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\def\eg{e.g.}
\let\leq=\leqslant
\let\geq=\geqslant
\def\rank{\mathop{\rm rank}}
\def\diam{\mathop{\rm diam}}
\def\Ker{\mathop{\rm Ker}}
\def\Im{\mathop{\rm Im}}
\def\Hom{\mathop{\rm Hom}}
\def\Herm{\mathop{\rm Herm}}

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\footline{\hfill}
\headline{\ifnum\pageno=1\hfil\else\ifodd\pageno{\sevenbf\hfil
Magnetic fields and Morse inequalities for d"-cohomology\hfil\folio}%
\else{\sevenbf\folio\hfil Annales de l'Institut Fourier 35,
(1985) 189--229\hfil}\fi\fi}

{\sevenrm\baselineskip=9pt
Magnetic fields and Morse inequalities for d"-cohomology, {\sevenit English
translation of:\\
Champs magnétiques et inégalités de Morse pour la d"-cohomologie}\\
Annales de l'Institut Fourier, tome {\sevenbf 35}, n${}^{\scriptscriptstyle\circ}$4 (1985), p.~189--229.
\vskip1.5cm}

\def\statement{\bf}
\def\frac#1#2{{#1\over #2}}
\def\card{\mathop{\rm card}}

\centerline{\hugebf MAGNETIC FIELDS}
\bigskip
\centerline{\hugebf AND MORSE INEQUALITIES}
\bigskip
\centerline{\hugebf FOR d"-COHOMOLOGY}
\bigskip
\centerline{\bf by Jean-Pierre DEMAILLY}
\smallskip
\centerline{\vbox{\hrule width 2cm}}
\vskip1.5cm

\footnote{}{\sevenrm\baselineskip=9pt
{\sevenit Keywords~}: Morse inequalities, {\sevenit d}''-cohomology,
Hermitian vector bundle, curvature form, magnetic field,
Schrödinger operator, Bochner-Kodaira-Nakano identity, 
Moi\v{s}ezon space.\vskip-\parskip}

\centerline{\bigbf 0. Introduction.}

Let $X$ be a compact complex analytic manifold of dimension~$n$,
$F$ a holomorphic vector bundle of rank $r$, and $E$ a holomorphic
line bundle with a Hermitian structure of class $\cC^{\infty}$ on $X$.
Let $D=D'+D''$ be the canonical connection of $E$ and
$c(E)=D^{2}=D'D''+D''D'$ be the curvature tensor of this connection.
Let us denote by $X(q)$, $0\leq q\leq n$, the open subset of points
of $x\in X$ that are of index $q$, i.e.\ points $x$ at which the
$(1,1)$ curvature form $ic(E)(x)$ has exactly $q$ negative and
$n-q$ positive eigenvalues. We also put
$$
X(\leq q)=X(0)\cup X(1)\cup\ldots\cup X(q).
$$ 
We then prove the following Morse inequalities, which bound
the dimension of the cohomology groups $H^{q}(X,E^{\otimes k}\otimes F)$
in terms of integral invariants of the curvature of $E$.
\medskip

{\statement Theorem 0.1.\pointir} {\it For all degrees $q=0,1, \ldots, n$,
the following asymptotic inequalities hold when $k$ tends to $+\infty$.
{\parindent=6.5mm
\vskip2pt
\item{\rm (a)} Morse inequalities~$:$ 
$$
\dim H^{q}(X,E^{\otimes k}\otimes F)\leq r\frac{k^{n}}{n!}\int_{X(q)}(-1)^{q}
\Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}).
$$ 
\vskip2pt
\item{\rm(b)} Strong Morse inequalities~$:$ 
$$
 \sum_{j=0}^{q}(-1)^{q-j}\dim H^{j}(X,E^{\otimes k}\otimes F)\leq r\frac{k^{n}}{n!}
\int_{X(\leq q)}(-1)^{q}\Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}).
$$
\vskip2pt
\item{\rm(c)} Asymptotic Riemann-Roch formula~$:$ 
$$
 \sum_{q=0}^{n}(-1)^{q}\dim H^{q}(X,E^{\otimes k}\otimes F)=r\frac{k^{n}}{n!}\int_X
\Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}) .
$$}}
\medskip

