\input broccod.tex
\input ams.tex

%Special macros for this tex file

\def\le{\leqslant}
\def\ge{\geqslant}
\font \bftwelve=cmbx10 at 12pt
\def\pav #1|{\par\vskip 17pt{\bftwelve #1.}~~\vskip7pt}
\def\la{\longrightarrow}
\def\ld {, \ldots ,}
\def\finpr{\hbox{$~\square$}}
\def\ort{\kern3pt\hbox{\vrule width 4.5pt height 0.4pt depth 0pt
                       \vrule height 6pt depth 0pt\kern5pt}}

\def\dr{{\cal D}}
\def\hr{{\cal H}}
\def\cb{{\Bbb C}}
\def\nb{{\Bbb N}}
\def\qb{{\Bbb Q}}
\def\rb{{\Bbb R}}
\def\zb{{\Bbb Z}}
\def\ssubset{\subset\!\subset}
\long\def\theo#1|#2|#3|#4\fintha{\par\vskip12pt
        {\bf #2.~#1}{\rm #3\pointir}{\sl #4}\par\vskip 5pt}
\def\remark#1|#2|{\par\vskip12pt{\bf #2.~#1}\pointir}
\def\proof{\par{\sl Proof}\pointir}
\def\ov{\overline}
\def\bu{{\scriptstyle\bullet}}
\def\wh{\widehat}
\def\wt{\widetilde}
\def\Ll{\langle\kern-2pt\langle}
\def\Gg{\rangle\kern-2pt\rangle}
\def\ci{C^\infty}

$\,\,$
\vskip2.5cm
 
\centerline{\bftwelve HOLOMORPHIC MORSE INEQUALITIES}
\centerline{\bftwelve ON $Q$-CONVEX MANIFOLDS}
\vskip5pt
\centerline{\bftwelve after the work of Thierry Bouche}
\vskip20pt
{\noindent\hangindent4cm\hangafter-1
{\sl by Jean-Pierre DEMAILLY

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e Joseph Fourier Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\omini$\,$188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip 30pt

\pav 1. Statement of results and applications|

This note is a report about a recent work of Thierry Bouche
(Univ. Joseph Fourier, Grenoble). Bouche [Bo] obtained
an extension of the holomorphic Morse inequalities proved in Demailly [D3]
for strongly $q$--convex manifolds.

Let $X$ be a complex manifold of dimension $n$~. We assume that
$X$ is strongly $q$--convex in the sense of Andreotti-Grauert [A-G],
i.e.\ that there exists a $\ci$ exhaustion function $\psi$ on $X$
such that the complex Hessian $id'd''\psi$ has at least $n-q+1$
positive eigenvalues outside a compact subset of $X$~. In this case,
Andreotti-Grauert's theorem [A-G] asserts that all cohomology groups
$H^m(X,{\cal F})$ with values in a coherent analytic sheaf are finite
dimensional for $m\ge q$~.

Let $L\la X$ be a holomorphic line
bundle with a hermitian metric of class $\ci$ and $E\la X$ a holomorphic
vector bundle of rank $r$~. Let $D=D'+D''$ be the Chern connection of $L$
and $c(L)=D^2$ the associated curvature form. We denote by $X(m,L)$ the
set of points of $X$ where the real $(1,1)$--form $ic(L)$ is non degenerate 
and has exactly $m$ negative eigenvalues. We also set 
$$X(\le m,L)=\bigcup_{\nu\le m}X(\nu,L)~,~~~~
  X(\ge m,L)=\bigcup_{\nu\ge m}X(\nu,L)~.$$
Then the cohomology groups $H^m(X,E\otimes L^k)$ are finite dimensional
for $m\ge q$~, and the following theorem gives an asymptotic estimate
of $\dim H^m(X,E\otimes L^k)$ as $k$ tends to $+\infty$~.\eject

\theo Theorem| 1.1||Assume that $X$ is strongly $q$--convex and that 
the curvature form $ic(L)$ has at least $n-p+1$ nonnegative eigenvalues
outside a compact subset of $X$~. Then for all $m\ge p+q-1$
the following asymptotic inequalities hold~:
 
\noindent$({\rm a}_m)$~~Weak Morse inequalities$~:$
$$\dim H^m(X,E\otimes L^k)\le r{k^n\over n!}
\int_{X(m,L)}(-1)^m\Big({i\over 2\pi }c(L)\Big)^n+
{\rm o}(k^n)~.$$
 
\noindent$({\rm b}_m)$~~Strong Morse inequalities$~:$
$$\sum_{m\le\nu\le n}(-1)^{\nu-m}\dim H^\nu (X,E\otimes L^k)
\le r{k^n\over n!}\int_{X(\ge m,L)}(-1)^m
\Big({i\over 2\pi }c(L)\Big)^n+{\rm o}(k^n)~.$$\fintha

