%Article J.P. Demailly 
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\vskip1.4cm
\noindent {\bf HOLOMORPHIC MORSE INEQUALITIES}
\vskip0.5cm
\noindent {\sl Jean-Pierre DEMAILLY, Universit\'e de Grenoble I}
\vskip0.5cm
\noindent {\sl Series of Lectures given at the AMS Summer Institute on
Complex Analysis held in Santa Cruz, California, July 1989.}


\titr 1. Introduction|

Let $M$ be a compact $C^\infty$ manifold, $\dim_\rr M=m$, and $h$ a
Morse function, \ie a function such that all critical points are non
degenerate. The standard Morse inequalities relate the Betti numbers
$b_q=\dim H^q_{DR}(M,\rr)$ and the numbers
$$s_q = \# \hbox{~critical points of index~} q~,$$
where the index of a critical point is the number of negative
eigenvalues of the Hessian form $(\partial^2h/\partial x_i\partial
x_j)$. Specifically, the following ``strong Morse inequalities" hold~:
$$b_q-b_{q-1}+\cdots +(-1)^qb_0\le s_q-s_{q-1}+\cdots
+(-1)^qs_0\leqno(1.1)$$
for each integer $q\ge 0$. As a consequence, one recovers the ``weak
Morse inequalities" $b_q\le s_q$ and the expression of the
Euler-Poincar\'e characteristic
$$\chi (M) = b_0-b_1+\cdots + (-1)^mb_m = s_0-s_1+\cdots
+(-1)^ms_m~.\leqno(1.2)$$

The purpose of these lectures is to explain what are the complex
analogues of these inequalities for $\overline \partial$-cohomology
groups with values in holomorphic line (or vector) bundles, and to
present a few applications.

Let $X$ be a compact complex manifold, $n=\dim _\cc X$ and $E,F$
holomorphic vector bundles over $X$ with rank$\,E=1$, rank$\,F=r$. 
Assume that $E$ is endowed with a $C^\infty$ hermitian metric and denote
by $c(E)$ its curvature form. Then $ic(E)$ is a real $(1,1)$-form on $X$
(\cf \S2). We consider the $q$-index sets
$$\eqalign{
X(q,E) &=\left\{x\in X~;~ic(E)_x \hbox{~has~~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\}~,\cr
X(\le q,E) &= \bigcup_{1\le j\le q} X(j,E)~.\cr}$$
Observe that $X(q,E)$ and $X(\le q,E)$ are open subsets of $X$~.
Finally, we denote $h^q(F) = \dim H^q(X,{\cal O}(F))~.$

\tha Main Theorem|The cohomology groups with values in the tensor powers 
$E^k\otimes F$ satisfy the following asymptotic estimates as $k\to +\infty$.

\noindent {\rm (1.3) Weak Morse inequalities}~:
$$h^q(E^k\otimes F)\le r {k^n\over n!}\int_{X(q,E)} (-1)^q \big({i\over 2\pi
}c(E)\big)^n + o(k^n)~.$$

\noindent {\rm (1.4) Strong Morse inequalities}~:
$$\sum_{0\le j\le q} (-1)^{q-j}h^j(E^k\otimes F) \le r {k^n\over n!}
\int_{X(\le q,E)}(-1)^q\big({i\over 2\pi }c(E)\big)^n+o(k^n)~.$$

\noindent {\rm (1.5) Asymptotic Riemann-Roch formula}~:
$$\chi (E^k\otimes F) = r{k^n\over n!}\int_X \big({i\over 2\pi
}c(E)\big)^n + o(k^n)~.$$\fintha

Observe that (1.5) is in fact a weak consequence of the
Hirzebruch-Riemann-Roch formula. In that case, the lower order term
$o(k^n)$ is a polynomial of degree $n-1$ in $k$ whose coefficients
can be expressed in terms of the first Chern class $c_1(E)=
\{{i\over2\pi}c(E)\}$ and of the Todd class of $X$. For the individual
cohomology groups $H^q$, the lower order term is extremely complicated
and involves derivatives of the curvature tensors of $X,E,F$.

The above theorem was first proved in [De2] in 1985. It was largely
motivated by Siu's solution of the Grauert-Riemenschneider conjecture
([Siu2], 1984), which we will reprove below as a special case of a
stronger statement. The basic tool is a spectral theorem which describes
the eigenvalue distribution of the complex Laplace-Beltrami operators.
The original proof of [De2] was based partly on Siu's techniques and
partly on an extension of Witten's analytic proof [Wi, 1982] for the
standard Morse inequalities. Somewhat later Bismut [Bi] and quite
recently Getzler [Ge3] gave new proofs, both relying on an analysis of
the heat kernel in the spirit of Atiyah-Bott-Patodi's proof of the
Atiyah-Singer index theorem [A-B-P]. Although the basic idea is simple,
Bismut used deep results arising from probability theory (the Malliavin
calculus), while Getzler relied on his supersymmetric symbolic calculus
for spin pseudodifferential operators [Ge1].

We present here a simple heat equation proof, based essentially on 
Mehler's formula and elementary asymptotic estimates (\cf \S3 and \S4).
Our treatment is rather close to the proof of Bismut and includes
some technical arguments which have been explained to us in an illuminating 
way by Mohan Ramachandran [Ram] at the University of Chicago.  The next 
sections deal with various generalizations and applications~:

{\parindent=10pt
\item{$\bullet$ }in \S5, we obtain as a consequence a strong
characterization of Moishezon varieties in terms of the existence of
line bundles satisfying some integral positivity hypothesis of the
curvature. In particular, this gives a solution of
the Grauert-Riemenschneider conjecture. We also give a generalization to
all projective algebraic manifolds of G. Kempf's distortion inequalities
for ample line bundles over abelian varieties [Kem].

\item{$\bullet$ }in \S6, the case of $q$-convex manifolds is considered.
As shown by Thierry Bouche [Bou1], Morse inequalities then hold in
degrees $\ge q$. By an argument of Siu [Siu3,Siu4], these inequalities
imply a general a priori estimate for the Monge-Amp\`ere operator.

\item{$\bullet$ }in \S7, we present (with a different and more
elementary proof) Getzler's extension [Ge1] of holomorphic Morse
inequalities to the vector bundle case. Estimates for the
number of sections of all irreducible tensor powers are obtained.

\item{$\bullet$ }\S8 briefly discusses related open problems,
especially the degenerate case when $ic(E)$ has rank $<\dim X$ 
everywhere. Then the leading term in the asymptotic estimates is 
identically zero and a computation of the lower order terms is
desired.\par}

\titr 2. Hermitian connections, curvature and Laplace-Beltrami
operators|

Let $F$ be a complex vector bundle of rank $r$ over a smooth
differentiable manifold $M$. We denote by $C^\infty_q(M,F)$ the space of
smooth $q$-forms with values in $F$, \ie smooth sections of $\Lambda^q
T^\star M\otimes F$~. To fix notations, we recall here the basic
terminology used. A {\sl connection} $D$ on $F$ is a linear differential 
operator
$$D:C^\infty_q(M,F) \to C^\infty_{q+1}(M,F)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\hbox{deg}~f}f\wedge Du\leqno(2.1)$$
for all forms $f\in C^\infty_p(X,\cc)$, $u\in C^\infty_q(X,F)$. On an
open set $\Omega \subset M$ where $F$ is trivial, $F_{|\Omega }\simeq
\Omega \times \cc^r$, a connection $D$ can be written
$$Du=du+\Gamma \wedge u$$
where $\Gamma \in C^\infty_1(\Omega ,\hbox{Hom}(\cc^r,\cc^r))$ is an
arbitrary matrix of $1$-forms and $d$ acts componentwise. It is then
easy to check that
$$D^2u=(d\Gamma +\Gamma \wedge \Gamma )\wedge u \hbox{~on~} \Omega ~,$$
so that $D^2u=c(D)\wedge u$ for some global $2$-form
$$c(D) \in C^\infty_2(M,\hbox{Hom}(F,F))$$
called the curvature of $D$.

Assume now that $F$ is endowed with a $C^\infty$ hermitian metric along
the fibers and that the isomorphism $F_{|\Omega }\simeq \Omega \times
\cc^r$
is given by a $C^\infty$ frame $(e_\lambda )$. We then have a canonical
sesquilinear pairing
$$\matrix{
C^\infty_p(M,F) &\longrightarrow  &C^\infty_{p+q}(M,\cc)\cr
(u,v) &\longmapsto &\{u,v\}\cr}$$
given by
$$\{u,v\} = \sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda ,e_\mu \rangle \hbox{~~~~for~} u=\sum u_\lambda \otimes e_\lambda
~,~ v=\sum v_\mu \otimes e_\mu ~.$$
The connection $D$ is said to be {\sl hermitian} if it satisfies the
additional property
$$d\{u,v\} = \{Du,v\} + (-1)^{\hbox{deg~}u}\{u,Dv\}~.\leqno(2.2)$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma $. In this case
$c(D)^\star=-c(D)$, thus
$$ic(D) \in C^\infty_2(M,\hbox{Herm}(F,F))~.$$
\medskip

\noindent (2.3) {\sl Special case}. For a bundle $E$ of rank 1, the
connection form $\Gamma $ of a hermitian connection $D$ can be written
$\Gamma =-iA$ where $A$ is a {\sl real} $1$-form. Then we have $c(D) =
d\Gamma  = -idA$ and we will denote
$$B = ic(D) = dA~.$$
A phase change $u=ve^{i\theta }$ in the isomorphism $E_{|\Omega }\simeq
\Omega \times \cc$ replaces $A$ by the new connection form $A+d\theta $.