Estimates 0.1 (a), (b) are new as far as we know, even 
in the case of projective varieties. The asymptotic equality
0.1~(c), on the other hand, is a weakened version of the
Riemann-Roch formula, which is itself a special case of Hirzebruch's
general formula or, alternatively, of the Atiyah-Singer
index theorem [1]. Indeed, the Riemann-Roch formula expresses
the Euler-Poincaré characteristic
$$
\chi(X,E^{\otimes k}\otimes F)=\sum_{q=0}^{n}(-1)^{q}\dim H^{q}(X,E^{\otimes k}\otimes F)
$$
under the form
$$
\chi( X,E^{\otimes k}\otimes F)=r\frac{k^{n}}{n!}c_{1}(E)^{n}+P_{n-1}(k)\,,
\leqno(0.2)
$$ 
where $P_{n-1}(k)\in\bQ[k]$ is a polynomial of degree${}\leq n-1$ 
and $c_{1}(E)\in H^{2}(X,\bZ)$ is the first class of Chern de $E$, 
represented in De Rham's cohomology by the closed $(1,1)$-form 
${i\over 2\pi}c(E)$ (see for example [16]). It can be observed that
the numerical constant of inequality 0.1~(a) is optimal, as shown
by the example of the line bundle given by the total tensor product
$\cO(1)^{\stimes n-q}\stimes\cO(-1)^{\stimes q}$
over $X=(\bP^{1}(\bC))^{n}$. For this line bundle, one indeed gets
$X(q)=X$ and
$$
\eqalign{
&\dim H^{q}(X,E^{\otimes k})=(k+1)^{n-q}(k-1)^{q},\qquad k\geq 1,\cr
&\int_{X}\Big(\frac{i}{2\pi}c(E)\Big)^{n}=(-1)^{q}n!\,.\cr}
$$ 
The existence of an inequality of the form 0.1(a) has been
conjectured by Y.T. Siu, who has first proved the case
where $ic(E)$ is${}>0$ in the complement of a set of measure zero (cf.\ [16]),
and then the case where $ic(E)$${}\geq 0$ over $X$ (cf.\ [17]). A~substantial
part of the methods used here have been inspired by Siu's work, especially
in~\S3 and~\S5. The~proof of Theorem~0.1 is based on an analytic
technique recently introduced by E.~Witten~[18], [19]. This
technique allows, among other things, to reprove the
classical Morse inequalities  $b_{q}\leq m_{q}$ on any
compact differentiable variety $M$, where $b_{q}$ 
denotes the $q$-th number of Betti and $m_{q}$ the number of
critical points of index $q$ of a given (arbitrary) Morse function on~$M$.
In our situation, the role of the Morse function is held
by the choice of the Hermitian metric on $E$. We also equip
$X$ and $F$ with arbitrary Hermitian metrics which, in the end,
will only play a role in the expression of the $o(k^{n})$ terms in
the final estimates. Given a real number $\lambda\geq 0$, we
consider a sub-complex $\cH_{k}^\bullet(\lambda)$ of the
Dolbeault complex $(\cC_{0,\bullet}^{\infty}(X,E^{\otimes k}\otimes F),d'')$,
consisting in degree $q$ of the subspace of $(0,q)$-forms of
class $\cC^{\infty}$ on $X$ with values in
$E^{\otimes k}\otimes F$, that is spanned by the eigenfunctions
of the anti-holomorphic Laplacian $\Delta''$, of eigenvalues${}\leq k\lambda$.
The cohomology groups of the $\cH_{k}^\bullet(\lambda)$ complex are 
then isomorphic to the groups $H^{q}(X,E^{\otimes k}\otimes F)$ (Proposition 4.1).
As a consequence, it is sufficient to estimate the dimensions of these
spaces $\cH_{k}^{q}(\lambda)$. For this, two tools are used in an
essential manner. The first tool consists of a Weitzenböck type formula
$$
\frac{2}{k}\int_{X}\langle\Delta''u,u\rangle=\int_{X}\frac{1}{k}
|\nabla_{k}u+Su|^{2}-\langle Vu,u\rangle+\frac{1}{k}\langle\Theta
u,u\rangle \leqno(0.3)
$$ 
prove in \S3, and derived from non-Kähler
Bochner-Kodaira-Nakano identity given in [6]. Here, $\nabla_{k}$ 
denotes the natural Hermitian connection on
the bundle $\Lambda^{0.q}T^{*}X\otimes E^{\otimes k}\otimes F$, 
$V$~is a linear potential of order $0$ related to the
curvature of the line bundle $E$, and finally, $S$ and $\Theta$ are
linear operators of order $0$ depending on the torsion of the
Hermitian metric on $X$ and on the curvature of $F$. The study of
the spectrum of $\Delta''$ is then reduced to the study of
the spectrum of the self-adjoint operator
$\nabla_{k}^{*} \nabla_{k}$ associated  with the real connection
$\nabla_{k}$. The second fundamental tool precisely consists of
a very general spectral theorem relative to elliptic
operators of the form $\nabla^*\nabla$. Let $(M,g)$ be a $\cC^\infty$
Riemannian manifold of real dimension $n$, $E$ a complex line bundle
over~$X$, equipped with a Hermitian connection~$\nabla$.
If $\nabla_k$ denotes the connection induced by $\nabla$ on $E^{\otimes k}$,
one then studies the spectrum of the quadratic form
$$
Q_{k}(u)=\int_{\Omega}\Big(\frac{1}{k}|\nabla_{k}u|^{2}-V|u|^{2}\Big)d\sigma,
\qquad u\in L^{2}(\Omega,E^{\otimes k})
\leqno(0.4)
$$ 
for the Dirichlet problem, where $\Omega$ is a relatively open-ended 
compact in~$M$, and where $V$ is a continuous scalar potential on~$M$.
From a physical point of view, this is equivalent to studying the spectrum of 
the operator of Schrödinger $\frac{1}{k}(\nabla_{k}^{*}\nabla_{k}-kV)$ 
associated with the electric field $kV$ and the magnetic field $kB$, 
where $B=-i\nabla^{2} $ is none other than the curvature $2$-form of
of the connection~$\nabla$. With respect to Witten's arguments
explained in [18], [19], our main contribution consists of analyzing
the role of the magnetic field; in the case of De Rham cohomology, on
the other hand, one can consider the magnetic field to be zero,
as a consequence of the fact that $d^2=0$.

At any point $x\in X$, let $2s=2s(x)\leq n$ be the rank of the $2$-form
$B(x)$, and let $B_{1}(x) \geq\ldots\geq B_{s}(x)>0$ the absolute values of
the non-zero eigenvalue of the associated skew-symmetric endomorphism.
We define a function $\nu_{B(x)}(\lambda)$ of the pair
$(x,\lambda)\in M\times\bR$, that is left-continuous in $\lambda$,
by setting
$$
\nu_{B}(\lambda)=\frac{2^{s-n}\pi^{-\frac{n}{2}}}{\Gamma(\frac{n}{2}-s+1)}
B_{1}\ldots B_{s}\sum_{(p_{1},\ldots,p_{s})\in\bN^{s}}
\big[\lambda-\sum(2p_{j}+1)B_{j}\big]_+^{\frac{n}{2}-s}
\leqno(0.5) 
$$ 
with the convention that $0^0=0$. Finally, if
$\lambda_1\leq\lambda_2\leq\ldots$ denote the eigenvalues of $Q_{k}$
(counted with  multiplicity), we consider the counting function
$N_k(\lambda)=\card\{j\,;\;\lambda_j\leq\lambda\}$, for $\lambda\in\bR$.
\medskip

{\statement Theorem 0.6.\pointir}{\it If the boundary $\partial\Omega$ is 
of measure zero, there exists a countable subset $\cD\subset\bR$ such that
$$
\lim_{k\to+\infty}k^{-\frac{n}{2}}N_{k}(\lambda)=\int_{\Omega}\nu_{B}(V+\lambda)\,
d\sigma
$$ 
for all $\lambda\in \bR\ssm \cD$.}
\medskip