Observe that $({\rm a}_m)$ is obtained simply by adding $({\rm b}_m)$
and $({\rm b}_{m+1})$. The special case when $X$ is compact ($p=q=0$) has been
first proved in [D3]. Then $({\rm a}_m)$ and $({\rm b}_m)$ are valid for all 
$m\ge -1$~. As the strong Morse inequalities $({\rm b}_0)$~, $({\rm b}_{-1})$
are identical except for the sign of their terms, we see that the case $m=0$ 
is an asymptotic equality~; this equality is in fact a weak form of the 
Riemann-Roch-Hirzebruch formula for the Euler-Poincar\'e characteristic 
$\chi(X,E\otimes L^k)$~. Subtracting inequality $({\rm b}_2)$ from
$({\rm b}_0)$~, we get the lower bound
$$\dim H^0(X,E\otimes L^k)-\dim H^1(X,E\otimes L^k)
\ge r{k^n\over n!}\int_{X(\le 1,L)}
\Big({i\over 2\pi }c(L)\Big)^n+{\rm o}(k^n)~.\leqno(1.2)$$
This result is used in [D3] in order to obtain a new proof of the 
Grauert-Riemenschneider conjecture [G-R], which had been solved in the 
affirmative by Y.T. Siu [S2] soon before. More precisely, we get the 
following stronger version of the conjecture.

\theo Theorem| 1.3||Let $X$ be a connected $n$--dimensional compact
manifold. If $X$ carries a hermitian line bundle $L$ such that 
$\int_{X(\le 1,L)}\big(ic(L)\big)^n>0$~, then $X$ is Moi$\check{\sl s}$ezon.
\fintha

Let us recall that the {\sl algebraic dimension} $a(X)$ is by definition
the trans\-cendence degree of the field of meromorphic functions of $X$~. 
This degree always verifies $0\le a(X)\le n$ and $X$ is said to be a
{\sl Moi$\check{\sl s}$ezon} variety if $a(X)=n$~. Inequality (1.2) shows 
that $\dim H^0(X,L^k)\ge ck^n$ for some constant $c>0$ and $k\ge k_0$~;
by a standard method of Poincar\'e, Serre [Se] and Siegel [Si], it follows
that the field of meromorphic functions generated by quotients of sections 
in $H^0(X,L^k)$~, $k\in\nb$~, has transcendence degree $\ge n$~, hence 
$a(X)=n$~.~\finpr\vskip8pt

Another consequence of Morse inequalities is a general a priori
estimate for Monge-Amp\`ere operator $(id'd'')^n$ on
$q$--convex manifolds. 

\theo Theorem| 1.4||Let $X$ be a strongly $q$--convex manifold and $\varphi$
a ${\cal C}^\infty$ function on $X$~, weakly $p$-convex outside a
compact subset $K\subset X$~. For $0\le\nu \le n$~, let $G_\nu$ be the
set of points of $X$ where $id'd''\varphi $ is non degenerate and
admits $\nu$ negative eigenvalues. Then for all $m\ge p+q-1$ one has
$$\sum_{m\le\nu\le n}\int_{G_\nu}(-1)^m(id'd''\varphi )^n\ge 0~.$$\fintha

This result has been first obtained by Y.T. Siu [S4] 
for $q$--convex domains $\Omega$ in a Stein manifold $X$~. Siu's method
is rather sophisticated and rests merely upon the compact case of
theorem 1.1~: $X$ is ``approximated" by affine algebraic manifolds 
$X_{\rm alg}$~, and the proof is then reduced to an application of Morse 
inequalities on a compactification of $X_{\rm alg}$~. The general statement 
given above appears in fact as a straightforward consequence of
the $q$--convex case of theorem 1.1~:
take for $L$ the trivial bundle $X\times\cb$ endowed with the curvature
form defined by the weight $e^{-\varphi}$~. Then $ic(L)=id'd''\varphi$~.
Since $H^m(X,L^k)=H^m(X,\cb)$ is independent of $k$ and finite
dimensional for $m\ge q$~, theorem 1.1 (b) implies
$$k^n\sum_{m\le\nu\le n}\int_{G_\nu}(-1)^m(id'd''\varphi)^n\ge {\rm constant}$$
for all $k\ge k_0$ and $m\ge p+q-1$~, whence the result.~\finpr

Following Th. Bouche [Bo], we give now an account of the main ideas 
occurring in the proof of theorem 1.1. We refer to [Bo] and to [D3] for
the details.

\pav 2. The Bochner technique|

The cohomology groups $H^m(X,E\otimes L^k)$ are computed by means of
the Dolbeault complex $D'':\ci_{0,\bu}(X,E\otimes L^k)$~, where
$\ci_{s,t}(X,E)$ denotes the space of $\ci$ sections of the bundle
$\Lambda^{s,t}T^\star X\otimes E$~. This complex is isomorphic to
the complex $\ci_{n,\bu}(X,\wt E\otimes L^k)$ with $\wt E=\Lambda^nTX$~,
thus after changing $E$ into $\wt E$~, we can work with forms of bidegree
$(n,m)$ instead of $(0,m)$ (note that the estimates of theorem 1.1 do not 
depend on $E$ except for the rank $r$).

Assume that the bundle $E$ is endowed with a $\ci$ hermitian metric,
and let $D=D'+D''$~, $c(E)=D^2$ be the corresponding Chern connection and
curvature form. Let $\omega$ be a complete hermitian metric on $X$
(to be chosen later in a suitable way), $\langle ?,?\rangle$
the pointwise inner product of differential forms on $X$ with values in $E$ 
and $\Ll ?,?\Gg$ the global $L^2$ inner product obtained after integration 
with respect to the volume form $dV=\omega^n/n!$~. 
The corresponding norms are denoted $|\,?\,|$ and 
$||\,?\,||$~, and $\Lambda$ stands as usual for the (pointwise) adjoint
of the wedge multiplication by $\omega$~. Finally, we denote by
$\delta =\delta '+\delta ''$ the formal adjoint of $D$~. 

