (97b:14004)

18. C. S. Seshadri, ”Fibrés vectoriels sur les courbes algébriques”, Astérisque 96 (1982). MR0699278 (85b:14023)

19. C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. IHES 79 (1994), 47–129. MR1307297 (96e:14012)

20. C. Simpson, A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kempen, Preprint math. AG 9704006.

21. C. Simpson, Algebraic aspects of higher nonabelian Hodge theory, Preprint math. AG 9902067. MR1978713 (2004m:14010)

22. B. Toen, Dualité de Tannaka supérieure I: Structures monoidales, MPI Bonn, Preprint June 10, 2000.

23. G. W. Whitehead, ”Elements of homotopy theory”, Grad. Texts in Math. 61, Springer-Verlag, Berlin-Heidelberg-New York, 1978. MR0516508 (80b:55001)

MR1809550 (2002a:32009) 32G13 14J60 32G05 Kosarew, S. ^{(}F-GREN^{)}; Lupascu, P. ^{(}CH-ZRCH^{)} Moduli of simple holomorphic pairs and effective divisors. (English. English summary) Abh. Math. Sem. Univ. Hamburg 70 (2000), 265–274. The main subject of this paper is the comparison of two complex structures (one is given by algebraic geometry and the other by gauge theory) on the moduli space of simple pairs of a holomorphic vector bundle coupled with its holomorphic section on a fixed compact complex manifold. (A pair is said to be simple if the identity is its only automorphism.) The existence of the moduli space of simple pairs is deduced by an algebro-geometric argument in a paper by S. Kosarew and C. Okonek [Publ. Res. Inst. Math. Sci. 25 (1989), no. 1, 1–19; MR0999347 (90k:32069)] or a gauge-theoretic one (Theorem 2.4). Then the following comparison theorems are proved. Theorem 2.5. These two complex structures on the moduli space of simple pairs are biholomorphic to each other. Theorem 1.1. The moduli space of simple pairs of rank one is isomorphic to the space of all effective divisors on the complex manifold.

Kimio Miyajima (Kagoshima)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. M. Atiyah, N. Hitchin, I. Singer, Self-duality in fourdimensional Riemannian geometry, Proc. R. Soc. Lond. A 362 (1978), 425– 461. MR0506229 (80d:53023)

2. J. Bingener, Darstellbarkeitskriterien für analytische Funktoren, Ann. scient. Éc. Norm. Sup. 4e série, t. 13 (1980), 317–347. MR0597743 (82i:32047)

3. J. Bingener, Offenheit der Versalität in der analytischen Geometrie, Math. Z. 173 (1980), 241–281. MR0592373 (82h:32025)

4. J. Bingener, S. Kosarew, Lokale Modulräume in der analytischen Geometrie I,II, Aspekte der Mathematik, Vieweg Verlag, 1987. MR0912099 (89h:32041)

5. V. Brînzănescu, Holomorphic Vector Bundles over Compact Complex Surfaces, Lect. Notes Math. 1624, Springer, 1996. MR1439504 (98g:32053)

6. A. Douady, Le problème des modules pour les sous-espaces analytiques compacts d’une espace analytique donné, Ann. Inst. Fourier Grenoble, 16 (1966), 1–95. MR0203082 (34 #2940)

7. G. Fischer, Complex Analytic Geometry, Lect. Notes Math. 538, Springer, 1976. MR0430286 (55 #3291)

8. H. Flenner, Eine Bemerkung über relative Ext-Garben, Math. Ann. 258 (1981), 175–182. MR0641823 (83i:32019)

9. A. Fujiki, Projectivity of the Space of Divisors on a Normal Compact Complex Space, Publ. R.I.M.S., Kyoto Univ. 18 (1982), 1163–1173. MR0688950 (85b:32045)