\medskip
\noindent (2.4) {\sl Complex analytic case}. If $M$ is a complex
manifold $X$, every connection $D$ can be splitted in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection~:
$$D=D'+D''~.$$
In a local trivialization given by a $C^\infty$ frame, one can write 
$$\eqalign{
D'u &= d'u + \Gamma '\wedge u~,\cr
D''u &=d''u + \Gamma ''\wedge u~,\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star $ in any orthonormal frame. Thus there
exists a unique hermitian connection corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the bundle $E$ itself has a {\sl holomorphic} structure. The unique
hermitian connection for which $D''=\overline
\partial$ is called the {\sl Chern connection} of $F$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$~, the metric is given
by some hermitian matrix $H=(h_{\lambda \mu })$ where $h_{\lambda \mu }
= \langle e_\lambda ,e_\mu \rangle$. Easy computations yield the
expression of the Chern connection~:
$$\cases{
D'u&$=\partial u+\overline H^{-1}\partial \overline H \wedge u$\cr
d''u &$=\overline \partial u$\cr
c(F) &$\buildu \hbox{def}\under = c(D) = -\overline \partial(\overline
H^{-1}\partial \overline H)$~.\cr}$$
For a rank 1 bundle $E$, the matrix $H$ is simply a positive weight
function $e^{-\varphi }$, $\varphi \in C^\infty (\Omega ,\rr)$, and we
find
$$c(E) = \partial \overline \partial \varphi ~.$$

\medskip
\noindent (2.5) {\sl Hodge theory}. Assume now that $M$ is a riemannian
manifold with metric $g=\sum g_{ij} dx_i\otimes dx_j$~. Given a $q$-form
$u$ on $M$ with values in $F$~, we consider the global $L^2$ norm
$$\|u\|^2 = \int_M |u(x)|^2d\sigma (x)$$
where $|u|$ is the pointwise hermitian norm and  $d\sigma $ the
riemannian volume form. The Laplace Beltrami operator associated to the
connection $D$ is
$$\Delta =DD^\star +D^\star D$$
where $D^\star$ is the (formal) adjoint of $D$~; the complex Laplace
operators $\Delta '$ and $\Delta ''$ are defined similarly. In degree $0$
we simply have $\Delta =D^\star D$~.

When $M$ is compact, the elliptic operator $\Delta $ has a discrete
spectrum
$$\lambda _1\le \lambda _2\le \cdots \le \lambda _j\le \cdots$$
and corresponding eigenfunctions $\psi _j\in C^\infty_q(M,F)\cap L^2$.
Our main goal is to obtain asymptotic formulae for the eigenvalues. For
that, we make use of the {\sl heat operator} $e^{-t\Delta }$. In the above
setting, the heat operator is the bounded hermitian operator associated
to the {\sl heat kernel}
$$K_t(x,y) =\sum^{+\infty}_{j=1} e^{-\lambda _jt} \psi _j(x) \otimes
\psi ^\star _j(y)~,$$
and $K_t \in C^\infty(]0,+\infty[\times M\times M,\hbox{Hom}(F,F))$. The
asymptotic distribution of eigenvalues can be recovered from the well
known (obvious) formula
$$\sum^{+\infty}_{j=1} e^{-\lambda _jt} = \int_M \hbox{tr}_F K_t(x,x)
d\sigma (x)~.$$

\titr 3. Asymptotic formulae for the heat kernel|
\vskip-40pt
\titr and the eigenvalue distribution|

Let $E,F\to M$ be complex vector bundles equipped with hermitian
connections, $D_k$ the associated connection on $E^k\otimes F$ and
$\Delta _k = D^\star _k D_k$ the Laplace-Beltrami operator acting on sections
of $E^k\otimes F$ (degree $0$). Finally, let $V\in C^\infty(M,\hbox{Herm}\,
(F,F))$ be an arbitrary hermitian endomorphism of $F\,$; we still denote by
$V$ the operator $\hbox{Id}_{E^k}\otimes V$ acting on $E^k\otimes F$~.

If $\Omega \subset\!\subset M$ is a smoothly bounded relatively compact
open subset of $M$, we consider the quadratic form
$$Q_{k,\Omega }(u) = \int_\Omega  {1\over k} |D_ku|^2 - \langle
Vu,u\rangle,$$
with domain equal to the Sobolev space $W^1_0(\Omega ,E^k\otimes F)=$
closure of the space of smooth sections with compact support in $\Omega$,
in the space $W^1_{\hbox{loc}}(M,E^k\otimes F)$ of sections which have
$L^2_{\hbox{loc}}$ coefficients as well as their first derivatives
(Dirichlet boundary conditions). This
quadratic form is associated to the second order elliptic operator
$$\square_k = {1\over k} D^\star _k D_k - V,$$
which is densely defined and selfadjoint on the Hilbert space
$W^1_0(\Omega ,E^k\otimes F)$.

We want to study the asymptotic eigenvalue distribution of $\square_k$
as $k\to +\infty$, as well as an asymptotic formula for the heat kernel
$e^{-t\square_k}$. The basic idea is extremely simple and reduces the
proof into two steps~:

{\parindent=10pt
\item{$\bullet$} show that the asymptotic estimates are purely local (up
to error terms of lower order) and can be obtained by freezing the
coefficents of the operators involved at any given point.

\item{$\bullet$} compute explicitly the heat kernel in the case of
connections with constant curvature, assuming that $M$ has a flat metric.
\par}

In order to see that the situation is local, let $(\psi _j)$ be a
partition of unity relative to an arbitrarily fine covering of
$\overline \Omega $, such that $\sum \psi ^2_j = 1$ near $\overline\Omega $. As
$\sum \psi _jd\psi _j = 0$ on $\Omega $, we then find
$$
\sum_j Q_{k,\Omega }(\psi _ju) - Q_{k,\Omega }(u) = {1\over k}
\int_\Omega  (\sum |d\psi _j|^2)|u|^2
\le O({1\over k}) |u|^2~.$$
By the minimax principle, it follows that the eigenvalues are shifted by
at most $O(1/k)$, thus the asymptotic distribution is not modified as
$k\to +\infty$.

Now, let $x^0\in M$ be a given point. We can choose coordinates $(x_1\ld
x_m)$ centered at $x^0$ such that $(\partial/\partial x_1\ld
\partial/\partial x_m)$ is orthonormal at $x^0$ and such that $B=ic(E)$
is ``diagonal" at $x^0$~:
$$B(x^0) = \sum^s_{j=1} B_jdx_j\wedge dx_{j+s}~;$$
here $2s\le m$ is the rank of the skew-symmetric $2$-form $B(x^0)$,
which may depend on $x^0$, and
$B_1\ge B_2\ge \cdots \ge B_s\ge 0$ are its non-zero eigenvalues.

Let us consider the operators obtained by ``freezing" the coefficients
at $x^0$. More specifically, we assume that

{\parindent=10pt
\item{$\bullet$} $E$ has constant curvature $B=\sum^s_{j=1}B_jdx_j\wedge
dx_{j+s}$. Then there is a local trivialization in which
$$\eqalign{D_Eu &= du - iA\wedge u~,\cr
A &= \sum^s_{j=1} B_jx_jdx_{j+s}~.\cr}$$

\item{$\bullet$} $D_F$ is flat.

\item{$\bullet$} $\Omega \simeq \rr^n$ and the metric $g$ is flat~:
$g=\sum dx_j\otimes dx_j$~.