If $A$ and $B$ are endomorphisms of respective degrees $a,b$ of the 
graded module $\ci_\bu(X,E)$~, we set $[A,B]=AB-(-1)^{ab}BA$~.
The holomorphic and conjugate-holomorphic Laplace-Beltrami operators
are then defined by 
$$\Delta'=[D',\delta']~,~~~~\Delta''=[D'',\delta'']~.$$
These operators are related by a formula of Weitzenb\"ock type,
known as the Bochner-Kodaira-Nakano identity~; we refer to [D1] for
the explicit formulation given below.\eject

\theo Proposition| 2.1||Let $\tau=[\Lambda,d'\omega]$~, let $\tau^\star$
be its pointwise adjoint,\break $\Delta'_\tau =[D'+\tau,\delta'+\tau^\star]$ 
and $T_\omega =[\Lambda ,[\Lambda ,{i\over 2}d'd''\omega ]]-
[d'\omega ,(d'\omega) ^\star]$~. Then 
$$\Delta''=\Delta'_\tau+[ic(E),\Lambda]+T_\omega~.$$\fintha

Now, assume that $\psi$ is an exhaustion function on $X$ that is
strongly $q$--convex on $X\setminus K$ and that $ic(L)$ has
at least $(n-p+1)$ nonnegative eigenvalues on $X\setminus K$~,
for some compact set $K$ in $X$~. In order to take into account the 
$q$--convexity of $X$ in formula 2.1, we multiply the metric of $L$ 
by a weight $e^{-\chi\circ\psi}$ where $\chi$ is a convex increasing function.
The resulting hermitian line bundle is denoted $L_\chi$~. Formula 2.1 
is now applied to the bundle $E\otimes L^k_\chi$~, whose curvature form
is $c(E\otimes L_\chi^k)=c(E)+kc(L_\chi)\otimes{\rm Id}_E$~. For
every compactly supported smooth form with values in $E\otimes L^k$
one gets
$$\Ll \Delta''u,u\Gg=\Ll\Delta'_\tau u,u\Gg+\int_X 
\Big(k\langle[ic(L_\chi),\Lambda]u,u\rangle+\langle\Theta u,u\rangle\Big)dV
\leqno(2.2)$$
where $\Theta=[ic(E),\Lambda]+T_\omega$ is a $\ci$ endomorphism 
independent of $\chi$ and $k$~. Now, we have $\Ll\Delta''u,u\Gg=
||D''u||^2+||\delta''u||^2$ and a similar formula for $\Delta'_\tau$~,
in particular $\Ll\Delta'_\tau u,u\Gg$ is nonnegative.

For every $c\in\rb$~, we consider the sublevel sets
$$X_c=\{x\in X\,;\,\psi(x)<c\}~.$$
Select $c_0$ such that $K\subset X_{c_0}$~. One
can choose a hermitian metric $\omega^0$ on $X$ in such a way that
the eigenvalues $\gamma^0_1\le\ldots\le\gamma^0_n$ of $id'd''\psi$ with
respect to $\omega^0$ verify 
$$-1/n\le\gamma^0_1\le\ldots\le\gamma^0_{q-1}\le 1~~~{\rm and}~~~
\gamma^0_q=\ldots=\gamma^0_n=1~~~{\rm on}~~X\setminus X_{c_0}~;$$
this can be achieved by taking $\omega^0$ equal to $id'd''\psi$ on
a $\ci$ subbundle of $TX$ of rank $n-q+1$ on which $id'd''\psi$ is positive,
and $\omega^0$ large on the orthogonal complement.
We select $\omega=e^\rho\omega_0$ where $\rho$ is a function increasing
fast enough at infinity so that $\omega$ is complete. Let $\alpha$ be
the lowest eigenvalue of $ic(L)$ with respect to $\omega$~. The eigenvalues
$\beta_1\le\ldots\le\beta_n$ of 
$$ic(L_\chi)=ic(L)+id'd''(\chi\circ\psi)\ge
ic(L)+\chi'\circ\psi\,id'd''\psi$$
with respect to $\omega$ verify the inequality
$$\beta_j\ge\alpha+\chi'\circ\psi\,\gamma^0_j/e^\rho~.$$
Let $u$ be a $(n,m)$--form. A standard computation gives
$$\langle[ic(L_\chi),\Lambda]u,u\rangle\ge(\beta_1+\ldots+\beta_m)|u|^2~.
\leqno(2.3)$$
Moreover, we have
$$\beta_1+\ldots+\beta_m\ge m\alpha+e^{-\rho}\chi'\circ\psi(\gamma^0_1+\ldots
+\gamma^0_m)\ge\alpha+{1\over n}e^{-\rho}\chi'\circ\psi~~~{\rm on}~~
X\setminus X_{c_0}$$
because $\gamma^0_j\ge -1/n$ and $\gamma^0_m=1$ ($m\ge q$)~.
It follows that one can choose $\chi$ increasing very fast in such a way
that (2.2) and (2.3) imply
$$\Ll\Delta''u,u\Gg\ge k\int_{X\setminus X_{c_0}}A|u|^2dV-
C\int_{X_{c_0}}|u|^2\,dV~,\leqno(2.4)$$
where $A\ge 1$ is a function tending to $+\infty$ at infinity on $X$
and where $C$ is a large constant. The completeness of the metric $\omega$
shows that inequality (2.4) is still valid for any $u$ in the domain 
of $\Delta''$ acting on the Hilbert space $L^2_{n,m}(X,E\otimes L^k_\chi)$~.