In order to prove Theorem~0.6, one starts with the simple case 
where $M=\bR^n$, the magnetic field $B$ being constant and $V=0$.
When $\Omega$ is a cube, a partial Fourier transform reduces the
problem to the classical harmonic oscillator problem in one variable,
and one can then compute explicitly the eigenfunctions. The idea of
this calculation was strongly inspired by Y.~Colin de Verdière's
papers [3], [4]. The generalization of this result to the case of a
vayring magnetic field elaborates on an idea developped by Siu in [16],
consisting of paving $\Omega$ by sufficently small cubes. 
Our method is nevertheless different from the one used in [16], since
we work directly on harmonic forms, whereas Siu's argument was based
on a use of holomorphic cochains via the Dolbeault isomorphism.
In this way, the estimates become substantially more accurate.
The side of the cubes must then be ajusted to a value comprised
the orders of magnitude $k^{-\frac{1}{2}}$ and $k^{-\frac{1}{4}}$,
for instance $k^{-\frac{1}{3}}$~: $k^{-\frac{1}{2}}$ is indeed
the wavelength of the the first eigenfunction, so that
the effect of the magnetic field is not sensible at a lower scale
lower~; on the other hand, at a scale larger than $k^{-\frac{1}{4}}$,
the oscillation of $B$ becomes too strong. We finally use the 
minimax principle to compare the eigenvalues of the quadratic
form on $\Omega$ to the eigenvalues obtained on the small cubes.
In the previous method employed in [16] (also reproduced
in [7]), the side of the cubes was chosen equal to 
to $k^{-\frac{1}{2}}$~; one can easily see that this choice ss 
critical to bound the effects of the magnetic field
independently of~$k$, but the exact determination of the spectrum 
then became impossible. The last section of the present paper
is devoted to a study of geometric characterizations of 
Moi\v{s}ezon spaces [13]. Recall that an irreducible compact complex
space $X$ is called a Moi\v{s}ezon space if the field
$K(X)$ of meromorphic functions on $X$ has a transcendence
degree equal to $n=\dim_{\bC}X$. The Grauert-Riemenschneider
conjecture [10] states that $X$ is Moi\v{s}ezon if and only if
there exists a torsion free quasi-positive sheaf $\cE$ of rank $1$
over~$X$.

By using a desingularization, one is reduced to the case where $X$
is smooth and  where $\cE$ is the locally free sheaf of sections of
a hermitian holomorphic line bundle $E$ of strictly positive curvature
on a dense open subset~$X$. Y.T.~Siu [17] recently solved the conjecture,
and strengthened it by merely assuming that $ic(E)$ is semi-positive
and positive in at least one point. The use of theorem 0.1~(b) makes
it possible to find even weaker conditions, which do not require the 
pointwise semi-positivity of $ic(E)$, but only the 
positivity of a certain integral of the curvature form. For $q=1$,
inequality 0.1~(b) indeed implies a lower bound of the number of
holomorphic sections of $E^{\otimes k}$, namely:
$$
\dim H^{0}(X,E^{\otimes k})\geq\frac{k^{n}}{n!}\int_{X(\leq 1)}\Big(\frac{i}{2\pi}
c(E)\Big)^{n}-o(k^{n}).
\leqno(0.7)
$$ 
On the other hand, one can show, using a classical arument of Siegel
[15] also reinvestigated by [16], that
$\dim H^{0}(X,E^{\otimes k})\leq{\rm Const}\cdot k^{n-1}$ 
if $X$ is not Moi\v{s}ezon (cf.\ Theorem~5.1). From there,
one gets\medskip

{\statement Theorem 0.8.\pointir} {\it Let $X$ a compact 
$\bC$-analytic manifold of complex dimension~$n$. Then $X$ is
Moi\v{s}ezon as soon as it possesses a Hermitian holomorphic 
line bundle satisfying one of the hypotheses {\rm (a), (b), (c)} 
below.
{\parindent=6.5mm
\vskip2pt
\item{\rm(a)} $\int_{X(\leq 1)}(ic(E))^{n}>0$.
\vskip2pt
\item{\rm(b)} $c_{1}(E)^{n}>0$, and there is no point where
$ic(E)$ has a non zero even index.
\vskip2pt
\item{\rm(c)} $ic(E)$ is semi-positive at any point of $X$, and 
positive definite in at least one point of~$X$.\vskip-\parskip}}
\medskip

The results of this work have been published in a short
Comptes Rendus note [8] with the same title. This paper is an
improved version of an earlier work [7], in which techniques
closer to Siu's initial approach where employed. In the latter,
inequality 0.1~(a) had been proved only up to a certain constant numerical
factor, and  as a consequence, the estimates 0.1~(b) and (c)
remained inaccessible.

The author addresses warm thanks to MM.\ Gérard Besson, Alain Dufresnoy,
Syl\-ves\-tre Gallot and Yves Colin de Verdière, to whom he is
especially indebted, for stimulating discussions that greatly
contributed to shape up the ideas involved in this work, especially
in section~\S1.
\bigskip

\section{1}{Spectrum of the Schrödinger operator\\
associated with a constant magnetic field.}

Let $(M,g)$ be a Riemannian manifold of class $\cC^\infty$, of 
real dimension~$n$, and $E\to M$ a complex line bundle
over~$M$, equipped with a $\cC^\infty$ Hermitian metric. We denote by
$\cC_{q}^{\infty}(M,E)$ the space of $\cC^{\infty}$ sections of the 
vector bundle $\Lambda^{q}T^{*}M\otimes E$, and $(?|?)$ the 
canonical sesquilinear pairing
$$
\cC_{q}^{\infty}(M,E)\times \cC_{q}^{\infty}(M,E)\to
\cC_{p+q}^{\infty}(M,\bC).
$$ 
We assumed given a smooth Hermitian connection $D$ on $E$, 
that is, a linear differential operator of order one
$$
D:\cC_{q}^{\infty}(M,E)\to \cC_{q+1}^{\infty} (M,E),\qquad 0\leq q<n,
$$ 
satisfying identities
$$
\leqalignno{
&D(f\wedge u)=df\wedge u+(-1)^{m}f\wedge Du,&(1.1)\cr
&d(u|v)=(Du|v)+(-1)^{p}(u|Dv),&(1.2)\cr}
$$ 
for all sections $f\in\cC_m^{\infty}(M,\bC)$, $u\in \cC_p^{\infty}(M,E)$,
$v\in \cC_q^{\infty}(M,E)$. Let us consider an isometric trivialization 
$\theta:E_{|W}\to W\times\bC$ of $E$ over an open $W\subset M$.
The Hermitian connections of $E_{|W}$ are then given by a
formula of the type
$$
Du=du+iA\wedge u,
$$ 
where $u\in \cC_{q}^{\infty}(W,E)\simeq \cC_{q}^{\infty}(W,\bC)$ and
$A\in \cC_{1}^{\infty}(W,\bR)$ is an arbitrary {\it real} $1$-form.
The {\it magnetic field} (or curvature form) associated with 
the connection $D$ is the real closed $2$-form $B=dA$ such that
$$
D^{2}u=iB\wedge u
$$ 
for all $u\in \cC_{q}^{\infty}(M,E)$. Therefore, $B$ depends only on the 
connection~$D$, but not on trivialization $\theta$ that has been chosen.
A phase change $u=ve^{i\varphi}$ in $\theta$ leads to replace $A$ by
$A+d\varphi$. The choice of a trivialization of $E$ and of a $1$-form $A$ 
can be interpreted physically as the choice of a particular
``vector potential'' of the magnetic field~$B$.