10. S. Kosarew, Local Moduli spaces and Kuranishi maps, preprint, 1999. MR1962536 (2004c:32026)

11. S. Kosarew, Ch. Okonek, Global Moduli Spaces and Simple Holomorphic Bundles, Publ. R.I.M.S., Kyoto Univ. 25 (1989), 1–19 MR0999347 (90k:32069)

12. K. Miyajima, Kuranishi Family of Vector Bundles and Algebraic Description of the Moduli Space of Einstein-Hermitian Connections, Publ. R.I.M.S., Kyoto Univ. 25 (1989), 301–320. MR1003790 (90k:32065)

13. Ch. Okonek, A. Teleman, The Coupled Seiberg-Witten Equations, Vortices, Moduli Spaces of Stable Pairs, Internat. Journ. Math. vol.6 6 (1995), 893–910. MR1354000 (97a:32012)

14. Y. Suyama The analytic moduli space of simple framed holomorphic pairs, Kyushu J. Math. 50 (1996), no. 1, 65–82. MR1447909 (98i:32027)

MR1623408 (99d:32015) 32E10 32C37

Kosarew, Siegmund ^{(}F-GREN-F^{)}

Nonabelian duality on Stein spaces. (English. English summary) Amer. J. Math. 120 (1998), no. 3, 637–648. Let X be a Stein complex space and Γ(X, 𝒪) the C-algebra of global holomorphic functions on X. Then the “Gel′fand” evaluation map φ: X → Spec(Γ(X, 𝒪)), φ(x) = φ_{x}, with φ_{x}(f) = f(x) is defined and it is well known that the Gel′fand map is homeomorphic by the theorem of Igusa-Remmert-Forster. Let G = Mor(X, GL_{n}(C)) be the topological group of holomorphic maps from X to GL_{n}(C) and G^{0} the connected component of the identity in G. Then the “Gel′fand” evaluation map φ: X → Hom_{e}(G, GL_{n}(C)), φ(x) = φ_{x}, with φ_{x}(A) = A(x) is also defined, where Hom_{e}(G, GL_{n}(C)) denotes the set of continuous group homomorphisms from G to GL_{n}(C) which are in addition GL_{n}(C)-left and -right equivariant.

In this paper, replacing the algebra C by the complex Lie group GL_{n}(C) (n ≥ 2), bijectivity of the Gel′fand map is considered and the following interesting duality theorem is proved. Theorem: The Gel′fand map is injective and, moreover, for any π ∈ Hom_{e}(G, GL_{n}(C)), there are a point x_{0} ∈ X, a continuous group homomorphism δ: Γ(X, 𝒪^{∗}) → C^{∗} with δ(c) = 1 for any locally constant holomorphic function c, and a homomorphism α: G/G^{0} → C^{∗} such that π(A) = A(x_{0})δ(det(A))α(A · G^{0}) for A ∈ G = GL_{n}(Γ(X, 𝒪)). If π commutes with interchanging entries of matrices, we have δ = 1.

First, the theorem is proved in the case n = 2. For any π ∈ Hom_{e}(G, GL_{2}(C)), there exists a point x_{0} ∈ X such that π is the evaluation map in x_{0} on all matrices in SL_{2}(Γ(X, 𝒪)) where at least one entry belongs to Γ(X, 𝒪^{∗}), and δ is constructed. The theorem of Igusa-Remmert-Forster, the classical Bruhat decomposition and the GL_{2}(C)-equivariance are essentially used in the proof. Put π′: G → GL_{2}(C), π′(A) = A(x_{0})δ(det(A)); then it is shown that π and π′ coincide on G^{0} and the difference between π, π′ is described by the homomorphism α: G/G^{0} → C^{∗}.

In the general case n ≥ 3, the theorem is proved by the same method with some technical modifications.