\theo Lemma| 2.5||When $\chi$ increases sufficiently fast at infinity,
the space\break $\hr^{n,m}(X,E\otimes L^k_\chi)$ of $\Delta''$--harmonic forms 
in $L^2_{n,m}(X,E\otimes L^k_\chi)$ is isomorphic to the cohomology group 
$H^m(X,\Lambda^nT^\star X\otimes E\otimes L^k)$ for all $k\in\nb$ and
$m\ge q$~.\fintha

As $A$ tends to $+\infty$ in inequality (2.4), Rellich's lemma easily
shows that $\Delta''$ has a compact resolvent. Hence the spectrum of 
$\Delta''$ is discrete and its eigenspaces are finite dimensional.
When $X$ is compact, lemma 2.5 is true for all $m\ge 0$ and for any choice
of $\chi$ (a very well-known fact of Hodge theory). The $q$--convex 
case is slightly more involved~; it rests essentially on the above
finiteness result and on an approximation theorem of Runge type
deduced from (2.4) (cf.\ for example [D2]).~\finpr

\pav 3. Witten's complex|

We come now to one of the main ideas introduced by E. Witten [Wi].
This idea led Witten to an analytic new proof of the standard Morse 
inequalities for Betti numbers of a compact differentiable manifold.
In our case, we consider the Dolbeault complex instead of the De Rham
complex, but the method is similar by many aspects.

Since $D''$ commutes with $\Delta''$~, the sequence of eigenspaces
$E^m(\lambda)\subset L^2_{n,m}(X,E\otimes L^k_\chi)$ corresponding
to a given eigenvalue $\lambda$ is a {\sl subcomplex} of
the Dolbeault complex $D'':\ci_{n,\bu}(X,E\otimes L^k)$~.
Moreover we have $D''\delta''+\delta''D''=\Delta''=\lambda\,{\rm Id}$\break
on $E^\bu(\lambda)$~, thus ${1\over\lambda}\delta''$ is a homotopy operator
for $D''$ and $E^\bu(\lambda)$ is acyclic for $\lambda\ne 0$~.\break
For any $\lambda\ge 0$~, let 
$$\hr^m_{X,k}(\lambda)~,~~~~m\ge q~,\leqno(3.1)$$
be the sum of the eigenspaces of $\Delta''$ corresponding to eigenvalues
$\le k\lambda$ (the reason for this choice will become apparent later). 
By what we have said at the end of \S 2, the sequence $\hr^m_{X,k}(\lambda)$~, 
$m\ge q$~, is a finite dimensional subcomplex of the Dolbeault complex. 
Its cohomology groups are equal to the desired groups 
$H^m(X,\Lambda^nT^\star X\otimes E\otimes L^k)$~, 
except perhaps for $m=q$~~. In order to get the correct group for $m=q$, 
one possibility is to add an extra term
$$\hr^{q-1}_{X,k}(\lambda)=\delta''\Big({\rm cycles~in~~}\hr^q_{X,k}(\lambda)
\Big)~.$$
The following elementary lemma of homological algebra is then involved.

\theo Lemma| 3.2||Let $0\la C^0\buildo d^0\over \la C^1
\buildo d^1\over \la \cdots \la C^n\buildo d^n\over \la 0$
be a complex of finite dimensional vector spaces of dimensions $c^q$
over a field $K$~. Let $h^q=\dim H^q(C^\bullet)$~. Then for
every index $m$ the following ``strong Morse inequalities" hold~:
$$h^m-h^{m+1}+\cdots +(-1)^{n-m}h^n\le
c^m-c^{m+1}+\cdots +(-1)^{n-m}c^n~.$$\fintha

In our case, $C^m=\hr^m_{X,k}(\lambda)$ and $H^m(C^\bu)=H^m(X,\Lambda^n
T^\star X\otimes E\otimes L^k)$
for $m\ge q$~. The proof of theorem 1.1 is therefore reduced to finding
an asymptotic estimation of the dimensions $\dim\hr^m_{X,k}(\lambda)$~.
The first step is to show that this can be done after replacing $X$
by relatively compact open sets $\Omega\ssubset X$~.

For a domain $\Omega\ssubset X$~, we consider the quadratic form
$$Q^m_{\Omega,k}(u)={1\over k}\int_\Omega\big(|D''u|^2+|\delta''u|^2\big)\,dV$$
associated to ${1\over k}\Delta''$ with Dirichlet conditions on $\partial\Omega$~;
this means that the domain of $Q^m_{\Omega,k}(u)$ is the Sobolev space
$W^1_0(\Omega,\Lambda^{n,m}T^\star X\otimes E\otimes L^k)$
of sections which are in $L^2(\Omega)$ so as their first derivatives,
and which can be approximated in this topology by smooth sections with
compact support in $\Omega$~. We denote by $\hr^m_{\Omega,k}(\lambda)$
the direct sum of all eigenspaces of $Q^m_{\Omega,k}$ corresponding to
eigenvalues $\le\lambda$ (i.e.\ $\le k\lambda$ for $\Delta''$).