Let us denote by $|u|$ the pointwise norm of any element
$u\in\Lambda^{q}T^{*}M\otimes E$ for metric defined as the tensor product
of the respective metrics on $M$ and $E$. If $\Omega$ is an open subset
of~$M$, we let $L^{2}(\Omega,E)$ (resp.\ $L_q^{2}(\Omega,E)$) be the
space of $L^{2}$ sections of $E$ (resp.\ of $\Lambda^{q}T^{*}M\otimes E$)
above~$\Omega$, equipped with the norm
$$
\Vert u\Vert_{\Omega}^{2}=\int_\Omega|u|^{2}d\sigma,
$$
where $d\sigma$ is the Riemannian volume element on~$M$.

Let $D_{k}$ be the connection induced by $D$ on the $k$-th tensor power
$E^{\otimes k}$, and $V$ a scalar potential on~$M$, i.e.\ a continuous
function with real values. Given a relatively compact open subset
$\Omega\subset M$, our goal is to find an asymptotic estimate,
as $k$ tends to $+\infty$, of the spectrum of the quadratic form
$$
Q_{\Omega,k}(u)=\int_{\Omega}\Big(\frac{1}{k}|D_{k}u|^{2}-V|u|^{2}\Big)d\sigma
\leqno(1.3)
$$ 
where $u\in L^{2}(\Omega,E^{\otimes k})$, with Dirichlet condition
$u_{|\partial\Omega}=0$. The domain of $Q_{\Omega,k}$ is therefore the
Sobolev space $W_{0}^{1}(\Omega,E^{\otimes k})={}$closure of the space
$\cD(\Omega,E^{\otimes k})$ of $C^{\infty}$ sections of $E^{\otimes
k}$ with compact support in $\Omega$ in the space $W^{1}(M,E^{\otimes
k})$. From a physical point of view, this amounts to study the
spectrum of the Schrödinger operator
\hbox{$\frac{1}{k}(D_{k}^{*}D_{k}-kV)$} asso\-ciated with the magnetic
field $kB$ and the electric field $kV$, when $k$ tends
towards~$+\infty$. We refer the reader to the classical article [2]
for a general study of the spectrum of Schrödinger operators, and to
papers [3], [4], [5], [9], [12] for the study of asymptotic problems
that are closely related to the above one.
\medskip

{\statement Definition 1.4.\pointir} {\it Let $N_{\Omega,k}(\lambda)$
denote the number of eigenvalues of
the quadratic form $Q_{\Omega,k}$ that do not exceed $\lambda$.}
\medskip

We first study a simple case that will serve as a model for the 
general case in~\S2. We consider the following situation~: $M=\bR^{n}$ 
with the constant metric $g= \sum_{j=1}^{n}dx_{j}^{2}$, the open set
$\Omega$ is the cube of side length $r\,$:
$$
\Omega=\Big\{(x_{1},\ldots,x_{n})\in R^{n}\,;~|x_{j}|<\frac{r}{2}, 
~1\leq j\leq n\Big\},
$$ 
$V=0$, and finally the magnetic field $B$ is constant, equal to 
the alternate $2$-form of rank $2s$ given by
$$
B=\sum_{j=1}^{s}B_{j}\,dx_{j}\wedge dx_{j+s},
$$ 
where $B_{1}\geq B_{2}\geq\cdots\geq B_{s}>0$, $s \leq\frac{n}{2}$.
One can then select a trivialization of $E$ whose vector potential
associated to $B$ is
$$
A = \sum_{j=1}^{s}B_{j}x_{j}\,dx_{j+s}.
$$ 
The connection of $E^{\otimes k}$ can then be written as
$$
D_{k}u=du+ikA\wedge u,
$$ 
and the quadratic form $Q_{\Omega,k}$ is given by
$$
Q_{\Omega,k}(u)=\frac{1}{k}\int_{\Omega}\Bigg[\sum_{1\leq j\leq s}
\Big(\Big|\frac{\partial u}{\partial x_{j}}\Big|^2 
+\Big| \frac{\partial u}{\partial x_{j+s}}+ikB_{j}x_{j}u\Big|^{2}\big)+
\sum_{j>2s}\Big|\frac{\partial u}{\partial x_{j}}\Big|^{2}\Bigg]\,d\mu
$$ 
where $d\mu$ is the Lebesgue measure on~$\bR^n$. If we perform
the homothety $X_{j}=\sqrt{k}\,x_{j}$, we are reduced to 
study the eigenvalues of the quadratic form
$$
\int_{\sqrt{k}\Omega}\Bigg[
\sum_{1\leq j\leq s}\Big(\Big|\frac{\partial u}{\partial X_{j}}\Big |^2 
+\Big|\frac{\partial u}{\partial X_{j+s}}+iB_{j}X_{j}u\Big|^{2}\Big)+
\sum_{j>2s}\Big|\frac{\partial u}{\partial X_{j}}\Big|^{2}\Bigg]\,d\mu
$$ 
on the cubes $\sqrt{k}\,\Omega$ of side $\sqrt{k}\,r$. With the field~$B$,
we associate the function of the real variable $\lambda$ defined by
$$
\nu_{B}(\lambda)=\frac{2^{s-n}\pi^{-\frac{n}{2}}}{\Gamma(\frac{n}{2}-s+1)}
B_{1}\ldots B_{s}\sum_{(p_{1},\ldots,p_{s})\in\bN^s}\Big[\lambda-
\sum(2p_{j}+1)B_{j}\Big]_+^{\frac{n}{2}-s}
\leqno(1.5)  
$$ 
where we agree to put $\lambda_+^0=0$ if $\lambda\leq 0$ and 
$\lambda_+^{0}=1$ if $\lambda>0$. The function $\nu_B$ is then 
non decreasing and left continuous on~$\bR$~; let us observe that $\nu_{B}$ 
is actually continuous if $s< \frac{n}{2}$. The spectrum of $Q_{\Omega,k}$ is 
then described asymptotically by the following theorem, the idea
of which was suggested to us by Y.~Colin of Verdière [4].\medskip