Kunio Takijima (J-SAI)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. A. Borel, Linear Algebraic Groups, 2nd. enlarged ed., Springer-Verlag, New York, 1991. MR1102012 (92d:20001)

2. O. Forster, Uniqueness of topology in Stein algebras, Function Algebras, Proc. Internat. Sympos. Tulane Univ., 1965 (F. T. Birtel, ed.), Scott, Foresmann and Co., Chicago, 1966, pp. 157– 163. MR0197772 (33 #5935)

3. J. E. Humphreys, Linear Algebraic Groups, 2nd ed., Springer-Verlag, New York, 1981. MR0396773 (53 #633)

4. S. Kosarew, Non-abelian dualities in complex geometry, in preparation.

5. T. A. Springer, Linear Algebraic Groups, Birkhäuser, Boston, 1981. MR0632835 (84i:20002)

MR1214659 (94b:32045) 32J05 32C36 32C37 32J17 Kosarew, Siegmund; Peternell, Thomas Addendum to the paper: “Formal cohomology, analytic cohomology and nonalgebraic manifolds” [Compositio Math. 74 (1990), no. 3, 299–325; MR1055698 (91g:32035)]. Compositio Math. 86 (1993), no. 1, 121. The authors note an omission and correct Theorem 5.1 in the original paper.

MR1144555 (93d:32027) 32G05 32S30

Kosarew, Siegmund ^{(}D-GTN-GA^{)}

Maximal modular subspace in semi-universal deformation of holomorphic mapping germs. Analyse complexe (Bucharest, 1989). Rev. Roumaine Math. Pures Appl. 36 (1991), no. 5-6, 231–252. In this paper, the author shows the existence of maximal modular subspaces in semi-universal deformations of morphisms between germs of graded complex spaces. The main result says that for a germ f: X → Y of a morphism between graded complex spaces, the assumptions which already imply the existence of a semi-universal deformation 𝒳 → 𝒴 of f over the germ R are sufficient for obtaining a maximal modular subgerm M ⊂ R. Applications of this result are the existence of such subspaces in semiuniversal deformations of isolated singularities of germs of complex spaces or coherent sheaves. The method used is first to construct an infinite-dimensional graded Lie algebra and then to apply the construction of the subspaces given in a paper of the author and H. Stieber [Abh. Math. Sem. Univ. Hamburg 60 (1990), 17–36; MR1087115 (92k:32033)]. An essential point is the verification of the smoothness of the infinite-dimensional stabilizer “subgroups”.

Vasile Brînzănescu (Bucharest)

MR1123547 (92g:32056) 32J25 14C30 14F17 32C35 32L20 Bauer, Ingrid ^{(}D-GTN-GA^{)}; Kosarew, Siegmund ^{(}D-GTN-GA^{)} Some aspects of Hodge theory on noncomplete algebraic manifolds. Prospects in complex geometry (Katata and Kyoto, 1989), 281–316, Lecture Notes in Math., 1468, Springer, Berlin, 1991. Let U be a smooth algebraic variety. The authors continue their study of the Dolbeault and de Rham cohomology groups on U which admits a compactification X of a certain type. The situation is divided into two cases, which correspond to the notion of concavity and convexity in complex analytic geometry. The first one is “Y := X − U is of ‘high’ codimension in X” and the other is “Y is a divisor, X is smooth and the normal bundle N_{Y/X} satisfies a suitable positivity condition”. Their results contain finiteness theorems, Lefschetz type theorems, Hodge theory, and vanishing theorems of Akizuki-Nakano type. An interesting counterexample is also given. The paper is written in a survey style.

Takeo Ohsawa (J-NAGO)

MR1122178 (92i:14018) 14F32 14C30 14F40 32J25

Kosarew, Siegmund ^{(}D-GTN-GA^{)}

The hard Lefschetz theorem for concave and convex algebraic manifolds. Complex analysis (Wuppertal, 1991), 175–187, Aspects Math., E17, Vieweg, Braunschweig, 1991. Let U be a smooth complex algebraic variety, together with a compactification X and an ample line bundle L on X. The author shows the validity of the “hard Lefschetz” isomorphism

L^{r}: H_{c}^{dim X−r}(U^{an}, C) → H_{c}^{dim X+r}(U^{an}, C)

for r ≥ dim_{C}(X − U) + 1. The first proof reduces it to the (certainly nonelementary) hard Lefschetz isomorphism in intersection cohomology (due to O. Gabber). The second proof reduces it to the case where dim(X − U) = 0, for which the author uses a theorem of Navarro Aznar. If (X − U) is a divisor, the author relates the validity of the hard Lefschetz isomorphism to the “k-ampleness” of the normal bundle of (X − U) by reducing it to the case k = 0, for which he uses again a theorem of Navarro Aznar.