\theo Lemma| 3.3||For every $\lambda\ge 0$ and $\varepsilon>0$~,
there exists a domain $\Omega\ssubset X$ and an integer $k_0$
such that
$$\dim\hr^m_{\Omega,k}(\lambda)\le \dim\hr^m_{X,k}(\lambda)
\le \dim\hr^m_{\Omega,k}(\lambda+\varepsilon)~~~~{\rm for}~~k\ge k_0~.$$\fintha

\proof As we work on a complete manifold, the eigenvalues of 
${1\over k}\Delta''$ on $L^2_{m,n}(X,E\otimes L^k_\chi)$ are the same as 
those of the quadratic form $Q^m_{X,k}$~. The left hand inequality is then a 
consequence of the minimax principle, because the domain of $Q^m_{\Omega,k}$
is contained in that of $Q^m_{X,k}$ (the sections of $W^1_0(\Omega)$ can be
extended by $0$ on $X\setminus\Omega$).

For the other inequality, we proceed in several steps. Let $u\in
\hr^m_{X,k}(\lambda)$~. Then
$$\Ll\Delta''u,u\Gg=||D''u||^2+||\delta''u||^2\le k\lambda||u||^2~,$$
and a combination of this estimate with (2.4) yields
$$k\int_{X\setminus X_{c_0}}A|u|^2\,dV-C\int_{X_{c_0}}|u|^2\,dV\le
k\lambda \int_X|u|^2\,dV~.$$
Consider the compact set $P=\ov{X_{c_0}}\cup\{A(x)\le a\}$~. Then 
$A(x)>a$ on $X\setminus P$ and we find
$$\int_{X\setminus P}|u|^2\,dV\le{C+k\lambda\over ka}\int_X|u|^2\,dV~.$$
If we choose $a$ large enough, we get $\int_{X\setminus P}|u|^2\,dV\le
\varepsilon||u||^2$~. Let $\varphi\in\dr(X)$ be a cut-off function equal 
to $1$ on $P$ and $\Omega\ssubset X$ an open set containing 
the support of $\varphi$~. The section $\varphi u$ is in the domain of
$Q^m_{\Omega,k}$ and we have
$$\eqalign{
Q^m_{\Omega,k}(\varphi u)&={1\over k}\int_\Omega\big(|d''\varphi\wedge u
+D''u|^2+|\delta''u-d'\varphi\ort u|^2\big)\,dV\cr
&\le(1+\varepsilon)Q^m_{X,k}(u)+{C'\over k}(1+1/\varepsilon)||u||^2\cr
&\le\Big((1+\varepsilon)\lambda+{C'\over k}(1+1/\varepsilon)\Big)||u||^2~.\cr}$$
As $||\varphi u||^2\ge\int_P|u|^2\,dV\ge(1-\varepsilon)||u||^2$~, we infer
$$Q^m_{\Omega,k}(\varphi u)\le\Big({1+\varepsilon\over 1-\varepsilon}
\lambda+{C(\varepsilon)\over k}\Big)||\varphi u||^2~.$$
If $\varepsilon$ is replaced by a smaller number and $k$ taken large 
enough, we obtain $Q^m_{\Omega,k}(v)\le (\lambda+\varepsilon)||v||^2$ 
when $v$ describes the space $\varphi\hr^m_{X,k}(\lambda)$~. The right 
hand inequality in lemma 3.3 follows by the minimax principle.~\finpr\vskip8pt

In order to handle properly the quadratic form $Q^m_{\Omega,k}$~,
we need another calculation derived from the Bochner-Kodaira-Nakano
identity (2.2) (cf.\ [D3])~:
$$2Q^m_{\Omega,k}(u)=\int_\Omega\Big({1\over k}|\nabla u+Su|^2-\langle Vu,
u\rangle+{1\over k}\langle\Theta u,u\rangle\Big)\,dV~,\leqno(3.4)$$
where $\nabla$ is the canonical hermitian connection on the bundle
$\Lambda^{n,m}T^\star X\otimes E\otimes L^k_\chi$ deduced from the
Chern connections on $\Lambda T^\star X$~, $E$ and $L_\chi$~. The
operators $S$ and $\Theta$ are hermitian endomorphisms which act only
on the component $\Lambda^{n,m}T^\star X\otimes E$ and are independent
of $k$~. Finally $V$ is a curvature endomorphism acting only
on the component $\Lambda^{n,m}T^\star X$~. Let $(\xi_1\ld\xi_n)$~,
$(e_1\ld e_r)$ and $l$ be orthonormal frames of $TX$~, $E$~, $L_\chi$
respectively, and assume that $ic(L_\chi)$ is diagonalized in $(\xi_j)$~. 
One finds
$$\leqalignno{
u&=\sum_{|J|=m,\,1\le h\le r}u_{J,h}\,\xi^\star_1\wedge\ldots\wedge\xi^\star_n
\wedge\ov\xi^\star_J\otimes e_h\otimes l^k~,\cr
\langle Vu,u\rangle&=\sum_{|J|=m,\,1\le h\le r}(\beta_{\complement J}-\beta_J)
|u_{J,h}|^2~,&(3.5)\cr}$$
where $\beta_1\ld\beta_n$ are the eigenvalues of $ic(L_\chi)$
and $\beta_J=\sum_{j\in J}\beta_j$~.