\item{$\bullet$} the hermitian form $V$ is constant. We choose an
orthonormal frame of $F$ in which $V$ is diagonal, \ie
$$\langle Vu,u\rangle = \sum_{1\le \lambda \le r} V_\lambda |u_\lambda
|^2~.$$
\par}

The connection $D_k$ on $E^k\otimes F$ can then be written
$D_ku=du-ikA\wedge u$ and the quadratic form $Q_{k,\Omega }$ is given by
$$\displaylines{
Q_{k,\Omega }(u) = \int_{\rr^m} {1\over k} \left[
\sum_{{1\le j\le s}\atop {1\le \lambda \le r}}\Bigg(\Big|{\partial
u_\lambda \over \partial x_j}\Big|^2+\Big|{\partial u_\lambda \over
\partial x_{j+s}}-ikB_jx_ju_\lambda \Big|^2\Bigg) + \sum_{{j>2s}\atop
{1\le \lambda \le r}} \Big|{du_\lambda \over dx_j}\Big|^2\right]\cr
\hfill -\sum_{1\le \lambda \le r} V_\lambda |u_\lambda |^2~.\cr}$$
In this situation, $Q_{k,\Omega }$ is a direct sum of quadratic forms
acting on each component $u_\lambda $ and the computation of
$e^{-t\square_k}$ is reduced to the following simple cases (a), (b)~:
$$Q(f) = \int_{\rr} \big|{df\over dx}\big|^2~,~~~\square f= - {d^2f\over
dx^2}\leqno\hbox{\rm(a)}$$
As is well known the heat kernel is given in this case by
$$K_t(x,y) = {1\over \sqrt {4\pi t}} e^{-(x-y)^2/4t}~.$$
$$Q(f) = \int_{\rr^2} \Big|{df\over dx_1}\Big|^2 + \Big|{df\over
dx_2}-iax_1f\Big|^2~.\leqno\hbox{\rm(b)}$$
A partial Fourier transform in the $x_2$ variable gives 
$$Q(f) = \int_{\rr^2} \Big|{d\hat f\over dx_1}(x_1,\xi
_2)\Big|^2+a^2(x_1-{\xi _2\over a})^2|\hat f(x_1,\xi _2)|^2$$
and the change of variables $x'_1=x_1-x_2/a$, $x'_2 = \xi _2$ leads to
the so called ``harmonic oscillator" energy functional
$$\eqalign{
q(g) &= \int_{\rr}\big|{dg\over dx}\big|^2+a^2x^2|g|^2~,\cr
\square &= - {d^2\over dx^2} + a^2x^2~.\cr}$$
The heat kernel of this operator is given by Mehler's formula~:
$$k_t(x,y) = \sqrt{a\over 2\pi \sinh 2at} \exp \Big(-{a\over 2}(\coth
2at)(x-y)^2 - a(\tanh at)xy\Big)~.$$
One way of obtaining this relation is to observe that the eigenfunctions
of $\square$ are
$$(2^pp!\sqrt{{\pi \over a}})^{-1/2} \Phi _p(\sqrt a
x)~,~p=0,1,2,\ldots~,$$
with associated eigenvalues $(2p+1)a$, where $(\Phi _p)$ is the sequence
of functions associated to Hermite polynomials~:
$$\Phi _p(x) = e^{x^2/2}{d^p\over dx^p} (e^{-x^2})~.$$
Therefore we have
$$k_t(x,y) = \sqrt{a\over \pi } e^{a(x^2+y^2)/2} \sum^{+\infty}_{p=0}
{e^{-(2p+1)at}\over 2^pp!a^p} {d^p\over dx^p} (e^{-ax^2}) {d^p\over dy^p}
(e^{-ay^2})$$
and the summation $\Sigma  (x,y)$ can be computed from its Fourier
transform
$$\widehat \Sigma (\xi ,\eta ) = e^{-at} \exp (-{1\over 2a} e^{-2at} \xi
\eta )\cdot {1\over \sqrt {2a}} e^{-(\xi ^2+\eta ^2)/4a}~.$$
The heat kernel operator of $Q$ is thus given by
$$(e^{-t\square}f)^\wedge (x_1,\xi _2) = \int_{\rr} k_t\big(x_1-{\xi
_2\over a},y_1-{\xi _2\over a}\big) \hat f(y_1,\xi _2)dy_1~.$$
By an inverse Fourier transform we obtain the desired heat kernel~:
$$\displaylines{
k_t(x_1,x_2;y_1,y_2)={a\over 4\pi  \sinh at} \exp \Big(-{a\over
4}(\coth at)\big((x_1-y_1)^2 + (x_2-y_2)^2\big)\Big)\cr
\hfill \times \exp
\big({i\over 2}a(x_1+y_1)(x_2-y_2)\big)~.\cr}$$
The heat kernel associated to a sum of pairwise commuting operators
$\square_1\ld \square_m$ acting on disjoint sets of variables is the
product of all heat kernels $e^{-t\square_j}$. Let $K^\lambda _t$ be the heat kernel
of $Q_{k,\Omega }$ acting on a single component $u_\lambda $. The factor
in the heat kernel corresponding to the pair of variables
$(x_j,x_{j+s})$, $1\le j\le s)$ is obtained when substituting $kB_j$ to
$a$ and $t/k$ to $t$. Therefore
$$\displaylines{
K^\lambda _t(x,y) = \prod^s_{j=1} {kB_j\over 4\pi  \sinh B_jt} \exp
\Big(-{kB_j\over 4} (\coth B_jt)\big((x_j-y_j)^2 +
(x_{j+s}-y_{j+s})^2\big)\cr
\hfill + {i\over 2} kB_j (x_j+y_j)(x_{j+s}-y_{j+s})\Big)\cr
\hfill \times e^{tV_\lambda }\times {1\over (4\pi t/k)^{m-2s}/2} \exp
\big(-k\sum_{j>2s} (x_j-y_j)^2/4t\big)~.\cr}$$
On the diagonal of $M$, the global heat kernel $K_t$ is thus given by
$$K_t(x,x) = k^{m/2} {e^{tV}\over (4\pi )^{m/2}t^{m/2-s}} \prod^s_{j=1}
{B_j\over \sinh B_jt}~.$$

\tha Theorem 3.1|In the variable coefficient case, the heat kernel of
$\square_k$ admits the asymptotic estimate
$$K^k_t(x,x) \sim k^{m/2} {e^{tV(x)}\over (4\pi )^{m/2}t^{m/2-s}}
\prod^s_{j=1} {B_j(x)\over \sinh B_j(x)t}$$
as $k\to +\infty$, where $\sim$ is uniform with respect to $x\in\Omega'
\subset\!\subset\Omega$ and
$t$ in a bounded interval $[t_0,t_1] \subset ]0,+\infty[$~.\fintha

\remarque Proof|The only thing one has to get convinced of is that
$e^{-t\square_k} - e^{-t\square^0_k}$ is $o(k^{m/2})$ at the point
$(x^0,x^0)\in M\times M$~, where $\square^0_k$ is the operator
$\square_k$
``freezed" at $x^0$. This can be cheecked by means of the well known
formula
$$e^{-t\square_k} - e^{-t\square^0_k} = \int^1_0
e^{-(1-u)t\square^0_k}(\square^0_k-\square_k)e^{-ut\square^0_k} du~,$$
once the singularity of $e^{-t\square^0_k}$ along the diagonal is known.
We also use the estimate
$$\square_k-\square^0_k = O\Big({1\over k}|x-x^0|\nabla^2+\big({1\over
k}+|x-x^0|^2\big)\nabla + |x-x^0|+{1\over k}\big)~.\finpr$$

By the localization argument already discussed, we obtain as a
consequence the following estimate for the eigenvalues~:

\tha Corollary 3.2|The eigenvalues $\lambda ^{k,\Omega }_j$ of
$Q_{k,\Omega }$ satisfy for every $t>0$ the estimate
$$\sum^{+\infty}_{j=1} e^{-t\lambda ^{k,\Omega }_j} \sim k^{m/2}
\int_\Omega  {\tr (e^{tV(x)})\over (4\pi )^{m/2} t^{m/2-s}} \prod^s_{j=1}
{B_j(x)\over \sinh B_j(x)t} d\sigma (x)~.$$\fintha

This result can be also interpreted in terms of the counting function
$$N_{k,\Omega }(\lambda ) = \# \{j~;~\lambda ^{k,\Omega }_j \le \lambda
\}$$
and of the spectral density measure (a sum of Dirac measures)
$$\mu _{k,\Omega } = k^{-m/2} {d\over d\lambda } N_{k,\Omega }(\lambda )~.$$
In these notations, corollary 3.2 can be restated~:
$$\lim_{k\to +\infty} \int^{+\infty}_{-\infty} e^{-t\lambda } d\mu
_{k,\Omega }(\lambda ) = \int_\Omega  {\tr (e^{tV(x)})\over (4\pi
)^{m/2} t^{m/2-s}} \prod^s_{j=1} {B_j(x)\over \sinh B_j(x)t} d\sigma
(x)~.$$
We thus see that the sequence of measures $\mu _{k,\Omega }$ converges
weakly to a measure $\mu _\Omega $ whose Laplace transform is given by
the right hand side. Inverting the formula, we obtain~:

\tha Corollary 3.3|For almost all $\lambda \in \rr$
$$\lim_{k\to +\infty} k^{-m/2} N_{k,\Omega }(\lambda ) = \mu _\Omega
(]-\infty,\lambda ]) = \int_\Omega \sum^r_{j=1} \nu _{B(x)}
(V_j(x)+\lambda )d\sigma (x),$$
where $\nu _{B(x)}(\lambda )$ is the function on $M\times \rr$ defined
by
$$\nu_B(\lambda ) = {2^{s-m}\pi ^{-m/2}\over \Gamma ({m\over 2}-s+1)}
B_1\cdots B_s \sum_{(p_1\ld p_s)\in \nn^s} [\lambda -\Sigma
(2p_j+1)B_j]^{{m\over 2}-s}_+~.$$\fintha


\titr 4. Proof of the holomorphic Morse inequalities|

Let $X$ be a compact complex manifold, $E$ and $F$ holomorphic hermitian
vector bundles of rank $1$ and $r$ over $X$. If $X$ is endowed with a
hermitian metric $\omega $, Hodge theory shows that the Dolbeault
cohomology group $H^q(X,E^k\otimes F)$ can be identified with the space
of harmonic $(0,q)$-forms with respect to the Laplace-Beltrami operator
$\Delta''_k =D''_k D^{\prime\prime\star}_k+D^{\prime\prime\star}_k D''_k$
acting on $E^k\otimes F$~. We thus have to estimate the zero-eigenspace of
$\Delta ''_k$~.