{\statement Theorem 1.6.\pointir}{\it Given a real number $R>0$, we let
$$
P(R)=\Big\{x\in\bR^{n}\,;\;|x_{j}|<\frac{R}{2}\Big\}
$$ 
be the cube of side $R$ and consider the quadratic form $Q_R$ such that
$$
Q_{R}(u)=\int_{P(R)}\Bigg[\sum_{1\leq j\leq s}\Big(\Big|
\frac{\partial u}{\partial x_{j}}\Big|^2 +
\Big|\frac{\partial u}{\partial x_{j+s}}+iB_{j}x_{j}u\Big|^{2}\Big)+
\sum_{j>2s}\Big|\frac{\partial u}{\partial x_{j}}\Big|^{2}\Bigg]\,d\mu.
$$ 
Finally, we let $N_R(\lambda)$ be the number of eigenvalues${}\leq\lambda$
of $Q_R$ for the Dirichlet problem. Then for all $\lambda\in\bR$, we have
$$
\lim_{R\to+\infty}R^{-n}N_{R}(\lambda)=\nu_{B}(\lambda).
$$}\medskip

For $s= \frac{n}{2}$, $\nu_{B}$ is a step function. The 
eigenvalues of $Q_{R}$ are then grouped in packets around the 
values $\sum(2p_{j}+1)B_{j}$, with approximate multiplicity 
$(2\pi)^{-s}B_{1}\ldots B_{s}R^{n}$. This can be interpreted physically
as a phenomenon of quantification of the eigenstates.
Coming back to our initial eigenvalue problem for the quadratic form 
$Q_{\Omega,k}$, we get
\medskip

{\statement Corollary 1.7.\pointir} Asymptotically,
$\displaystyle \lim_{k\to+\infty}k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)=
r^{n}\nu_{B}(\lambda)$.\hfill$\square$\medskip

{\it Proof of theorem 1.6. --} 
First we seek for an upper bound of $N_{R}(\lambda)$. In this direction,
given $u\in W_0^1(P(R))$, we express $u$ as a partial Fourier series
with respect to the variables $x_{s+1},\ldots,x_{n}$~:
$$
u(x)=R^{-\frac{1}{2}(n-s)}\sum_{\ell\in\bZ^{n-s}}u_{\ell}(x')\exp
\Big(\frac{2\pi i}{R}\ell\cdot x''\Big),
$$ 
where $u_\ell\in W_{0}^{1}(\bR^{s}\cap P(R))$, with the notation
$$
\eqalign{
&x'=(x_{1}, \ldots,x_{s}),\quad x''=(x_{s+1}, \ldots,x_{n}),\cr
&\ell\cdot x''=\ell_{1}x_{s+1}+\cdots+\ell_{n-s}x_{n}.\cr}
$$ 
The hypothesis $u\in W_{0}^{1}(P(R))$ implies that the series
$$
\sum|\ell|^{2}|u_{\ell}(x')|^{2}
$$ 
is in $L^{2}(\bR^{s})$. Let us put $\ell'=(\ell_{1}, \ldots,\ell_{s})$, 
$\ell''=(\ell_{s+1}, \ldots,\ell_{n-s})$. The norm $\Vert u\Vert_{P(R)}$ and 
the quadratic form $Q_R$ are given by
$$
\eqalign{
\Vert u\Vert_{P(R)}^{2}
&=\sum_{\ell\in\bZ^{n-s}}\int_{\bR^{s}}|u_{\ell}(x')|^{2}\,d\mu(x'),\cr
 Q_{R}(u)&=\sum_{\ell\in\bZ^{n-s}}\int_{\bR^{s}}\Bigg[
\sum_{1\leq j\leq s}\!\bigg(\Big|\frac{\partial u_\ell}{\partial x_{j}}\Big|^{2}
+\Big(\frac{2\pi}{R}\ell_{j}+B_{j}x_{j}\Big)^{2}|u_{\ell}|^{2}\bigg)
+\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}|u_{\ell}|^{2}\Bigg]\,d\mu(x').\cr}
$$ 
We are therefore led to consider a Dirichlet problem with
``separate variables'' on the cube 
$\bR^{s}\cap P(R)$. By putting $t=x_j+\frac{2\pi\ell_j}{RB_j}$, we are 
reduced to studying the spectrum of the quadratic form in one variable
$$
q(f)=\int_{R}\Big(\Big|\frac{df}{dt}\Big|^{2}+B_{j}^{2}t^{2}|f|^{2}\Big)\,dt,
$$ 
with $f \in \smash{W_{0}^{1}\big(\,]-\frac{R}{2}\frac{R}{2}[{}
+\frac{2\pi\ell_{j}}{RB_{j}}\big)}$. The latter question is the
classical harmonic oscillator problem (see\ for example [14], Vol.~I, p.~142). 
On $\bR$, i.e.\ without support condition for $f$, the sequence of values
of $q$ is the suite $(2m+1)B_{j}$, $m\in\bN$, and the eigenfunctions of $q$
are given by $\Phi_{m}(\sqrt{B_{j}}\,t)$ where
$\Phi_{0}$, $\Phi_{1},\ldots$ are the Hermite functions$\,$:
$$
\Phi_{m}(t)=e^{t^{2}/2}\frac{d^{m}}{dt^{m}}(e^{-t^{2}}) .
$$ 
For all $p_{j}\in\bN$, let $\Psi_{p_{j},\ell_{j}}(x_{j})$ be the $p_{j}$-th
eigenfunction of the quadratic form
$$
q(f)= \int_{R}\bigg(\Big|\frac{df}{dx_{j}}\Big|^{2}+
\Big(\frac{2\pi}{R}\ell_{j}+B_{j}x_{j}\Big)^{2}|f|^{2}\bigg)\,dx_{j}
\leqno(1.8)
$$ 
for $f \in W_{0}^{1}(\,]-\frac{R}{2}\frac{R}{2}[\,)$, and
let $\lambda_{p_{j},\ell_{j}}$ be
the corresponding eigenvalue. We can then decompose each 
function $u_{\ell}$ as a series of eigenfunctions. Then we see that
$u$ can be written under the form
$$
u(x)= R^{-\frac{1}{2}(n-s)}\sum_{(p,\ell)\in\bN^{s}\times\bZ^{n-s}}u_{p,\ell}
\Psi_{p,\ell'}(x')\exp\Big(\frac{2\pi i}{R}\ell\cdot x''\Big)
\leqno(1.9) 
$$ 
with
$$
u_{p,\ell}\in\bC,\qquad 
\Psi_{p,\ell'}(x')=\prod_{1\leq j\leq s}\Psi_{p_{j},\ell_{j}}(x_{j}).
$$
One should pay attention to the fact that
$\Psi_{p,\ell'}(x')\exp(\frac{2\pi i}{R}\ell \cdot x'')$ is not 
not a genuine eigenfunction for the Dirichlet problem, since the
exponential  term takes non-zero values at the boundary points 
$x_{j}= \pm\frac{R}{2}$, $j>s$. Therefore, the coefficients
$(u_{p,\ell})$ are not arbitrary for a given function
$u\in W_{0}^{1}(P(R))\;$; they must satisfy the relevant vanishing
boundary conditions, namely
$$
\sum_{t_{j}\in\bZ}(-1)^{\ell_{j}}u_{p,\ell}=0
\leqno(1.10)
$$ 
for all $j=1, \ldots, n-s$ and all indices else than $\ell_{j}$ 
fixed~:
$$
p\in \bN^{s},\quad \ell_{1},\ldots,\ell_{j-1},\;\ell_{j+1},\ldots,\ell_{n-s}\in\bZ.
$$ 
Thanks to formula (1.9), the $L^{2}$ norm and the quadratic form $Q_{R}$ 
can be expressed under the form
$$
\Vert u\Vert_{P(R)}^{2}=\sum|u_{p,\ell}|^{2},\qquad
Q_{R}(u)=\sum\Big(\lambda_{p,\ell'}+
\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\Big)|u_{p,\ell}|^{2},
$$ 
where $\lambda_{p,\ell'}=\sum_{1\leq j\leq s}\lambda_{p_{j},\ell_{j}}$. The 
minimax principle 1.20~(b) recalled below implies that
$$
N_{R}(\lambda)\leq\card\Big\{(p,\ell)\in\bN^{s}\times\bZ^{n-s}\,;\;
\lambda_{p,\ell'} + \frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda\Big\}.
\leqno(1.11) 
$$ 
It is therefore sufficient to obtain an appropriate lower bound of 
$\lambda_{p_{j},\ell_{j}}$.\medskip