\pav 4. A spectral theorem for Schr\"odinger operators|

The main tool for the proof of Morse inequalities is a spectral 
theorem which describes very precisely the asymptotic distribution 
of eigenvalues for quadratic forms similar to $Q^m_{\Omega,k}$~.

Let $M$ be a $n$--dimensional Riemannian manifold of class $\ci$ and
$F\la M$~, $L\la M$ hermitian vector bundles of respective ranks
$t$ and $1$~. We assume that $F,L$ are endowed with hermitian
connections $\nabla_E$~, $\nabla_L$~, and we let $\nabla_k$ be the 
resulting connection on $F\otimes L^k$~. Finally let
$V$ and $S$ be hermitian endomorphisms of $F$~; we still denote
by $V$ and $S$ their extensions $V\otimes{\rm Id}$~, $S\otimes{\rm Id}$
to $F\otimes L^k$~. For any domain $\Omega\ssubset M$~, we consider the 
quadratic forms
$$Q_{\Omega,k}(u)=\int_\Omega\Big({1\over k}|\nabla_ku+Su|^2-\langle Vu,
u\rangle\Big)\,dV~,~~~~u\in W^1_0(\Omega,F\otimes L^k)\leqno(4.1)$$
with Dirichlet condition on $\partial\Omega$~. For every
real number $\lambda$~, we denote by $N_{\Omega,k}(\lambda)$ the number
of eigenvalues of $Q_{\Omega,k}$ which are $\le\lambda$~; these
eigenvalues are of course counted with multiplicity. We shall see soon that 
$N_{\Omega,k}(\lambda)$ can be estimated in terms of the curvature of $L$ and 
of the potential $V$~.

The curvature form
$B=ic(L)=i\nabla_L^2$ is a real $2$--form on $X$~; at each point
$x\in M$~, this form can be written
$$B(x)=\sum_{1\le j\le s}B_j(x)\,\xi^\star_{2j-1}\wedge\xi^\star_{2j}~,~~~~
s\le n/2\leqno(4.2)$$
where $(\xi^\star_1\ld\xi^\star_n)$ is an orthonormal basis of $T^\star_x M$
and $B_1(x)\le\ldots\le B_s(x)>0$ the modules of the non zero eigenvalues of 
$B(x)$~, viewed as an antisymmetric endomorphism on $T_xM$~. Let
$$\nu_B(\lambda)={2^{s-n}\pi^{-n/2}\over\Gamma\big({n\over 2}-s+1\big)}
B_1\ldots B_s\sum_{(p_1\ld p_s)\in\nb^s}\Big(\lambda-\sum_{1\le j\le s}
(2p_j+1)B_j\Big)_+^{(n/2)-s}~.\leqno(4.3)$$
Here $a_+$ means the maximum of $a$ and $0$~, with the
special convention that $a_+^0=0$ for $a\le 0$ and $a_+^0=1$ for $a>0$~.
It is clear that $\nu_B(\lambda)$ is a non decreasing function of $\lambda$
that is continuous on the left side~; one can also check that 
$\nu_{B(x)}(\lambda)$ is lower semi-continuous in $x$~. Let
$\ov\nu_B(\lambda)=\lim_{\varepsilon\to 0+}\nu_B(\lambda)$ be the associated
increasing function continuous on the right side. The asymptotic eigenvalue 
distribution of $Q_{\Omega,k}$ is then given by the following fundamental 
theorem.

\theo Theorem| 4.4||Let $V_1(x)\ld V_t(x)$ be the eigenvalues of the
endomorphism $V(x)\in{\rm End}(F_x)$~. Then
$$\eqalign{
\liminf_{k\to+\infty}~~k^{-n/2}\,N_{\Omega,k}(\lambda)&\ge
\sum_{1\le j\le t}\int_\Omega\nu_B(V_j+\lambda)\,dV~,\cr
\limsup_{k\to+\infty}~~k^{-n/2}\,N_{\Omega,k}(\lambda)&\le
\sum_{1\le j\le t}\int_{\ov\Omega}\ov\nu_B(V_j+\lambda)\,dV~.\cr}$$\fintha

Assume that $\partial\Omega$ has measure zero. As the integrals are 
increasing functions of $\lambda$ and as the upper bound is the limit
of the lower bound at $\lambda+0$~, we easily conclude~:

\theo Corollary| 4.5||There is a countable subset $\dr\subset\rb$ such that
$$N_{\Omega,k}(\lambda)\sim k^{n/2}
\sum_{1\le j\le t}\int_\Omega\nu_B(V_j+\lambda)\,dV~,~~~~
\forall\lambda\in\rb\setminus\dr~.$$\fintha

