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		\else\ifodd\pageno\rightheadline
	\else\leftheadline \fi \fi}
\def\leftheadline{\folio\hfil {\sl Chapter 0}}
\def\rightheadline{{\sl Holomorphic Morse inequalities}
\hfil\folio}
\voffset=2\baselineskip
 
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\vglue 4pc
\centerline{\arab{0}}
\smallskip
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\vglue 1pc
 
\centerline{\chap{Holomorphic Morse Inequalities}}
\vglue .5pc
\centerline{{\chap{on q-Convex Manifolds}}\footnote*
{After the work of Thierry Bouche}}
 
\vglue 1pc
\centerline{\sl Jean--Pierre Demailly}
 
\vglue 4pc
\centerline{\bf 1. Statement of Results and Applications}
\vglue 1pc
 
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This note is a report about a recent work of Thierry Bouche [Bo],
in which he obtains 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 $E\la X$ be a holomorphic
vector bundle of rank $r$. Let $D=D'+D''$ be the Chern connection of $L$
and $c(L)=D^2$ be 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 nondegenerate
and has exactly $m$ negative eigenvalues. We also set
$$
X(\le m,L)=\bigcup_{\nu\le m}X(\nu,L),~~~~
  X(\ge m,L)=\bigcup_{\nu\ge m}X(\nu,L).$$
\noindent Then the cohomology groups $H^m(X,E\otimes L^k)$ are finite
dimensional for\break $m\ge q$, and the following theorem gives an asymptotic
estimate of the dimension of $H^m(X,E\otimes L^k)$ as $k$ tends to $+\infty$.
 
\vglue 1pc
\proclaim$\quad\,\,${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:
$$\sum_{m\le\nu\le n}\!\!(-1)^{\nu-m}\dim H^\nu (X,E\otimes L^k)
\le r{k^n\over n!}\int_{X(\ge m,L)}\!\!\!\!\!\!\!(-1)^m
\Big({i\over 2\pi }c(L)\Big)^n+{\rm o}(k^n).$$
 
\smallskip
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
$$\dim H^0(X,E\otimes L^k)-\dim H^1(X,E\otimes L^k)
\ge r{k^n\over n!}\int_{X(\le 1,L)}
\Big({i\over 2\pi }c(L)\Big)^n+{\rm o}(k^n).\eqno(1.2)$$
\medskip
 
\noindent 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:
 
\vglue 1pc
\proclaim$\quad\,\,${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\v sezon.
 
\smallskip
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 satisfies $0\le a(X)\le n$ and $X$ is said to be a
{\sl Moi\v sezon} 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 at least equal to
$n$, hence $a(X)=n$.~\finpr\vskip8pt
 
Another consequence of Morse inequalities is a general {\sl a priori}
estimate for the Monge-Amp\`ere operator $(id'd'')^n$ on
$q$--convex manifolds.
 
\vglue 1pc
\proclaim$\quad\,\,${Theorem 1.4}. Let $X$ be a strongly $q$--convex
manifold and $\varphi$ be 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
nondegenerate and admits $\nu$ negative eigenvalues. Then for all
$m\ge p+q-1$ one has
$$\sum_{m\le\nu\le n}\int_{G_\nu}(-1)^m(id'd''\varphi )^n\ge 0.$$
 
\smallskip
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 $({\rm b}_m)$ 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
\vskip8pt

Following Th. Bouche [Bo], we now give an account of the main ideas
occurring in the proof of Theorem 1.1. We refer to [Bo] and to [D3] for
the details.
\vfil\eject
 
 
\centerline{\bf  2. The Bochner Technique}
\smallskip
 
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,
let $D=D'+D''$ and $c(E)=D^2$ 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); let $\langle ?,?\rangle$ be
the pointwise inner product of differential forms on $X$ with values in $E$
and let $\Ll ?,?\Gg$ be the global $L^2$ inner product obtained after 
integration with respect to the volume form $dV=\omega^n/n!$.
The corresponding norms are denoted by $|\,?\,|$ 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.
 