The author also gives the interpretation of the isomorphisms L^{r} in terms of the cohomology of differential forms via the Hodge to de Rham spectral sequence.

Hélène Esnault (D-DUES2)

MR1014252 (92c:32014) 32C35 32G07 32G13

Kosarew, Siegmund ^{(}D-GTN-GA^{)}

Grothendieck’s existence theorem in analytic geometry and related results. Trans. Amer. Math. Soc. 328 (1991), no. 1, 259–306. A powerful tool for the solution of many moduli problems in algebraic geometry is Grothendieck’s existence theorem. This theorem gives a general algebraicity criterion for coherent formal sheaves.

The author states and proves several kinds of “analytification” theorems for formal coherent sheaves and formal complex spaces in the spirit of Grothendieck’s theorem. The straightforward translation into the context of analytic geometry is not possible. The formulation of the results is derived from deformation theory and especially from M. Artin’s work on representability of functors. The methods of proof depend heavily on a deeper study of the cotangent complex and resolutions and use the methods of V. P. Palamodov [Russian Math. Surveys 31 (1976), no. 3, 129–197; MR0508121 (58 #22671); Trudy Sem. Petrovsk. No. 4 (1978), 173–226; MR0524531 (80j:32019); translated in Amer. Math. Soc. Transl. (2) 122 (1984), 119–171; see MR0760261 (85g:00031)].

As applications one can deduce the convergence of formal versal deformations in diverse situations.

V. S. Retakh (Brooklyn, NY)

MR1087115 (92k:32033) 32G05 32G07 Kosarew, S. ^{(}D-GTN-GA^{)}; Stieber, H. ^{(}D-RGBG^{)} A construction of maximal modular subspaces in local deformation theory. Abh. Math. Sem. Univ. Hamburg 60 (1990), 17–36. Introduction: “In general, semi-universal deformations of analytic objects are not universal. This is one major difficulty in constructing global moduli spaces from local pieces. But at least there should be something like a maximal universal locus in the base space of a semi-universal deformation. In the case of deformations of compact complex spaces, this problem was first considered and solved by V. Palamodov. Quite recently a significant generalization has been obtained by Stieber [Math. Z. 203 (1990), no. 2, 263–277; MR1033436 (91f:32023)], using the approach he developed elsewhere [Existenz semiuniverseller Deformationen in der Komplexen Analysis, Vieweg, Braunschweig, 1988; MR0941144 (89f:32038)]. But still the objects which are deformed have to be compact in a certain sense, so the method does not apply to deformations of local objects. On the other hand, the formal aspects of this theory of ‘modular’ subspaces were treated extensively by O. A. Laudal and G. Pfister [Local moduli and singularities, Lecture Notes in Math., 1310, Springer, Berlin, 1988; MR0950171 (89h:14008)] for the case of deformations of singularities. Also some classes of examples were calculated there.