The proof of theorem 4.4 is made in several steps. The first step
is a localization procedure originated with the work of H. Weyl.
Weyl [We] introduced the localization procedure and the minimax principle
to get the asymptotic estimate of the distribution of eigenvalues
of linear partial differential equations. Assume that $\Omega$ is 
partitioned into a union of small cubes plus some extra set of small measure.
The intuition is that for the Dirichlet problem with zero boundary
value, the wave length of all eigenfunctions tends to $0$
as $k$ tends to infinity and the Dirichlet problem looks more and
more like the union of the Dirichlet problems for the small cubes.
If the side of the cubes is well chosen ($k^{-1/3}$ is
a suitable scale) the problem can be approximated by a simpler one
in which the potential $V$ and the curvature $B$ are constant.
At this point, an expansion of $u$ as a Fourier series shows
that the variables can be separated. Our Dirichlet problem
is then reduced to the $1$--dimensional harmonic oscillator problem, 
or to the Dirichlet problem for the operator $-d^2/dx^2$ on an interval.
In these cases, the eigenvalues and eigenfunctions can be computed 
explicitly. The general estimate follows.~\finpr\vskip8pt

To get the eigenvalue distribution of $Q^m_{\Omega,k}$ in (3.4),
we apply corollary 4.5 to the bundles $F=\Lambda^{n,m}T^\star X\times E$~,
$L=L_\chi$ and note that $\dim_\rb X=2n$~. When the eigenvalues $\beta_j$ 
are ordered in such a way 
that $|\beta_1|\ge\ldots\ge|\beta_s|>0=\beta_{s+1}=\ldots=\beta_n$~, we get
$B_j=|\beta_j|$~. By (3.5), the eigenvalues $V_j$ are the quantities
$\beta_{\complement J}-\beta_J$ when $J$ describes all multi-indices
of length $|J|=m$~, counted with multiplicity $r$~. Hence the eigenvalue 
distribution $N^m_{\Omega,k}(\lambda)$ of $Q^m_{\Omega,k}(\lambda)$ is
given by
$$N^m_{\Omega,k}(\lambda)\sim rk^n\sum_{|J|=m}\int_\Omega\nu_B(2\lambda+
\beta_{\complement J}-\beta_J)\,dV~,~~~~\forall\lambda\in\rb\setminus\dr~.$$
We use this estimate when $\lambda$ tends to $0+$~. Each integral converges 
to the limit $\int_\Omega\ov\nu_B(\beta_{\complement J}-\beta_J)\,dV$ and
we have
$$\eqalign{
\ov\nu_B(\beta_{\complement J}-\beta_J)={2^{s-2n}\pi^{-n}\over
\Gamma(n-s+1)}|\beta_1\ldots \beta_s|&\times\cr
\sum_{(p_1\ld p_s\in\nb^s)}\Big(\beta_{\complement J}-
\beta_J&-\sum_{1\le j\le s}(2p_j+1)|\beta_j|\Big)_+^{n-s}~.\cr}$$
The expression between $(~~~)_+$ is always $\le 0$~, and it can be zero
only when $p_1=\ldots=p_s=0$~, $\beta_j\le 0$ for $j\in J$ and $\beta_j\ge 0$
for $j\in\complement J$~. By our conventions\break $\lim_{\lambda\to 0+}~
(\lambda)_+^0=1$ and the result is non zero only when $s=n$. 
The above function is therefore equal to zero in all cases except when
$\beta_j<0$ for $j\in J$ and $\beta_j>0$ for $j\in\complement J$~; then
$$\ov\nu_B(\beta_{\complement J}-\beta_J)=(2\pi)^{-n}|\beta_1\ldots\beta_n|~.$$
This situation can only happen at a point $x\in X(m,L_\chi)$~, and the
corresponding multi-index $J$ is then unique. We find therefore
$$\eqalign{
\lim_{\lambda\to 0+}~~\lim_{k\to+\infty}~~k^{-n}N^m_{\Omega,k}(\lambda)&=
r\int_{X(m,L_\chi)}(2\pi)^{-n}\,|\beta_1\ldots\beta_n|\,dV\cr
&={r\over n!}\int_{X(m,L_\chi)}(-1)^m\Big({i\over 2\pi}c(L_\chi)\Big)^n\,dV~.
\cr}$$
This shows that theorem 1.1 is valid for all $m\ge q$ when the curvature
form $ic(L)$ is replaced by $ic(L_\chi)$~. Let $K\ssubset X$ 
be the exceptional compact set where the convexity properties of $\psi$
and $L$ are not satisfied. Take a convex increasing function $\chi$ 
such that $\chi=0$ on $]-\infty,c_0]$ with $X_{c_0}\supset K$~. Then
$$ic(L_\chi)=ic(L)+id'd''(\chi\circ\psi)$$
coincides with $ic(L)$ on $K$ and has at least $n-(p-1)-(q-1)$
positive eigenvalues on $X\setminus K$~, because $ic(L)$ and $id'd''\psi$
are both positive on subspaces of respective codimensions $p-1,q-1$ in $TX$~.
Hence $X(m,L_\chi)=X(m,L)$ for $m\ge p+q-1$ and $ic(L_\chi)=ic(L)$ on
these sets. Theorem 1.1 is proved.
\vfill\eject

%Macros for the bibliography

\newif\ifchiffre
\def\chiffre{\chiffretrue}
\chiffre
\newdimen\laenge
\def\lettre#1|{\setbox3=\hbox{#1}\laenge=\wd3\advance\laenge by 3mm
\chiffrefalse}