\vglue 1pc
\proclaim$\quad\,\,${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
$$\Delta''=\Delta'_\tau+[ic(E),\Lambda]+T_\omega.$$
\smallskip
 
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 Proposition 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 $L_\chi$. Proposition 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
$$\Ll \Delta''u,u\Gg=\Ll\Delta'_\tau u,u\Gg+\int_X
\Big(k\langle[ic(L_\chi),\Lambda]u,u\rangle+\langle\Theta u,u\rangle\Big)dV
\eqno(2.2)$$
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 $\omega^0$ satisfy
$$-1/n\le\gamma^0_1\le\ldots\le\gamma^0_{q-1}\le 1~~~{\rm and}~~~
\gamma^0_q=\ldots=\gamma^0_n=1~~~{\rm on}~~X\setminus X_{c_0}\,;$$
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 $\omega$ satisfy 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
$$\langle[ic(L_\chi),\Lambda]u,u\rangle\ge(\beta_1+\ldots+\beta_m)|u|^2.
\eqno(2.3)$$
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
$$\Ll\Delta''u,u\Gg\ge k\int_{X\setminus X_{c_0}}A|u|^2dV-
C\int_{X_{c_0}}|u|^2\,dV,\eqno(2.4)$$
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 $L^2_{n,m}(X,E\otimes L^k_\chi)$.
 
\vglue 1pc
\proclaim$\quad\,\,${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
$m\ge q$.
\smallskip
 
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) (see for example [D2]).~\finpr
 
\vglue 1.5pc
\centerline{\bf 3. Witten's Complex}
\smallskip
 
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 in many aspects.
 
Since $D''$ commutes with $\Delta''$, the sequence of eigenspaces
$E^m(\lambda)$ in $L^2_{n,m}(X,E\otimes L^k_\chi)$ 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 
$$D''\delta''+\delta''D''=\Delta''=\lambda\,{\rm Id}~~~~\hbox{\rm
on}~~E^\bu(\lambda),$$ 
thus ${1\over\lambda}\delta''$ is a homotopy operator
for $D''$ and $E^\bu(\lambda)$ is acyclic for $\lambda\ne 0$.
For any $\lambda\ge 0$, let
$$\hr^m_{X,k}(\lambda),~~~~m\ge q,\eqno(3.1)$$
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 section 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:
 
\vglue 1pc
\proclaim$\quad\,\,${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:
$$h^m-h^{m+1}+\cdots +(-1)^{n-m}h^n\le
c^m-c^{m+1}+\cdots +(-1)^{n-m}c^n.$$
\smallskip
 
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 $X$ has been replaced
by a relatively compact open set $\Omega\ssubset X$.
 
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 $L^2(\Omega)$, as with 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., to eigenvalues 
$\le k\lambda$ for $\Delta''$).
 
\vglue 1pc
\proclaim$\quad\,\,${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
$$\dim\hr^m_{\Omega,k}(\lambda)\le \dim\hr^m_{X,k}(\lambda)
\le \dim\hr^m_{\Omega,k}(\lambda+\varepsilon)~~~~{\rm for}~~k\ge k_0.$$
\medskip
 
{\sl 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 $X\setminus\Omega$).
\medskip
 
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 let $\Omega\ssubset X$ be 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) (see for example [D3]):
$$2Q^m_{\Omega,k}(u)=\int_\Omega\Big({1\over k}|\nabla u+Su|^2-\langle Vu,
u\rangle+{1\over k}\langle\Theta u,u\rangle\Big)\,dV,\eqno(3.4)$$
where $\nabla$ is the canonical hermitian connection on the $C^\infty$
vector 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 $TX$, $E$ and $L_\chi$,
respectively, and assume that $ic(L_\chi)$ is diagonalized in $(\xi_j)$.
We find
$$\eqalignno{
u&=\sum_{|J|=m,\,1\le h\le r}u_{J,h}\,\xi^\star_1\wedge\ldots\wedge\xi^\star_n
\wedge\ov\xi^\star_J\otimes e_h\otimes l^k,\cr
\langle Vu,u\rangle&=\sum_{|J|=m,\,1\le h\le r}(\beta_{\complement J}-\beta_J)
|u_{J,h}|^2,&(3.5)\cr}$$
where $\beta_1\ld\beta_n$ are the eigenvalues of $ic(L_\chi)$
and $\beta_J=\sum_{j\in J}\beta_j$.
 