In order to apply corollary 3.2, we first have to compute $\Delta ''_k$
in terms of the hermitian connection $\nabla_k$ on $E^k\otimes F\otimes
\Lambda ^{0,q}T^\star X$ deduced from the Chern connections of $E,F,TX$. What
plays now the role of $F$ is the bundle $F\otimes \Lambda ^{0,q}T^\star X$~.

The relationship between $\Delta ''_k$ and $\nabla_k$ is most easily
obtained by means of the Bochner-Kodaira-Nakano identity. In order to
simplify the exposition, we assume here that the metric $\omega $ on $X$
is {\sl K\"ahler}. For any hermitian holomorphic line bundle $G$ on $X$,
the operators $\Delta '$ and $\Delta ''$ of $G$ are related by the B-K-N
identity
$$\Delta '' = \Delta ' + [ic(G),\Lambda ]~.\leqno(4.1)$$
Here we have $c(E^k\otimes F) = kc(E) \otimes \id_F + c(F)$, thus 
$$\Delta ''_k = \Delta '_k + k[ic(E),\Lambda ] + [ic(F),\Lambda ]~.$$
At a given point $z^0\in X$~, we can find a coordinate system $(z_1\ld
z_n)$ such that $(\partial/\partial z_j)$ is an orthonormal basis of
$TX$ and
$$ic(E) = {i\over 2}\sum_{1\le j\le n} \alpha _jdz_j\wedge d\overline
z_j,$$
where $\alpha _1\ld \alpha _n$ are the curvature eigenvalues of $c(E)$
at $z^0$. A standard formula gives the expression of the curvature term
$[ic(E),\Lambda ]u$ for any $(p,q)$-form $u$. With $u=\sum u_{I,J,\lambda
}dz_I\wedge d\overline z_J\otimes e_\lambda $, we have
$$\langle [ic(E),\Lambda ]u,u\rangle = \sum_{I,J,\lambda }(\alpha
_J-\alpha _{\complement I})|u_{I,J,\lambda }|^2$$
where $\alpha _J = \sum_{j\in J}\alpha _j$~. In the case of a
$(0,q)$-form $u=\sum u_{J,\lambda }d\overline z_J \otimes e_\lambda $ we
simply have $\Delta '_ku= D^{\prime\star }_k D'_ku=\nabla^{\prime \star }_k \nabla'_ku$
and
$$\Delta ''_k=\nabla^{\prime \star }_k\nabla'_k - kV'+[ic(F),\Lambda
]~,\leqno(4.2')$$
$$\langle V'u,u\rangle = \sum_{J,\lambda }\alpha _{\complement J}
|u_{J,\lambda }|^2 ~\hbox{~~~(here~} I=\emptyset)~.$$
This is not yet what was needed, since only the $(1,0)$ part $\nabla'_k$
appears. To get the $(0,1)$ component, we consider $u$ as a $(n,q)$ form
with values in $E^k\otimes F\otimes \Lambda ^nTX$. We then get
$\Delta '_ku = D'_k D^{\prime \star }_ku$ where
$$D^{\prime \star }_ku= - \sum \partial u_{I,J,\lambda }/\partial \overline
z_jdz_1\wedge \cdots \widehat{dz_j}\cdots \wedge dz_n\wedge d\overline
z_J\otimes e_\lambda $$
in normal coordinates. Thus $\Delta '_ku =
\nabla^{\prime\prime\star }_k \nabla''_k u$ and
$$\Delta ''_k= \nabla^{\prime\prime \star }_k\nabla''_k + kV'' + [ic(F\otimes
\Lambda ^nTX),\Lambda ]~,\leqno (4.2'')$$
$$\langle V''u,u\rangle = \sum_{J,\lambda } \alpha _J|u_{J,\lambda }|^2
~~\hbox{~~(here~} I=\{1\ld n\})~.$$
If the metric $\omega $ is non K\"ahler, we get additional torsion
terms, but these terms are independent of $k$. A combination of $(4.2')$
and $(4.2'')$ yields
$${2\over k} \Delta ''_k = {1\over k} \nabla^\star _k\nabla_k - V + {1\over
k} \Theta, \leqno(4.3)$$
where $\Theta $ is a hermitian form independent of $k$ and
$$\langle Vu,u\rangle = \sum_{J,\lambda } (\alpha _{\complement J} -
\alpha _J)|u_{J,\lambda }|^2~.$$
Now apply theorem 3.1 and observe that $\Theta $ does not give any
significant contribution to the heat kernel as $k\to +\infty$~. We write
here $z_j = x_j+iy_j$, so that
$$B=ic(E) = \sum_{1\le j\le n} \alpha _jdx_j\wedge dy_j~.$$
The curvature eigenvalues are given by $B_j=|\alpha _j|$. We denote
$s=s(x)$ the rank of $B(x)$ and order the eigenvalues so that
$$|\alpha _1|\ge \cdots \ge |\alpha _s| >0 = \alpha _{s+1}=\cdots =
\alpha _n~.$$
The eigenvalues of $V$ acting on $F\otimes \Lambda ^nT^\star X$ are the
coefficients $\alpha _{\complement J} - \alpha _J$, counted with
multiplicity $r$~. Therefore

\tha Theorem 4.4|The heat kernel associated to $e^{-{2t\over k}\Delta''_k}$ 
in bidegree $(0,q)$ satisfies
$$K^k_t(x,x) \sim k^n {r\sum_{|J|=q}e^{t(\alpha _{\complement
J}(x)-\alpha _J(x))}\over (4\pi )^nt^{n-s}} \prod^s_{j=1} {|\alpha
_j(x)|\over \sinh |\alpha _j(x)|t}$$
as $k\to +\infty$~. In particular, if $\lambda ^{k,q}_1 \le \lambda
^{k,q}_2\le \cdots$ are the eigenvalues of ${1\over k}\Delta ''_k$ in
bidegree $(0,q)$, we have
$$\sum^{+\infty}_{j=1} e^{-2t\lambda ^{k,q}_j} \sim rk^n
\sum_{|J|=q}\int_X {e^{t(\alpha _{\complement J}(x)-\alpha _J(x))}\over
(4\pi )^nt^{n-s}} \prod^s_{j=1} {|\alpha _j(x)|\over \sinh|\alpha
_j(x)|t}$$
for every $t>0$~.\fintha

At this point, the main idea is to use the eigenspaces in order to
construct a finite dimensional subcomplex of the Dolbeault complex with
the same cohomology groups. This was already the basic idea in Witten's
analytic proof of the standard Morse inequalities. We denote by
$${\cal H}^{k,q}_\lambda ~~,~~\hbox{resp.}~~{\cal H}^{k,q}_{\le \lambda }$$
the $\lambda$--eigenspace of ${1\over k} \Delta ''_k$ acting on
$C^\infty_{0,q}(X,E^k\otimes F)$~, resp. the direct sum of eigenspaces
corresponding to all eigenvalues $\le \lambda $~. As $D''_k=\overline
\partial_k$ and $\Delta ''_k$ commute, we see that $\overline \partial
({\cal H}^{k,q}_\lambda ) \subset {\cal H}^{k,q+1}_\lambda $, thus
${\cal H}^{k,\bullet}_\lambda $ and ${\cal H}^{k,\bullet}_{\le \lambda }$
are finite dimensional subcomplexes of the Dolbeault complex
$$\overline\partial~: C^\infty_{0,\bullet}(X,E^k\otimes F)~.$$
Since $\overline \partial_k\overline \partial^\star _k + \overline
\partial^\star _k \overline \partial_k = \Delta ''_k = k\lambda \id$ on
${\cal H}^{k,\bullet}_\lambda $~, we find that ${\cal
H}^{k,\bullet}_\lambda $ has trivial cohomology for $\lambda \ne 0$~.
Since ${\cal H}^{k,\bullet}_0$ is the space of harmonic forms, we see
that
${\cal H}^{k,\bullet}_{\le \lambda }$ has the same cohomology as the
Dolbeault complex for $\lambda >0$~. We will call this complex the
Witten $\overline \partial$-complex. We need an elementary lemma of
linear algebra.