\def\article#1|#2|#3|#4|#5|#6|#7|%
    {{\ifchiffre\leftskip=7mm\noindent
     \hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\petcap #2}\pointir {\sl #3}, {\rm #4}, \nobreak{\bf #5} ({\oldstyle #6}), \nobreak #7.\par\else\noindent
\advance\laenge by 4mm \hangindent=\laenge\advance\laenge by -4mm\hangafter=1
\rlap{[#1]}\hskip\laenge{\petcap #2}\pointir
{\sl #3}, #4, {\bf #5} ({\oldstyle #6}), #7.\par\fi}}

\def\livre#1|#2|#3|#4|#5|%
    {{\ifchiffre\leftskip=7mm\noindent
    \hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\petcap #2}\pointir{\sl #3}, #4, {\oldstyle #5}.\par
\else\noindent
\advance\laenge by 4mm \hangindent=\laenge\advance\laenge by -4mm
\hangafter=1
\rlap{[#1]}\hskip\laenge{\petcap #2}\pointir
{\sl #3}, #4, {\oldstyle #5}.\par\fi}}

\def\divers#1|#2|#3|#4|%
    {{\ifchiffre\leftskip=7mm\noindent
    \hangindent=2mm\hangafter=1
     \llap{[#1]\hskip1.35em}{\petcap #2}\pointir #3, {\oldstyle #4}.\par
\else\noindent
\advance\laenge by 4mm \hangindent=\laenge\advance\laenge by -4mm
\hangafter=1
\rlap{[#1]}\hskip\laenge{\petcap #2}\pointir #3,{\oldstyle #4}.\par\fi}}

\centerline{\bf References}
\vskip15pt
\eightpoint
\lettre A-G|

\article A-G|a. Andreotti {\rm and} h. Grauert|Th\'eor\`emes de finitude
pour la cohomologie des espaces complexes|Bull. Soc. Math. 
France|90|1962|193--259|

\article Bi|j.-m. Bismut|Demailly's asymptotic inequalities~: a heat
equation proof|J. Funct. Anal.|72|1987|263--278|

\livre Bo|{\rm Th}. Bouche|In\'egalit\'es de Morse pour la
$d''$--cohomologie sur une vari\'et\'e holomorphe non compacte|Submitted
for publication in Ann. Scient. \'Ec. Norm. Sup.|1988|
 
\article D1|j.-p. Demailly|Champs magn\'etiques et in\'egalit\'es de 
Morse pour la $d''$-cohomolo\-gie|Ann. Inst. Fourier 
Grenoble|35|1985|189--229|

\article D2|j.-p. Demailly|Sur les th\'eor\`emes d'annulation et de finitude 
de T. Ohsawa et O. Abdelkader|S\'eminaire 
P.~Lelong~-- P. Dolbeault~-- H.~Skoda (Analyse) 
{\oldstyle1985{\rm/}86}, Lecture Notes in Math. n\omini~1295|\rm
Springer-Verlag|1987|48--58|
 
\livre D3|j.-p. Demailly|Sur l'identit\'e de Kodaira-Nakano en g\'eom\'etrie
hermitienne. {\rm S\'eminaire P. Lelong -- P. Dolbeault -- H. Skoda (Analyse)
(1983/84), 88-97}|Lecture Notes in Math. {\bf 1198}, Springer|1986| 

\article G-R|h. Grauert {\rm and} o. Riemenschneider|Verschwindungss\"atze
f\"ur analytische Kohomologiegruppen auf komplexen R\"aumen|Invent.
Math.|11|1970|263--292|

\divers Se|j.-p. Serre|{\sl Fonctions automorphes~: quelques
majorations dans le cas o\`u $X/G$ est compact}|{\rm S\'eminaire
Cartan {\oldstyle 1953}--{\oldstyle 1954}, 2--1 to 2--9}|
  
\article Si|c.l. Siegel|Meromorphe Funktionen auf kompakten
Mannigfaltigkeiten|Nach\-richten der Akademie der Wissenschaften
in G\"ottingen, Math.-Phys. Klasse|4|1955|71--77|

\article S1|y.t. Siu|A vanishing theorem for semi-positive line bundles 
over non-K\"ahler manifolds|J. Differential Geom.|19|1984|431--452|

\article S2|y.t. Siu|Some recent results in complex manifold theory related
to vanishing theorems for the semi-positive case|Proceedings of the Bonn
Arbeitstagung 1984, Lecture Notes in Math. n\omini~1111|\rm 
Springer|1985|169--192|

\divers S3|y.t. Siu|{\sl Asymptotic Morse inequalities for analytic 
sheaf cohomology}|{\rm~S\'em. Bourbaki, expos\'e n\omini~666, juin
{\oldstyle 1986}, 16 p}|

\livre S4|y.t. Siu|Calculus inequalities derived from holomorphic 
Morse inequalities|Pre\-print|1986{\rm, to appear in Math. Annalen}|

\article We|h. Weyl|Das asymptotische Verteilungsgesetz der Eigenwerte
linearer partielle Differentialgleichungen|Math. Ann.|71|1911|441--469|

\article Wi|e. witten|Supersymmetry and Morse theory|J. Differential
Geom.|17|1982|661--692|

\end