{\statement Lemma 1.12.\pointir}{\it We have an inequality
$$
\lambda_{p_{j},\ell_{j}}\geq\max\bigg((2p_{j}+1)B_{j}~,~\frac{4\pi^{2}}{R^{2}}
\Big[\Big(\frac{p_{j}+1}{2}\Big)^{2}+\Big(|\ell_{j}|-\frac{B_{j}R^{2}}{4\pi}
\Big)_{+}^{2}\Big]\bigg),
$$ 
which moreover is strict if $\ell_{j}\neq 0$ or if $\Phi_{p_j}(R\sqrt{B_j}/2)\neq 0$.}
\medskip

The lower bound $\lambda_{p_{j},\ell_{j}}\geq(2p_{j}+1)B_{j}$ follows in fact 
from the minimax and the fact that the eigenvalues of $q(f)$ on $\bR$ are
equal to $(2p_j+1)B_j$. In order to obtain the other inequality, we observe
that (1.8) dominates the quadratic form
$$
\widehat{q}(f)=\int_{x_{j}|<R/2}\bigg(\Big|\frac{df}{dx_{j}}\Big|^{2}+
\Big(\frac{2\pi}{R}|\ell_{j}|-B_{j}\frac{R}{2}\Big)_{+}^{2}|f|^{2}\bigg)dx_{j}.
$$ 
The eigenfunctions of $\widehat{q}$ are given explicitly by
$$
\sin\frac{\pi}{R}(p_{j}+1)\Big(x_{j}+\frac{R}{2}\Big),\qquad p_{j}\in\bN\;;
$$ 
$\lambda_{p_{j},t_{j}}$ is therefore bounded below by the corresponding
eigenvalue
$$
\frac{4\pi^{2}}{R^{2}}\Big[\Big(\frac{p_{j}+1}{2}\Big)^{2}+\Big(|t_{j}|-
\frac{B_{j}R^{2}}{4\pi}\Big)_{+}^{2}\Big].
$$ 
These inequalities are strict because on the one hand $q(f)>\widehat{q}(f)$ 
for any $f\neq 0$, and on the other hand, $\Phi_{p_j}(\sqrt{B_j}t)$
can be an eigenfunction of $q$ on $]{-}R/2, R/2[{}+2\pi\ell_{j}/RB_{j}$ only
when
$$
\Phi_{p_{j}}\big(\pm R\sqrt{B_{j}}/2+2\pi t_{j}/R\sqrt{B_{j}}\big)=0.
$$ 
Since the zeros of $\Phi_{p_{j}}$ are algebraic and $\pi$ is 
transcendental, this is only possible if
$$\ell_{j}=0\quad\hbox{and}\quad\Phi_{p_{j}}(R\sqrt{B_{J}}/2)=0.\eqno\square$$ 

{\statement Lemma 1.13.\pointir} {\it Let $\tau_n(\rho)$ be the number of
points of  $\bZ^{n}$ located in the closed ball $\overline{B}(0,\rho)
\subset\bR^{n}$. Then
$$
\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}\Big(\rho-
\frac{\sqrt{n}}{2}\Big)_{+}^{n}\leq\tau_{n}(\rho)\leq
\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}\Big(\rho+\frac{\sqrt{n}}{2}
\Big)^{n}.$$}\medskip

Indeed, the union of cubes of side $1$ centered at integral points
$x\in\bZ^{n}$ such that $|x|\leq\rho$ is contained in the ball 
$\overline{B}(0,\rho+\frac{\sqrt{n}}{2})$, and contains the ball 
$\overline{B}(0,\rho-\frac{\sqrt{n}}{2})$ if $\rho\geq\frac{\sqrt{n}}{2}$,
where $\frac{\sqrt{n}}{2}$ is half the diagonal of the cube~; the integer 
$\tau_{n}(\rho)$ is thus in the interval comprised between the volumes of 
the balls $\overline B(0,\rho\pm\frac{\sqrt{n}}{2})$.\hfill\square\medskip