“Now our intention was mainly to construct these so-called ‘maximal modular’ subspaces in the framework of operations of group germs. In the first section we treat the finite-dimensional case where the main idea is seen immediately. We also give a characterization of the maximal modular subspace by the smoothness of the relative stabilizer group. For the applications in deformation theory one has to treat, more generally, the infinite-dimensional case which is considered in Section 2. As usual, it is necessary to make certain splitting assumptions on the maps of Banach spaces which appear as tangent morphisms. In principle the construction itself is exactly the same as in the finite-dimensional situation. One main difference is that the stabilizer subgroup of the distinguished point has to be assumed to be smooth. This is a crucial point which needs to be verified in the applications. However, these will be treated in a second paper. For the formulations and definitions in Section 2, we stick to the context of PO-spaces and PO-Lie groups. The foundations of this theory were developed in a two-volume book [J. Bingener, Lokale Modulräume in der analytischen Geometrie. Band 1, Vieweg, Braunschweig, 1987; MR0912098 (89h:32040); Lokale Modulräume in der analytischen Geometrie. Band 2, Vieweg, Braunschweig, 1987; MR0912099 (89h:32041)], to which we must refer the reader.

“Many aspects of modular subspaces can be formulated in the context of general deformation groupoids. This is done in Section 3. The existence of a maximal modular subspace in the base of a semiuniversal deformation is guaranteed as soon as some automorphism functor is representable. From this we may recover at once the main result of the paper by Stieber cited above. Finally, in the last section we give an explicit calculation of the tangent space of the maximal modular subspace in the framework of group operations.”

MR1076130 (91j:14012) 14F17 32L20 Kosarew, Ingrid ^{(}D-GTN-GA^{)};

Kosarew, Siegmund ^{(}D-GTN-GA^{)}

Kodaira vanishing theorems on noncomplete algebraic manifolds. Math. Z. 205 (1990), no. 2, 223–231. In 1987, P. Deligne and L. Illusie published a purely algebraic proof of the Kodaira-Akizuki-Nakano vanishing theorem on nonsingular projective algebraic varieties [Invent. Math. 89 (1987), no. 2, 247– 270; MR0894379 (88j:14029)]. In the article under review, the authors develop Deligne and Illusie’s method to extend the vanishing theorem to noncomplete varieties. Part of their results are, as one expects, quite similar to those which have been proved by transcendental methods on weakly 1-complete manifolds [cf. the reviewer and K. Takegoshi, Publ. Res. Inst. Math. Sci. 17 (1981), no. 2, 723–733; MR0642658 (83e:32031)]. The reviewer expects that such a similarity is supporting evidence for the unity of mathematics.

Takeo Ohsawa (J-NAGO)

MR1055698 (91g:32035) 32J05 32C36 32C37 32J17 Kosarew, Siegmund ^{(}D-GTN-GA^{)};

Peternell, Thomas ^{(}D-BAYR^{)}

Formal cohomology, analytic cohomology and nonalgebraic manifolds. Compositio Math. 74 (1990), no. 3, 299–325. In a paper by Peternell and M. Schneider [Math. Ann. 280 (1988), no. 1, 129–146; MR0928301 (89g:32040)] a classification of compact complex manifolds which contain C^{3} was attempted from the point of view of existence of meromorphic functions, and this problem was reduced to a conjecture relating formal and analytic cohomology along the divisor at infinity. In the present paper, by analyzing some examples, the authors show that this conjecture has to be completed with additional assumptions.

The paper also contains comparison theorems between formal and convergent cohomology and between local and moderate local cohomology, obtained by using analytic duality.

A classification is given of all compact complex surfaces X containing a curve C such that C^{2} = 0 and X ∖ C is Stein. Finally the authors prove that any complex manifold, compactification of C^{3}, with irreducible divisor at infinity is Moishezon if it has at least one nonconstant meromorphic function.

Andrei Baran (Bucharest)

MR1006373 (90e:14019) 14F40 14F30 32C35 32F10 Bauer, Ingrid ^{(}D-GTN-GA^{)}; Kosarew, Siegmund ^{(}D-GTN-GA^{)} On the Hodge spectral sequence for some classes of noncomplete algebraic manifolds. Math. Ann. 284 (1989), no. 4, 577–593. Some of the significant results on complete algebraic varieties have natural extensions to noncomplete varieties. In this article the authors establish a beautiful method of extending Deligne-Illusie’s theory on the algebraic proof of the E_{1}-degeneration of the Hodge spectral sequence. Their main contribution seems to be in the step of transplanting the results for the positive characteristic case to those for C, which needs a rather delicate base change argument. It would be a matter of further interest whether M. Saito’s theory of Hodge modules can be extended by the same method.