\vglue 1.5pc
\centerline{\bf 4. A Spectral Theorem for Schr\"odinger Operators}
\smallskip
 
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
$F\la M$, $L\la M$ be 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
$$Q_{\Omega,k}(u)=\int_\Omega\Big({1\over k}|\nabla_ku+Su|^2-\langle Vu,
u\rangle\Big)\,dV,~~~~u\in W^1_0(\Omega,F\otimes L^k)\eqno(4.1)$$
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 as
$$B(x)=\sum_{1\le j\le s}B_j(x)\,\xi^\star_{2j-1}\wedge\xi^\star_{2j}\,,~~~~
s\le n/2\eqno(4.2)$$
where $(\xi^\star_1\ld\xi^\star_n)$ is an orthonormal basis of $T^\star_x M$,
$2s$ is the rank of $B(x)$ and $B_1(x)\le\ldots\le B_s(x)>0$ are 
the absolute values of the non zero eigenvalues of
$B(x)$, viewed as an antisymmetric endomorphism on $T_xM$. 
Introduce the ``spectral density function"
$$\nu_B(\lambda)={2^{s-n}\pi^{-n/2}\over\Gamma\big({n\over 2}-s+1\big)}
B_1\ldots B_s\sum_{(p_1\ld p_s)\in\nb^s}\Big(\lambda-\sum_{1\le j\le s}
(2p_j+1)B_j\Big)_+^{(n/2)-s}.\eqno(4.3)$$
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 $\lambda$
which is continuous on the left side; one can also check that
$\nu_{B(x)}(\lambda)$ is lower semicontinuous 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.
 
\vglue 1pc
\proclaim$\quad\,\,${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
$$\eqalign{
\liminf_{k\to+\infty}~~k^{-n/2}\,N_{\Omega,k}(\lambda)&\ge
\sum_{1\le j\le t}\int_\Omega\nu_B(V_j+\lambda)\,dV,\cr
\limsup_{k\to+\infty}~~k^{-n/2}\,N_{\Omega,k}(\lambda)&\le
\sum_{1\le j\le t}\int_{\ov\Omega}\ov\nu_B(V_j+\lambda)\,dV.\cr}$$
\smallskip
 
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 the following:
\vglue 1pc
 
\proclaim$\quad\,\,${Corollary 4.5}.
There is a countable subset $\dr\subset\rb$ such that
$$N_{\Omega,k}(\lambda)\sim k^{n/2}
\sum_{1\le j\le t}\int_\Omega\nu_B(V_j+\lambda)\,dV,~~~~
\forall\lambda\in\rb\setminus\dr.$$
\smallskip
 
The proof of Theorem 4.4 is made in several steps. The first step
is a localization procedure that originated with the work of H. Weyl [We].
Weyl 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
$$N^m_{\Omega,k}(\lambda)\sim rk^n\sum_{|J|=m}\int_\Omega\nu_B(2\lambda+
\beta_{\complement J}-\beta_J)\,dV,~~~~\hbox{\rm for all}~
\lambda\in\rb\setminus\dr.$$
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
$$\eqalign{
\ov\nu_B(\beta_{\complement J}-\beta_J)&={2^{s-2n}\pi^{-n}\over
\Gamma(n-s+1)}|\beta_1\ldots \beta_s|\cr
&\times\sum_{(p_1\ld p_s\in\nb^s)}\Big(\beta_{\complement J}-
\beta_J-\sum_{1\le j\le s}(2p_j+1)|\beta_j|\Big)_+^{n-s}.\cr}$$
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 $\lim_{\lambda\to 0+}~
(\lambda)_+^0=1$ 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.
 
\vglue 1.5pc
\centerline{\sl {Universit\'e Joseph Fourier, Grenoble I,
France}}
 
\vglue 1pc
\centerline{\bf References}
\frenchspacing
\smallskip
 
 
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\end