We now proceed to find an upper bound of $\lim\sup R^{-n}N_{R}(\lambda)$ 
by using (1.11) and lemmas 1.12, 1.13. For $p\in\bN^s$ fixed, the inequality 
$\lambda_{p,\ell'}+ \frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda$ implies
$$
|\ell''| \leq\frac{R}{2\pi}\Big(\lambda-\sum(2p_{j}+1)B_{j}\Big)^{\frac{1}{2}}_+,
\leqno(1.14) 
$$ 
and the inequality is strict for $R>R_0(p)$ large enough. When 
$s<n/2$ the number of corresponding multi-indices $\ell''\in\bZ^{n-2s}$ is 
therefore at most
$$
\leqalignno{
\frac{\pi^{\frac{n}{2}-s}}{\Gamma(\frac{n}{2}-s+1)}\bigg[\frac{R}{2\pi}
&\Big(\lambda-\sum(2p_{j}+1)B_{j}\Big)^{\frac{1}{2}}_++\frac{\sqrt{n}}{2}
\bigg]^{n-2s}\cr
&\mathop{\sim}\limits_{R\to+\infty}~~
\frac{2^{2s-n}\pi^{s-\frac{n}{2}}}{\Gamma(\frac{n}{2}-s+1)}R^{n
-2s}\Big(\lambda-\sum(2p_{j}+1)B_{j}\Big)^{\frac{n}{2}-s}_+.
&(1.15)\cr}
$$
When $s= \frac{n}{2}$, this number must be counted as $1$ 
if $\lambda-\sum(2p_{j}+1)B_{j}>0$ and $0$ otherwise, in conformity 
with the convention we adopted for the notation~$\lambda_{+}^{0}$.
The inequality $\lambda_{p,\ell'} \leq\lambda$ 
implies on the other hand
$$
|\ell_{j}| \leq\frac{R}{2\pi}\sqrt{\lambda_{+}}+\frac{B_{j}R^{2}}{4\pi},\qquad
1\leq j\leq s,
\leqno(1.16) 
$$ 
which asymptotically corresponds to a number of multi-indices 
$\ell'=(\ell_{1},\ldots,\ell_{s})\in\bZ^{s}$  equivalent to
$$
\prod_{j=1}^{s}\frac{B_{j}R^{2}}{2\pi}=2^{-s}\pi^{-s}B_{1}\ldots B_{s}R^{2s}.
\leqno(1.17)
$$ 
The upper bound $\lim\sup R^{-n}N_{R}(\lambda)\leq \nu_{B}(\lambda)$ 
is then obtained by multiplying (1.15) by (1.17), and taking the sum
over all $p\in\bN^{s}$ (the sum is finite).\hfill\square\medskip

For questions of convergence that will play a role in~\S2, we will also 
need an upper bound of $N_{R}(\lambda)$ that is independent of
the magnetic field~$B$. Such a uniform estimate is provided by 
the following proposition.
\medskip

{\statement Proposition 1.18.\pointir} 
$N_{R}(\lambda)\leq(R\sqrt{\lambda_{+}}+1)^{n}$.
\medskip

{\it Proof}.~-- For each index $j$, we bound the number 
of integers $p_j$ and $\ell_j$ such that the inequality
$$
\lambda_{p,\ell'}+\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda
$$ 
holds. Lemma 1.12 implies
$$
\card  \{p_{j}\}\leq\max(p_{j}+1)\leq\min\Big(\frac{\lambda_{+}}{B_{j}}\,,\,
\frac{R}{\pi}\sqrt{\lambda_{+}}\Big),\qquad 1\leq j\leq s,
$$ 
while (1.16) entails
$$
\card  \{l_{i}\}\leq\frac{R}{\pi}\sqrt{\lambda_{+}}+\frac{B_{j}R^{2}}{2\pi}+1, 
\qquad 1\leq j\leq s.
$$ 
For $1\leq j\leq s$, we therefore infer that
$$
\card  \{(p_{j},l_{j})\}
\leq\Big(\frac{R}{\pi}\sqrt{\lambda_{+}}\Big)^{2}+\frac{\lambda_{+}}{B_{j}}
\cdot\frac{B_{j}R^{2}}{2\pi}+\frac{R}{\pi}\sqrt{\lambda_{+}}\cdot 1
\leq\big(R\sqrt{\lambda_{+}}+1\big)^{2}.
$$ 
For $s<j\leq n-s$, inequality (1.14) gives on the other hand
$$
|\ell_{j}|<\frac{R}{2\pi}\sqrt{\lambda_{+}},
$$ 
hence $\card  \{l_{j}\}\leq\frac{R}{\pi}\sqrt{\lambda_{+}}+1$. 
Proposition 1.18 follows.\hfill\square\medskip

{\it End of the proof of Theorem} 1.6
(lower bound of $N_R(\lambda)$).

In order to get a lower bound of $N_R(\lambda)$, it is sufficient
by 1.20~(a) to construct a finite dimensional vector space on which 
$Q_{R}(u)\leq\lambda\Vert u\Vert_{P(R)}^{2}$. We consider for this 
the vector space $\cF_{\lambda}$ of linear combinations of 
``eigenfunctions'' of the type (1.9), subject to the 
additional boundary conditions (1.10), for which summations are
taken on indices $(p,\ell)\in\bN^{s}\times\bZ^{n-s}$ such as
$$
\lambda_{p,\ell'}+\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda.
$$ 
By the arguments used in Proposition 1.18, the number of conditions 
(1.10) to be realized is bounded above by
$$
\eqalign{
\sum_{j=1}^{s}\bigg[\card \{p_{j} \}&\times\prod_{1\leq i\leq s,\,i\neq j}
\card  \{(p_{i},\ell_{i})\}\times\prod_{s<i\leq n-s}\card \{\ell_{i}\}\bigg]\cr
&+ \sum_{s<j\leq n-s}\bigg[\prod_{1\leq i\leq s}\card  \{(p_{i},\ell_{i})\}
\times\prod_{s<i\neq j}\card \{\ell_{i}\}\bigg]
\leq n(R\sqrt{\lambda_{+}}+1)^{n-1}.\cr}
$$ 
The number $N_R(\lambda)$ is therefore bounded by
$$
\dim \cF_{\lambda}\geq
\card\Big\{(p,\ell)\in\bN^{s}\times\bZ^{n-s}\,;\;
\lambda_{p,\ell'} + \frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda\Big\}-O(R^{n-1}).
$$ 
A combination of Lemma 1.13 with the next lemma below shows that the
inequality $\liminf R^{-n}N_{R}(\lambda)\geq \nu_{B}(\lambda)$ 
can be obtained from calculations that are similar to those
used in the upper bound estimate of $N_{R}(\lambda)$.
\medskip

{\statement Lemma 1.19.\pointir} {\it Let $p\in\bN^{s}$ be a fixed multi-index.
Then there is a constant\break $C=C(p,B)\geq 0$ such that
$$
\lambda_{p,\ell'} \leq\Big(1+\frac{C}{R}\Big)\sum_{j=1}^{s}(2p_{j}+1)B_{j}
$$ 
when $|\ell_{j}| \leq\frac{B_{j}R^{2}}{4\pi}(1-R^{-\frac{1}{2}})$, 
$1\leq j\leq s$.}
\medskip