Takeo Ohsawa (J-NAGO)

MR0999347 (90k:32069) 32G13 32C35 Kosarew, Siegmund ^{(}D-GTN-GA^{)};

Okonek, Christian ^{(}D-GTN-GA^{)}

Global moduli spaces and simple holomorphic bundles. Publ. Res. Inst. Math. Sci. 25 (1989), no. 1, 1–19. The authors prove the existence of a coarse moduli space for simple coherent sheaves on a given compact complex space. In fact, the following relative version is established: Let f: X → S be a separated morphism of complex spaces. A coherent 𝒪_{X}-module ℱ, flat and proper over S, is called simple (over S) if the natural morphism S × C → End_{S}(ℱ) is an isomorphism. Isomorphism classes of such simple coherent sheaves on X/S have a coarse moduli space (not necessarily separated).

This is derived from an abstract criterion for representability of analytic functors. The latter relies on the construction of relative quotients for relative group operations on complex spaces.

Ciprian Borcea (Somerset, NJ)

MR0998418 (90g:32011) 32C35 32H35 55N30

Kosarew, Siegmund ^{(}D-GTN-GA^{)}

A note on the local homotopy type of direct image sheaf complexes. Arch. Math. (Basel) 52 (1989), no. 4, 393–401. Let f: X → S be a proper holomorphic map of complex spaces. By Grauert’s theorem the direct image sheaves ℝ^{i}f_{∗}(ℱ) are coherent 𝒪_{S}- modules for any coherent 𝒪_{X}-module ℱ. In the proof of Grauert’s theorem one constructs a quasi-isomorphism θ: ℒ → 𝒞(𝒰, ℳ) where ℒ is a complex of free 𝒪_{S}-modules and 𝒞(𝒰, ℳ) is a Čech representative of ℝ.f_{∗}(ℱ) given by a covering 𝒰 of X and a resolution ℳ of ℱ. In the present paper, by a suitable choice of 𝒞(𝒰, ℳ), it is proved that the quasi-isomorphism θ is a continuous homotopy equivalence of complexes of Fréchet sheaves. In order to get such a result the author uses the methods developed by himself and J. Bingener [Lokale Modulräume in der analytischen Geometrie, Vol. 1, Vieweg, Braunschweig, 1987; MR0912098 (89h:32040); Vol. 2; MR0912099 (89h:32041)].

Alessandro Tancredi (I-PERG-MI)

MR0944181 (89m:32022) 32C25 32G99 32J99

Kosarew, Siegmund ^{(}D-RGBG^{)}

Ein allgemeines Kriterium für das formale Prinzip. (German) [A general criterion for the formal principle] J. Reine Angew. Math. 388 (1988), 18–39. Let i: A → X and j: A → Y be imbeddings of a compact complex analytic space A into complex analytic spaces X and Y. The formal principle says that isomorphy of the formal completions of X and Y along A implies isomorphy of suitable neighborhoods of A in X and in Y. This principle is not in all cases true. Here the author formulates a rather general sufficient condition for its validity. For that he uses the deformations a and b over (C, 0) from i and j to the embedding of the subspace A into its respective normal cones. Further on he uses the functor \underline{Isom}_{(C,0)}(a, b), which to each germ T over (C, 0) attaches the set of all T -isomorphisms of the deformations a_{T} and b_{T} induced by a and b over T. Some applications are given. For instance the formal principle holds if the normal bundle N_{A|X} is q-concave with q + 1 < depth 𝒪_{A} + rkN_{A|X} and q < depth θ_{A} + rkN_{A|X} or if A and X are nonsingular and N_{A|X} is 1-positive (cf. a result of M. Commichau and H. Grauert [in Recent developments in several complex variables (Princeton, NJ, 1979), 101–126, Ann. of Math. Stud., 100, Princeton Univ. Press, Princeton, N.J., 1981; MR0627752 (84a:32018)]).