\tha Lemma 4.5|Set $h^q_k = \dim H^q(X,E^k\otimes F)$. Then for every
$t>0$ 
$$h^q_k-h^{q-1}_k+\cdots+(-1)^qh^0_k \le
\sum^q_{\ell=0}(-1)^{q-\ell}\sum^{+\infty}_{j=1}e^{-t\lambda
^{k,\ell}_j}~.$$
\fintha

\remarque Proof|The left hand side is the contribution of the $0$
eigenvalues in the right hand side. All we have to check is that the
contribution of the other eigenvalues is $\ge 0$. The contribution of
the eigenvalues such that $\lambda ^{k,\ell}_j = \lambda >0$ is
$$e^{-t\lambda }\sum^q_{\ell=0} (-1)^{q-\ell} \dim {\cal
H}^{k,\ell}_\lambda ~.$$
As ${\cal H}^{k,\bullet}_\lambda $ is exact, one easily sees that the last
sum is equal to the dimension of $\overline \partial{\cal H}^{k,q}_\lambda
\subset {\cal H}^{k,q+1}_\lambda $, hence $\ge 0$.\finpr

Combining theorem 4.4 with lemma 4.5, we get
$$\eqalign{
h^q_k&{}-h^{q-1}_k +\cdots + (-1)^q h^0_k\cr
&{}\le rk^n \sum^q_{\ell=0} (-1)^{q-\ell}\sum_{|J|=\ell} \int_X
{\prod_{j\le s}|\alpha _j|\cdot e^{t(\alpha _{\complement J}-\alpha
_J-\sum|\alpha _j|)}\over
2^{2n-s}\pi ^nt^{n-s} \prod_{j\le s} (1-e^{-2t|\alpha _j|})}+o(k^n)~.\cr}$$
This inequality is valid for any $t>0$, so we can let $t$ tend to
$+\infty$. It is clear that $\alpha _{\complement J}-\alpha _J -
\sum|\alpha _j|$ is always $\le 0$, thus the integrand tends to $0$ at
every point where $s<n$. When $s=n$, we have $\alpha _{\complement J}(x)
- \alpha _Jx) -\sum|\alpha _j(x)|=0$ if and only if $\alpha _j(x) >0$
for every $j\in \complement J$ and $\alpha _j(x)<0$ for every $j\in J$.
This implies $x\in X(\ell,E)$~; in this case there is only one
multi-index $J$ satisfying the above conditions and the limit is $(2\pi
)^{-n}|\alpha _1\cdots \alpha _n|$. By the monotone convergence theorem,
our sum of integrals converges to
$$\sum^q_{\ell=0} (-1)^{q-\ell}\int_{X(\ell,E)}(2\pi )^{-n} |\alpha
_1\cdots \alpha _n| d\sigma  = {1\over n!}\int_{X(\le q,E)} (-1)^q
({i\over 2\pi }c(E))^n~.$$

\titr 5. Applications to algebraic geometry|

Let $E$ be a holomorphic line bundle over a compact connected complex
manifold $X$ of dimension $n$ and $V_k = H^0(X,E^k)$. If $Z(V_k)$
denotes the common zero set of all sections in $V_k$, there is a
natural holomorphic map
$$\Phi _k~: X\backslash Z(V_k) \longrightarrow  \Bbb P(V^\star _k)$$
which sends a point $x\in X\backslash Z(V_k)$ to the hyperplane
$H\subset V_k$ of sections $\sigma \in V_k$ such that $\sigma (x)=0$.

When $E$ is $>0$, one can construct many sections of high tensor powers
$E^k$ (e.g. by H\"ormander's $L^2$ estimates). For $k\ge k_0$ large
enough, the ``base locus" $Z(V_k)$ is empty, the sections in $V_k$
separate any two points of $X$ and generate all $1$-jets at any point.
Then $\Phi _k$ gives an embedding of $X$ in some projective space $\Bbb
P^N$.This gives the famous {\sl Kodaira embedding theorem}~: a compact
complex manifold $X$ is projective algebraic if and only if $X$ has a
hermitian line bundle $E$ with positive curvature.

The Grauert-Riemenschneider conjecture [G-R] was an attempt to
characterize the more general class of Moishezon varieties in terms of
semi-positive line bundles. Let us first recall a few definitions. The
algebraic dimension $a(X)$ is the transcendence degree
of the field ${\cal M}(X)$ of meromorphic functions on $X$. A well-known
theorem of Siegel asserts that $0\le a(X)\le n$. A manifold (or variety) $X$ is
said to be {\sl Moishezon} if $a(X)=n$~.
\medskip

\tha Grauert-Riemenschneider conjecture (1970)|A compact complex variety
$Y$ is Moishezon if and only if there is
a proper non singular modification $X\to Y$ and a line bundle $E$ over
$X$ such that the curvature of $E$ is $>0$ on a dense open subset.\fintha

When $Y$ is Moishezon, it is well known that there exists a projective
algebraic modification $X$~; therefore $E$ can even be taken $>0$
everywhere on such an $X$. Siu [Siu2] solved the conjecture by proving
the converse statement in 1984~; he even showed that  $X$ is Moishezon as soon as $ic(E)\ge 0$
everywhere and $ic(E)>0$ in at least one point. We will see that Morse
inequalities give a still stronger criterion, requiring only the
positivity of some curvature integral.

Since ${\cal M}(Y) \simeq {\cal M}(X)$, we only have to show that $X$
itself is Moishezon. This will be done by producing many sections of
$E^k$. For $x=1$, the strong Morse inequality (1.4) gives
$$h^1(E^k) - h^0(E^k) \le - {k^n\over n!} \int_{X(\le 1,E)} \Big({i\over
2\pi }c(E)\Big)^n + o(k^n)~.$$
In particular, we get the lower bound
$$h^0(E^k) \ge {k^n\over (2\pi )^nn!} \int_{X(\le 1,E)}
(ic(E))^n-o(k^n)~.\leqno(5.1)$$
By definition, the {\sl Kodaira dimension} $\kappa (E)$ is the supremum of the
dimension of the images $Y_k=\Phi _k(X\backslash Z(V_k))\subset \Bbb
P(V^\star _k)$ for all integers $k>0$. Since the field of meromorphic
functions on $X$ obtained by restriction of rational functions of $\Bbb
P(V^\star _k)$ to $Y_k$ has transcendence degree $\dim Y_k$, we infer that
$\kappa (E) = \sup\dim Y_k \le a(X)$. The following elementary lemma is
needed.

\tha Lemma 5.2|For every line bundle $E$, there is a constant $C>0$ such
that
$$\dim H^0(X,E^k) \le C k^{\kappa (E)}~.$$\fintha

The proof proceeds as follows~: select a hermitian metric on $E$ and a
family of balls $B_j = B(z_j,r_j) \subset B'_j = B(z_j,2r_j)$ covering
$X$, on which $E$ is trivial. If $E^k$ had too many sections, one could
solve a linear system in many unknowns to get a section $s$ vanishing at
a high order $m$ at all centers $z_j$. Then Schwarz'lemma gives
$$\|s\|_\infty = \sup\|s\|_{B_j} \le 2^{-m} C^k \sup \|s\|_{B'_j} \le
2^{-m}C^k\|s\|,$$
where $C$ is the oscillation of the metric on $B'_j$. Thus $m\le k\log
C/\log 2$ if \hbox{$s\ne 0$.} Since the sections of $E^k$ are essentially
constant along the fibers of $\Phi _k$, only $m^{\dim Y_k}\# \{z_j\}$
equations are needed to make $s$ vanish at order $m$~. Therefore we can
choose $m\simeq C_1 h^0(E^k)^{1/\dim Y_k}$, so that
$$h^0(E^k) \le C_2 m^{\dim Y_k} \le C_3 k^{\kappa (E)}~.\finpr$$

Combining (5.1) and lemma 5.2, we get the following result which implies
the Grauert-Riemenschneider conjecture.

\tha Theorem 5.3|If a hermitian line bundle $E$ satisfies the integral
condition $\int_{X(\le 1,E)}(ic(E))^n>0$, then $\kappa (E)=n$, in
particular $X$ is Moishezon.\finpr\fintha

Another application of the heat kernel estimates is a generalization of G.
Kempf's distortion inequalities ([Kem], [Ji]) to all projective
algebraic manifolds.

Let $E$ be a positive hermitian line bundle over a projective manifold
$X$, equip\-ped with a hermitian metric $\omega $. Then $V_k=H^0(X,E^k)$
has a natural hermitian metric given by the global $L^2$ norm of
sections.
For $k\ge k_0$ large enough, $\Phi _k$ is an embedding and $E^k$ can be
identified to the pull-back $\Phi ^\star _k O(1)$. We want to compare the
original metric $|~~~|$ of $E$ and the metric $|~~~|_{FS}$ induced by the
Fubini-Study metric of $O(1)$.

Let $(s_1\ld s_N)$ be an orthonormal basis of $H^0(X,E^k)$. It is not
difficult to check that
$$|\xi |^2_{FS} = {|\xi |^2\over |s_1(x)|^2+\cdots +|s_N(x)|^2}
\hbox{~for~} \xi \in E^k_x~,$$
thus all that we need is to get an estimate of $\sum |s_j(x)|^2$~.
However, this sum is the contribution of the $0$ eigenvalue in the heat
kernel
$$K^k_t(x,x) = \sum^{+\infty}_{j=1} e^{-2t\lambda ^k_j} |\psi_j(x)|^2$$
associated to ${2\over k}\square''_k$ in bidegree $(0,0)$. We observe
that non zero eigenvalues $\lambda ^k_j$ are also eigenvalues in
bidegree $(0,1)$, since $\overline \partial$  is injective on the
corresponding eigenspaces. The associated eigenfunctions are $\overline
\partial \psi_j/\sqrt{k\lambda ^k_j}$, for 
$$\|\overline
\partial \psi_j\|^2 = \langle \Delta ''_k\psi_j,\psi_j\rangle = k\lambda
^k_j~.$$
Thus the summation
$$\sum^{+\infty}_{j=1} e^{-2t\lambda ^k_j}|\overline
\partial \psi_j(x)|^2$$
is bounded by the heat kernel in bidegree $(0,1)$, which is itself
bounded by $k^ne^{-ct}$ with $c>0$ (note that $\alpha _{\complement
J}-\alpha _J - \sum|\alpha _j|<0$ on $X$ for $|J|=1$). Taking
$t=k^\varepsilon $ with $\varepsilon $ small, one can check that all
estimates remain uniformly valid and that the contribution of the non
zero eigenfunctions in $K^k_t(x,x)$ becomes negligible in $C^0$ norm.
Then theorem 4.4 shows that
$$\sum|s_j(x)|^2 \sim K^k_t(x,x) \sim k^n(2\pi )^{-n}|\alpha _1(x)\cdots
\alpha _n(x)|$$
as $t=k^\varepsilon \to +\infty$. For $\xi \in E^k_x$ we get therefore
the $C^0$ convergence
$${|\xi |^2\over |\xi |^2_{FS}} \sim \big({k\over 2\pi }\big)^n|\alpha
_1(x)\cdots \alpha _n(x)| \hbox{~~~~as~} k\to +\infty~.$$
As a consequence, the Fubini-Study metric on $E$ induced by $\Phi _k$
converges to the original metric. G. Tian [Ti] has proved that this last
convergence statement holds in fact in norm $C^4$.


\titr 6. The case of $q$-convex manifolds|

Thierry Bouche [Bou1] has obtained an extension of the holomorphic Morse
inequalities to the case of strongly $q$-convex manifolds. We explain
here the main ideas used.

A complex (non compact) manifold $X$ of dimension $n$ is strongly
$q$-convex in the sense of Andreotti and Grauert [A-G] if there exists a
$C^\infty$ exhaustion function $\psi$ on $X$ such that $i\partial
\overline\partial\psi$ has at least $n-q+1$ positive eigenvalues outside a
compact subset of $X$. In this case, the Andreotti-Grauert theorem shows
that all cohomology groups $H^m(X,{\cal F})$ with values in a coherent
analytic sheaf are finite dimensional for $m\ge q$.

\tha Theorem 6.1|Let $E,F$ be holomorphic vector bundles over $X$ with rank
$E=1$, rand $F=r$. Assume that $X$ is strongly $q$-convex and that $E$
has a metric for which $ic(E)$ has at least $n-p+1$ nonnegative
eigenvalues outside a compact subset. Then for all $m\ge p+q-1$ the
following strong Morse inequalities hold~:
$$\sum^n_{\ell=m} (-1)^{\ell-m} \dim H^\ell(X,E^k\otimes F) \le
r{k^n\over n!} \int_{X(\ge m,E)} (-1)^m \big({i\over 2\pi } c(E)\big)^n
+ o(k^n)~.$$\fintha

\remarque Proof|For every $c\in \rr$, we consider the sublevel sets
$$X_c=\{x\in X~;~\psi(x)<c\}~.$$
Select $c_0$ such that $i\partial\overline\partial\psi$ has $n-q+1$
positive eigenvalues on $X\backslash X_c$. One can choose a hermitian
metric $\omega _0$ on $X$ in such a way that the eigenvalues $\gamma
^0_1\le \cdots \le \gamma ^0_n$ of $i\partial\overline\partial\psi$ with
respect to $\omega _0$ satisfy
$$-{1\over n} \le \gamma ^0_1 \le \cdots \le \gamma ^0_{q-1}\le 1
\hbox{~~~and~~~} \gamma ^0_q = \cdots = \gamma ^0_n = 1 \hbox{~on~} X\backslash
X_{c_0}~;\leqno(6.2)$$
this can be achieved by taking $\omega _0$ equal to
$i\partial\overline\partial\psi$ on a $C^\infty$ subbundle of $TX$ of
rank $n-q+1$ on which $i\partial\overline\partial\psi$ is positive, and
$\omega _0$ very large on the orthogonal complement. We set $\omega
=e^\rho \omega _0$ where $\rho $ is a function increasing so fast at
infinity that $\omega $ is complete.

More important, we multiply the metric of $E$ by a weight $e^{-\chi
\circ \psi }$ where $\chi $ is a convex increasing function. The
resulting hermitian line bundle is denoted $E_\chi $. For any $(0,m)$
form $u$ with values in $E^k\otimes F$, viewed as an $(n,m)$ form with
values in $E^k\otimes F\otimes \Lambda ^nTX$, the Bochner-Kodaira-Nakano
formula implies an inequality
$$\langle \Delta ''_ku,u\rangle \ge \int_X k\langle [ic(E_\chi ),\Lambda
]u,u\rangle + \langle \Theta u,u\rangle,$$
where $\Theta $ depends only on the curvature of $F\otimes \Lambda ^nTX$
and the torsion of $\omega $. By the formulas of \S4, we have
$$\langle [ic(E_\chi ),\Lambda ]u,u\rangle \ge (\alpha _1+\cdots+\alpha
_m)|u|^2$$
where $\alpha _1\le \cdots \le \alpha _n$ are the eigenvalues of
$$ic(E_\chi ) = ic(E) + i\partial\overline\partial(\chi \circ \psi ) \ge
ic(E) +( \chi '\circ \psi) i\partial\overline\partial \psi ~.$$
If $\beta $ is the lowest eigenvalue of $ic(E)$ with respect to $\omega
$, we find
$$\eqalign{
\alpha _j &\ge \beta +(\chi '\circ \psi )\gamma ^0_j/e^\rho ~,\cr
\alpha _1+\cdots + \alpha _m &\ge m\beta  + (\chi '\circ \psi )(\gamma
^0_1+\cdots + \gamma ^0_m)/e^\rho ~,\cr}$$
and by (6.2) we get for all $m\ge q$~:
$$\alpha _1+\cdots + \alpha _m \ge m\beta  + {1\over n} e^{-\rho }\chi
'\circ \psi  \hbox{~on~} X\backslash X_{c_0}~.$$
It follows that one can choose $\chi $ increasing very fast in such a
way that the Bochner inequality becomes
$$\langle \Delta ''_ku,u\rangle \ge k\int_{X\backslash
C_{c_0}}A(x)|u(x)|^2 - C\int_X|u(x)|^2,\leqno(6.3)$$
where $A\ge 1$ is a function tending to $+\infty$ at infinity on $X$ and
$C\ge 0$. Now, Rellich's lemma easily shows that
$\Delta ''_k$ has a compact resolvent. Hence the spectrum of $\Delta
''_k$ is discrete and its eigenspaces are finite dimensional. Standard
arguments also show the following~:

\tha Lemma 6.4|When $\chi $ increases sufficiently fast at infinity, the
space\break
${\cal H}^m(X,E^k_\chi  \otimes F )$ of $L^2$-harmonic forms of bidegree
$(0,m)$ for $\Delta ''_k$ is isomorphic to the cohomology group
$H^m(X,E^k\otimes F)$ for all $k\in \nn$ and $m\ge q$~.\fintha

For a domain $\Omega \subset\!\subset X$, we consider the quadratic form
$$Q^{k,m}_\Omega (u) = {1\over k} \int_\Omega  |\overline \partial_ku|^2
+ |\overline\partial^\star _ku|^2,$$
with Dirichlet boundary conditions on $\partial \Omega $. We denote by
${\cal H}^{k,m}_{\le \lambda ,\Omega }$ the direct sum of all
eigenspaces of $Q^{k,m}_\Omega $ corresponding to eigenvalues $\le
\lambda $ (\ie eigenvalues $\le k\lambda $ for $\Delta ''_k$).

\tha Lemma 6.5|For every $\lambda \ge 0$ and $\varepsilon >0$, there
exists a domain $\Omega \subset \!\subset X$ and an integer $k_0$ such
that
$$\dim {\cal H}^{k,m}_{\le \lambda ,\Omega } \le \dim {\cal
H}^{k,m}_{\le \lambda ,X} \le \dim {\cal H}^{k,m}_{\le \lambda
+\varepsilon ,\Omega}~~~~\hbox{\rm for}~~k\ge k_0~.$$\fintha

\remarque Proof|The left hand inequality is a straightforward consequence
of the minimax principle, because the domain of the global quadratic
form $Q^{k,m}_\Omega $ is
contained in the domain of $Q^{k,m}_X$.