K. Wolffhardt (D-MNCH)

MR0925718 (89c:32063) 32G13 32F10

Kosarew, Siegmund ^{(}D-RGBG^{)}

Modulräume holomorpher Abbildungen auf konkaven komplexen Räumen. (German) [Moduli spaces of holomorphic mappings onto concave complex spaces] Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 285–310. Let X, Y be complex spaces and Hom(X, Y) the set of all holomorphic mappings X → Y. If X is compact, then, by a theorem of Douady, Hom(X, Y) can be equipped with a canonical complex structure. Here the author shows that for every concave complex space X the set Hom(X, Y) also possesses a complex structure. The relative case is treated more generally and the moduli problem is solved for concave mappings X → S.

The most important tools for the solution of the problem are the techniques of Banach-analytic spaces developed by Douady and Pourcin. The moduli problem is first solved locally, namely for space germs S and for relatively compact q-concave subsets in the fiber over S. For the globalization the author employs a representability criterion of Schuster and Vogt.

H. Kerner (Bayreuth)

MR0918518 (89c:32055) 32G05 Flenner, Hubert ^{(}D-GTN^{)}; Kosarew, Siegmund ^{(}D-RGBG^{)} On locally trivial deformations. Publ. Res. Inst. Math. Sci. 23 (1987), no. 4, 627–665. Let X be a compact complex space. A deformation X ↪ 𝒳 → (S, o) of X, parametrized by the germ (S, o), is said to be locally trivial if, for every x ∈ X, the deformation germ (𝒳, x) → (S, o) is isomorphic to the trivial deformation (X, x) × (S, o) of the germ (X, x). Such a deformation is said to be locally trivial semiuniversal if any other locally trivial deformation X ↪ 𝒳′ → (S′, o) fits into a Cartesian diagram

X \begin{matrix}𝒳′ & ⟶ & (S′, o) \\ ↓ & & ↓^{α} \\ 𝒳 & ⟶ & (S, o) \end{matrix}

with the tangent map of α at o uniquely determined.

The main purpose of the paper is to prove that, given X, a locally trivial semiuniversal deformation exists.

Given that a semiuniversal deformation for X is known to exist, an obvious candidate would be the restriction of this deformation to a “maximal locus” inside the parameter space, over which the deformation remains locally trivial. Thus, the authors convert their problem into one of a local nature: given holomorphic map germs φ: (𝒳, o) → (S, o), ψ: (𝒴, o) → (S, o) with special fibers (X, o) and (Y, o), produce criteria which ensure the existence of a maximal extension locus (S′, o) ⊂ (S, o) for a given map ξ_{o}: (X, o) → (Y, o), i.e.,

(1) ξ_{0} extends over S′ to ξ: (𝒳_{S′}, o) → (𝒴_{S′}, o); and (2) if ξ_{0} extends over a pullback by f: (T, o) → (S, o), then f factorizes through S′.

This is actually achieved in a general setting, using Banach analytic methods developed in works of A. Douady and G. Pourcin.

Applications to the case of locally trivial deformations of holomorphic mappings and coherent modules are also given.

Ciprian Borcea (Somerset, NJ)