For the other inequality, let $u\in {\cal H}^{k,m}_{\le \lambda ,X}$~.
Then (6.3) gives
$$k\int_{X\backslash X_{c_0}} A|u|^2-C\int_{X_{c_0}}|u|^2\le k\lambda
\int_X |u|^2~.$$
Choose $c_2>c_1>c_0$ so that $A(x) \ge a$ on $X\backslash X_{c_1}$ and a
cut-off function $\varphi $ with compact support in $X_{c_2}$ such that
$0\le \varphi \le 1$ and $\varphi =1$ on $X_{c_1}$. Then we find
$$\int_{X\backslash X_{c_1}} |u|^2 \le {C+k\lambda \over ka} \int_X
|u|^2~.$$
For a large enough, we get 
$\int_{X\backslash X_{c_1}}|u|^2\le \varepsilon \|u\|^2$. Set $\Omega
=X_{c_2}$. Then
$$\eqalign{
Q^{k,m}_\Omega (\varphi u) 
&= {1\over k} \int_\Omega  |\overline \partial \varphi \wedge u +
\varphi \overline \partial_ku|^2+|\varphi \overline\partial
^\star _ku-\partial \varphi \ort u|^2\cr
&\le (1+\varepsilon )Q^{k,m}_X(u) + {C'\over k}\Big(1+{1\over \varepsilon
}\Big)\|u\|^2\cr
&\le (1+\varepsilon )(\lambda +{C'\over k\varepsilon })\|u\|^2~.\cr}$$
As $\|\varphi u\|^2\ge \int_{X_{c_1}}|u|^2 \ge (1-\varepsilon
)\|u\|^2$~, we infer
$$Q^{k,m}_\Omega (\varphi u) \le {1+\varepsilon \over 1-\varepsilon
}\Big(\lambda +{C'\over k\varepsilon }\Big)\|\varphi u\|^2~.$$
If $\varepsilon $ is replaced by a suitable smaller number and $k$ taken
large enough, we obtain $Q^{k,m}_\Omega (v) \le (\lambda +\varepsilon
)\|v\|^2$ for all $v\in \varphi {\cal H}^{k,m}_{\le \lambda ,X}$~. Then
the right hand inequality in lemma 6.5 follows by the minimax principle.\finpr

Now, corollary 3.3 easily computes the counting function $N^{k,m}_\Omega
$ for the eigenvalues~:
$$\lim_{\lambda \to 0_+} \lim_{k\to +\infty} k^{-n} N^{k,m}_\Omega
(\lambda ) = {r\over n!}\int_{X(m,E_\chi)}(-1)^m ({i\over 2\pi }c
(E_\chi ))^n~.$$
Applying this to the Witten complex ${\cal H}^{k,\bullet}_{\le \lambda
,X}$, we easily infer the inequality of theorem 6.1, except that $c(E)$
is replaced by $c(E_\chi )$. However, up to now, the inequality is valid
for all $m\ge q$. Take the convex function $\chi $ equal to $0$ on
$]-\infty,c_0]$. Then
$$ic(E_\chi ) = ic(E) + i\partial \op (\chi \circ \psi )$$
coincides with $ic(E)$ on $X_{c_0}$ and has at most $(p-1)+(q-1)$
negative eigenvalues on $X\backslash X_{c_0}$. Hence $X(m,E_\chi )=
X(m,E)$ for $m\ge p+q-1$ and $ic(E_\chi ) = ic(E)$ on these sets. Theorem
6.1 is proved.\finpr

As a corollary, one obtains a general a priori estimate for the
Monge-Amp\`ere operator $(i\partial \op)^n$ on $q$-convex manifolds.

\tha Corollary 6.6|Let $X$ be a strongly $q$-convex manifold and
$\varphi $ a $C^\infty$ function on $X$, weakly $p$-convex outside a
compact subset of $X$. For $\ell = 0,1\ld n$, let $G_\ell$ be the open
set of points where $i\partial\op\varphi $ is non degenerate and admits
$\ell$ negative eigenvalues. Then for all $m\ge p=q-1$
$$\sum^n_{\ell=m} \int_{G_\ell}(i\partial\op\varphi)^m \hbox{~has the
sign of~} (-1)^m~.$$\fintha

This result has been first obtained by Y.T. Siu [Siu4] for $q$-convex
domains in a Stein manifold. At that time, the $q$-convex case of the
inequalities was not yet available and Siu had to rely on a rather
sophisticated approximation argument of Stein manifolds by algebraic
varieties~; the proof could then be reduced to the compact case.

The general statement given above is in fact a direct consequence of
theorem 6.1~: take for $E$ the trivial bundle $E={\cal O}_X$ equipped
with the metric defined by the weight $e^{-\varphi }$ and $F={\cal
O}_X$~. Since $H^m(X,E^k) = H^m(X,{\cal O}_X)$ is independent of $k$ and
finite dimensional, theorem 6.1 implies
$$k^n\sum^n_{\ell=m} \int_{G_\ell} (-1)^m (i\partial\op\varphi)^n\ge
\hbox{~constant~}- o(k^n)$$
for all $k\ge k_0$ and $m\ge p+q-1$, whence the result.\finpr 



\titr 7. Holomorphic Morse inequalities for vector bundles|

A natural question arising in connection with our Morse inequalities is
whether one can extend the inequalities for high tensor powers of a
vector bundle $E$ of rank $\ge 2$. Since $E^{\otimes k}$ is decomposable
for $k\ge 2$ (e.g. $E^{\otimes 2}=S^2E\oplus \Lambda ^2E)$ we are led to
consider only irreducible tensor powers of $E$, \ie the irreducible
representations of the linear group $\gl(E)$. This is done by Getzler
[Ge2], in the general framework of Lie group theory and
representations. As we are only dealing with the case of the full
linear group, we will give here an elementary presentation. We first
recall some ideas from Borel-Weil's theory and a special case of Bott's
formula [Bot].

Let $V$ be a complex vector space of dimension $r$ and $M(V)$ the flag
manifold of $V$, \ie the set of all $(r+1)$-tuples $z=(V_0,V_1\ld V_r)$
with $V=V_0\supset V_1\supset \cdots \supset V_r = \{0\}$ and
codim$V_j=j$. On $M(V)$ we have canonical line bundles $Q_j$ such that
$$Q_{j,z} = V_{j-1}/V_j~,~1\le j\le r^.$$
For any $r$-tuple $(a_1\ld a_r) \in \zz^r$, we set
$$Q^a = Q^{a_1}_1\otimes \cdots \otimes Q^{a_r}_r~.$$
As $\gl(V)$ acts equivariantly on $Q^a \to M(V)$, the spaces of
sections
$$\Gamma ^aV = H^0\big(M(V),Q^a\big)\leqno(7.1)$$
are equipped with a natural $\gl(V)$ action. Observe that $Q^{(1\ld
1)}$ is isomorphic to the trivial bundle $M(V)\times \det V$ (but of
course the action of $\gl(V)$ on $\det V$ is non trivial). To describe
$\Gamma ^aV$, we can therefore assume that all $a_j$ are nonnegative.
Then any section $\sigma \in \Gamma ^aV$ can be viewed as a polynomial
$P_\sigma (\xi _1\ld \xi _r)$ on $(V^\star )^r$ as follows~: if $\xi _1\ld
\xi _r\in V^\star $ are linearly independent, one can associate to $(\xi _j)$
the flag $z=(V_j)$ defined by $V_j = \xi ^{-1}_1(0) \cap \cdots \cap \xi
^{-1}_j(0)$. Then $\xi _j$ induces a well defined linear form $\tilde
\xi _j$ on $Q_{j,z} = V_{j-1}/V_j$ and we set
$$P_\sigma (\xi _1\ld \xi _r) = (\tilde \xi ^{a_1}_1\otimes \cdots
\otimes \tilde \xi ^{a_r}_r)\cdot \sigma (z)~.$$
It is clear that $P_\sigma $ remains locally bounded on a neighborhood
of the hypersurface $\det(\xi _1\ld \xi _r) = 0$~; therefore $P_\sigma $
extends to a polynomial on $(V^\star )^r$ that is homogeneous of degree $a_j$
in the variable $\xi _j$. Also, neither the flag $z$ nor the linear
forms
$\tilde \xi _j$ are modified if we replace $\xi _j$ by $\xi
_j+\sum_{k<j} \lambda _{jk}\xi _k$~. It follows that $P_\sigma $
satisfies the relation
$$P_\sigma (\xi _j+\sum_{k<j}\lambda _{jk}\xi _k) = P_\sigma (\xi _1\ld
\xi _r)~,~\forall \lambda _{jk}\in \cc~,$$
and conversely any polynomial $P$ of multidegree $(a_1\ld a_r)$
satisfying this condition yields a (unique) section $\sigma \in \Gamma
^aV$~. Hence $\Gamma ^aV$ is the subspace of tensors in
$S^{a_1}V\otimes\cdots \otimes S^{a_r}V$ enjoying the above additional
antisymmetry properties. In particular we have
$$\eqalign{
S^kV &= \Gamma ^{(k,0\ld 0)}V~,\cr
\Lambda ^kV  &=\Gamma ^{(1\ld 1,0\ld 0)}V~,~(k\hbox{~first
integers~}=1)~.\cr}$$
We will see soon that $\Gamma ^aV=\{0\}$ unless $a_1\ge a_2\ge \cdots
\ge a_r$. The spaces $\Gamma ^aV~(a_1\ge \cdots \ge a_r)$ can be seen to
be {\sl irreducible} representations of $\gl(V)$. As is well known in
representation theory, $(\Gamma ^aV)$ is in fact the complete list of
irreducible representations of $\gl(V)$ up to isomorphism.