MR0906064 (90a:32031) 32G99 32C35

Kosarew, Siegmund ^{(}D-RGBG^{)}

⋆Grothendieck’s existence theorem in analytic geometry and related results. Regensburger Mathematische Schriften [Regensburg Mathematical Publications], 14. Universität Regensburg, Fachbereich Mathematik, Regensburg, 1987. 83 pp. DM 15.00. ISBN 3-88246-120-9 The existence theorem in this paper is named after Grothendieck’s so-called existence theorem in algebraic geometry, which gives an equivalence of the category of coherent sheaves with proper supports on a relative scheme with that of the same sheaves on a formal completion of the scheme, see EGA III [A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. No. 17 (1963) (EGA III), Théorème 5.1.4; MR0163911 (29 #1210)]. Although it is not a direct analogue of the latter it gives a strong statement on the algebraic nature of formal sheaves or spaces with proper supports with applications to deformation theory. Its formulation is as follows: Let A be a formal C-algebra, i.e. an algebra C⟨x⟩^{∧}/, where C⟨x⟩ = C⟨x_{1}, ⋯, x_{m}⟩ is a convergent power series ring, C⟨x⟩^{∧} is the completion with respect to a proper ideal , and where is an ideal in the completion. Define the category An(A) of complex spaces over A f.e. by local models. A smooth local model over A is a domain D in some C^{n} with the structure sheaf 𝒪_{D} ∧ ≡ 𝒪_{D}⟨x_{1}, ⋯, x_{m}⟩^{∧}/𝒪_{D}⟨x_{1}, ⋯, x_{m}⟩^{∧}, and a local model over A is defined as Supp(𝒪_{D} ∧ /ℐ), where ℐ ⊂ 𝒪_{D} ∧ is some coherent sheaf of ideals. Note that the category An(A) is slightly more general than just the category of formal complex spaces over A. There is an obvious base change functor An(A) → An(B) for each homomorphism A → B of formal C-algebras, and there are finite fibre products in An(A). Let now (S, o) be a usual complex space germ and 𝒪_{S,o} → A be a formal algebra over 𝒪_{S,o}, and let X →_{f} S be a

morphism of complex spaces. If X_{A} ∈ An(A) denotes the pullback of X under base change, the following functors on the category of formal algebras over 𝒪_{S,o} are considered: Coh(X_{A}), Coh_{p}(X_{A}), Coh_{p,f}(X_{A}) are the categories of all coherent sheaves on X_{A}, those with compact supports, and those which are in addition flat over A, respectively. If B is any convergent 𝒪_{S,o}-subalgebra of A, f.e. one which is generated by an element, there is the pullback functor Coh(X_{B}) → Coh(X_{A}) whose image consists of pullbacks of usual coherent modules.

The existence theorem obtained essentially says that Coh_{p}(X_{A}) entirely consists of such pullbacks. More precisely: If a family A_{i}, i ∈ I, of usual analytic 𝒪_{S,o}-subalgebras of A is chosen such that A = \underrightarrow{lim} A_{i}, then the natural functor \underrightarrow{lim} Coh_{p}(X_{A_{i}}) → Coh_{p}(X_{A}) is an equivalence of categories. This theorem answers a question of P. Deligne and J. Bingener [see Bingener, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 3, 317–347; MR0597743 (82i:32047)] where the existence theorem for the functor Coh_{p,f} has been proved, which is related to the existence of flat versal deformations of coherent sheaves with compact supports.

Whilst the existence theorem for Coh_{p,f} has an elegant proof, the case Coh_{p} needs the enormous machinery of the so-called POspaces developed by Bingener [Lokale Modulräume in der analytischen Geometrie, Band 1, Vieweg, Braunschweig, 1987; MR0912098 (89h:32040); Band 2; MR0912099 (89h:32041)], using ideas of Palamodov. To follow the highly technical proof the reader has to know that apparatus. Details cannot be given in this review. However it should be mentioned that the same method also yields the existence theorem for the category An_{p}(A) of compact complex spaces over A: The functors \underrightarrow{lim} An_{p,f}(A_{i}) → An_{p,f}(A), \underrightarrow{lim} An_{p}(A_{i}) → An_{p}(A) are equivalences of categories. The author also considers the local situation with respect to X, which is not as satisfactory: If Coh′(X_{A}) is the category of germs of modules whose nonfree locus is finite, then \underrightarrow{lim} Coh′(X_{A_{i}}) → Coh′(X_{A}) is in general not an equivalence, but it is still surjective.

G. Trautmann (Zbl 641:32015)