Assume now that $V$ is equipped with a hermitian metric. Then any flag
$z^0\in M(V)$ is represented by an orthonormal basis $(e_1\ld e_r)$ such
that $V^0_j = \hbox{Vect}(e_{j+1}\ld e_r)$. Now $z^0$ is contained in the
affine chart of points $z=(V_j)$ with
$$V_j = \hbox{Vect}(v_{j+1}\ld v_r)~,~v_k = e_k+\sum_{j<k} z_{jk}e_j,$$
where $(z_{jk})\in \cc^{n(n-1)/2}$ are the affine coordinates of $z$~.
The canonical metric on $Q^a$ induced by $V$ has curvature
$$c(Q^a)_{z^0} = \sum_{1\le j<j\le n} (a_j-a_k)dz_{jk}\wedge d\overline
z_{jk}~;\leqno(7.2)$$
we will omit the easy (and standard) computation. By homogeneity, we
see that $Q^a$ is positive as soon as $a_1>a_2>\cdots >a_r$. On the
other hand, when $a_{j-1}<a_j$~, we see that $Q^a$ is negative along the
$\Bbb P^1$ line in $M(V)$ obtained by fixing all $V_k = V^0_k$ except
$V_j$. Therefore $\Gamma ^aV=H^0\big(M(V),Q^a\big) = \{0\}$ in this
case.

Assume from now on that $a_1\ge \cdots \ge a_r$, and more specifically
that
$$a_1 = \cdots = a_{s_1} > a_{s_1+1} = \cdots = a_{s_2} > \cdots >
a_{s_{m-1}+1} = \cdots = a_{s_m}~,$$
where $s_m = r$~. As $Q_{j+1,z} \otimes \cdots \otimes Q_{k,z} \simeq
\det(V_j/V_k)$, we see that $Q^a$ is the pull back of the bundle
$$Q^a_s = \det(V/V_{s_1})^{a_{s_1}}\otimes\cdots\otimes
\det(V_{s_{m-1}}/V_{s_m})^{a_{s_m}}$$
over the manifold $M_s(V)$ of partial flags
$$V\supset V_{s_1} \supset \cdots \supset V_{s_m} = \{0\}~,$$
via the obvious projection $\pi _s : M(V)\to M_s(V)$. On $M_s(V)$ we
have a formula completely analogous to (7.2), where the only indices
$(j,k)$ involved are those for which $a_j>a_k$. Thus $Q^a_s$ is ample
and $Q^a=\pi ^\star _sQ^a_s$, in particular
$$H^0\big(M_s(V),Q^a_s\big) = H^0\big(M(V),Q^a\big)=\Gamma ^aV~.\leqno(7.3)$$

Now let $E,F$ be holomorphic vector bundles over a compact manifold $X$
and let
$$n=\dim_{\cc}X~~,~~r=\hbox{rank~} E~~,~~r'=\hbox{rank~} F~.$$
We want to get asymptotic estimates for the dimension of cohomology
groups $H^q(X,\Gamma ^{ka}E\otimes F)$ as $k\to +\infty$. For that, we
introduce the flag bundle $$M_s(E) \to X~,$$
where $s=(s_1\ld s_m)$ is
defined as above, and we consider the universal line bundle $Q^a_s$ over
$M_s(E)$. As $Q^a_s$ is ample along the fibers of $\pi_s~: M_s(E) \to X$~,
the higher direct images
$$R^q(\pi_s)_\star  (Q^{ka}_s \otimes \pi^\star _s F),~~~~q\ge 1$$
vanish for $k\ge k_0~.$ By (7.3) we get
$$(\pi_s)_\star  (Q^{ka}_s \otimes \pi^\star _sF) = \Gamma ^{ka}E\otimes F~.$$
The Leray spectral sequence gives the isomorphism
$$H^q(X,\Gamma ^{ka}E\otimes F) \simeq H^q(Y,Q^{ka}_s \otimes \pi^\star _s
F)~,~~~~Y = M_s(E)~,$$
and we are reduced to applying Morse inequalities to tensor powers of
the line
bundle $Q^a_s$. We still need a formula for the curvature of $Q^a_s$
with the metric induced by a given hermitian metric on $E$.
Let $z^0\in M_s(E_{x^0})$ be a point in $M_s(E)$~. Choose a holomorphic
frame $(e_1\ld e_r)$ of $E$ such that the flag $z^0$ is given by the
basis $(e_1(x^0)\ld e_r(x^0))$ (supposed to be orthonormal). Assume also
$(e_\lambda )$ chosen such
that $De_\lambda (x^0)=0$ and consider the curvature tensor
$$c(E)_{x^0}=\sum_{{1\le j,k\le n}\atop{1\le\lambda,\mu\le r}}
c_{jk\lambda\mu}dx_j\wedge dx_k\otimes e_\lambda^\star\otimes e_\mu.$$
It can be shown that the associated curvature of $Q_s^a$ is
$$c(Q_s^a)_{z^0} = \sum_{{1\le j,k\le n}\atop{1\le \lambda \le r}}
a_\lambda c_{jk\lambda \lambda }dx_j\wedge dx_k + \sum_{a_\lambda >a_\mu
}(a_\lambda -a_\mu )dz_{\lambda \mu }\wedge d\overline z_{\lambda \mu
}$$
where $(z_{\lambda \mu })$ are the affine coordinates along the fiber
$M_s(E_{x^0})$. Finally, let $N(s)$ be the dimension of the fibers
$M_s(E_x)$. Using the isomorphism (7.4), the strong Morse inequality
becomes
$$\sum^q_{m=0}(-1)^{q-m} \dim H^m(X,\Gamma ^{ka}E\otimes F) \le
r' {k^{n+N(s)}\over (n+N(s))!} \int_{Y(\le q,Q^a_s)}\Big({i\over 2\pi }
c(Q^a_s)\Big)^{n+N(s)}.\kern-2pt$$
The most interesting case is the case of symmetric powers $S^kE$. Then
we simply have $M_s(E) = \Bbb P(E^\star)$, $N(s) = r-1$, $Q^a_s = O_E(1)$.

\titr 8. Related questions and open problems|

{\bf (a)} As in the case of the Riemann-Roch Hirzebruch formula, it
would be extremely interesting to get some insight on the error term
$o(k^n)$ of the estimates. However, this problem encounters two major
difficulties~:

$\bullet$ First, the lower order terms in the asymptotic expansion of
the heat kernel involve derivatives of the curvature of $E$, as well as
terms coming from the curvature of $F$ or of the manifold $X$. These
terms are very difficult to compute.

$\bullet$ Second, the open sets $X(\le q,E)$ involved in the estimates
may be extremely irregular, even when the metric of $E$ is smooth. This
leaves very little hope of being able to handle the error term.

One way to avoid the difficulties of the first point is to assume that
the dominant term is identically zero, \ie that $ic(E)$ has rank $<n$
everywhere and is generically of a given rank $s$. Concerning the second
point, a reasonable hypothesis is the following~: assume that there is a
smooth foliation ${\cal F}$ of codimension $s$ in $X$ such that $ic(E)$
vanishes along the leaves. With these hypotheses, Th. Bouche [Bou2] has
shown that when $\rk E=\rk F=1$, $\ell\to+\infty$, $k/\ell\to+\infty$, then
$$\eqalign{
&\sum_{0\le m\le q} (-1)^{q-m} \dim H^m(X,E^k\otimes F^\ell \otimes G)\cr
&\le {k^s\over s!} {\ell^{n-s}\over (n-s)!}
(\rk G) \int_{X(\le q,E,F)} \Big({i\over 2\pi }c(E)\Big)^s \wedge
\Big({i\over 2\pi } c(F)\Big)^{n-s}+o(k^s\ell^{n-s}),\cr}$$
where the index set $X(m,E,F)$ is the set where $(ic(E))^s\wedge
(ic(F))^{n-s}$ has sign $(-1)^m$. It would be very interesting to get
rid of the tensor powers $F^\ell$, but then the difficulty is that the
estimates cannot be localized along the leaves of the foliation.

{\bf (b)} A natural problem is of course to extend the Morse
inequalities to cohomology groups associated to other operators than
$\op$. This has been carried out by Getzler [Ge3] for the operator
$\op_b$ on a compact strongly pseudoconvex $CR$ manifold.

{\bf (c)} Jean Varouchas has drawn my attention on the following
interesting question of Fujiki related to the Grauert-Riemenschneider
conjecture. The G-R conjecture was an attempt to characterize Moishezon
manifolds, \ie manifolds which have a projective algebraic modification.

Another interesting class of manifolds is the Fujiki class ${\cal C}$,
that is, the class of compact complex manifolds which have a K\"ahler
modification (for instance, such manifolds possess Hodge decomposition).
Assume that $X$ is compact and has a closed semi-positive $(1,1)$-form
$\omega $ such that $\omega $ is positive definite on a dense open
subset (say, on the complement of an analytic subset). Then, is $X$ in
the Fujiki class ${\cal C}$~?


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\end