{\it Proof}.~-- We use again the minimax principle and the fact
that the Hermite functions $\Phi_{p}(\sqrt{B_{j}}t)$ are good 
approximations of the eigenfunctions of $q$ on any sufficiently large
interval centered at $0$. When $|\ell_{j}| \leq\frac{B_{j}R^{2}}{4\pi}
(1-R^{-\frac{1}{2}})$ and $x_{j} \in{}]-\frac{R}{2},\frac{R}{2}[$, 
the parameter $t=x_{j}+ \frac{2\pi\ell_{j}}{B_{j}R}$ that appears in (1.8) 
indeed runs on an interval containing $]-\frac{\sqrt{R}}{2},
\frac{\sqrt{R}}{2}[$. Therefore, we have 
$\lambda_{p_{j},\ell_{j}}\leq\widetilde\lambda_{p_{j}}$ where 
$(\widetilde\lambda_{m})_{m\in\bN}$ is the sequence of eigenvalues of the 
quadratic form
$$
\widetilde{q}(f)=\int\Big[\,\Big|\frac{df}{dt}\Big|^{2}+(B_{j}t)^{2}|f|^{2}\Big]dt,
\qquad
f\in W_{0}^{1}\Big(\,\Big]-\frac{\sqrt{R}}{2}, \frac{\sqrt{R}}{2}\Big[\,\Big).
$$ 
Let $\chi_{R}$ be a cut-off function with support in 
$\big[-\frac{\sqrt{R}}{2},\frac{\sqrt{R}}{2}\big]$, equal to $1$ on 
$\big[-\frac{\sqrt{R}}{4},\frac{\sqrt{R}}{4}\big]$, and whose derivative 
is bounded by~$5/\sqrt{R}$. For any linear combination
$$
f=\sum_{m\leq p_{j}}c_{m}\Phi_{m}(\sqrt{B_{j}}t),
$$ 
the exponential decay of the $\Phi_{m}$ functions at infinity
implies for $R$ large enough an inequality
$$
\Vert f\Vert \leq\Big(1+C_{1}\exp\Big(-\frac{R}{C_{1}}\Big)\Big)
\Vert\chi_{R}f\Vert 
$$ 
where $C_{1}=C_{1}(p_{j}, B_{j})>0$. Therefore, we obtain
$$
\eqalign{
\widetilde{q}(\chi_{R}f)
&\leq\widetilde{q}(f)+\int_{|t|>\sqrt{R}/4}\bigg(\frac{10}{\sqrt{R}}\Big|f
\frac{df}{dt}\Big|+\frac{25}{R}|f|^{2}\bigg)\,dt\cr
&\leq\widetilde{q}(f)+\int_{|t|>\sqrt{R}/4}\bigg(\frac{1}{R}\Big|\frac{df}{dt}\Big|^{2}
+25\Big(1+\frac{1}{R}\Big)|f|^{2}\bigg)\,dt\cr
&\leq\Big(1+\frac{C_{2}}{R}\Big)\widetilde{q}(f)
\leq\Big(1+\frac{C_{2}}{R}\Big)(2p_{j}+1)B_{j}\,\Vert f\Vert^{2}\cr
&\leq\Big(1+\frac{C}{R}\Big)(2p_{j}+1)B_{j}\,\Vert \chi_{R}f\Vert^{2}\cr}
$$ 
This gives $\lambda_{p_{j},\ell_{j}}\leq\widetilde\lambda_{p_{j}}
\leq\big(1+\frac{C}{R}\big)(2p_{j}+1)B_{j}$.\hfil\square
\medskip

For the reader's convenience, we now state
the minimax principle in the exact form it has been applied above.
\medskip

{\statement Proposition 1.20 {\rm (minimax principle, see\ [14], Vol.~IV, 
p.~76 and 78)}{\bf.\pointir}} {\it Let $Q$ be a quadratic form with 
dense domain $D(Q)$ in a Hilbert space $\cH$. We assume that $Q$ is
bounded from below, i.e.\ $Q(f)\geq-C\Vert f\Vert^{2}$ for
$f\in D(Q)$, that $D(Q)$ is complete for the norm $\Vert f\Vert_{Q}=[Q(f)+
(C+1)\Vert f\Vert^{2}]^{\frac{1}{2}}$, and finally that the injection 
morphism $(D(Q), \Vert~~\Vert_{Q})\hookrightarrow(\cH,\Vert~~\Vert)$ is 
compact. Then $Q$ has a discrete spectrum 
$\lambda_{1}\leq\lambda_{2}\leq\ldots~$, and we have equalities 
{\parindent=6.5mm
\vskip3pt
\item{\rm(a)}
$\displaystyle\lambda_{p}=\min_{F\subset D(Q)}~~\max_{f\in F,\,\Vert f\Vert =1}Q(f),$ 
\vskip2pt
where $F$ runs over the set of subspaces of dimension $p$ of $D(Q)\;;$ 
\vskip3pt
\item{\rm(b)}
$\displaystyle\lambda_{p+1}=\max_{F\subset D(Q)}~~\min_{f\in F,\,\Vert f\Vert =1}Q(f),$ 
\vskip2pt
where $F$ runs over the set of $Q$-closed subspaces
of codimension $p$ of~$D(Q)$.\vskip-\parskip}}
\bigskip

\section{2}{Asymptotic distribution of the spectrum\\
(case of a variable field).}

We consider again the general framework described at the beginning of 
section~\S1. Our goal is to study the spectrum of the quadratic form 
$Q_{\Omega,k}$ (see (1.3)) in the case of arbitrary magnetic field
$B$ and electric field~$V$. At any point $a\in M$, let
$$
B(a)=\sum_{j=1}^{s}B_{j}(a)\,dx_{j}\wedge dx_{j+s}
\leqno(2.1)   
$$
be the normalized expression of $B(a)$ in a suitable orthonormal basis
$(dx_{1},\ldots,dx_{n})$ of $T_a^*M$, where $2s=2s(a)={}$rank
of $B(a)$, $2s\leq n$, and $B_{1}(a)\geq B_{2}(a)\geq\ldots\geq B_{s}(a)>0$ 
are the positive eigenvalues of the associated
antisymmetric endomorphism. The defining equality 1.5
allows us to view $\nu_{B}(\lambda)$ as a function of the
pair$(a,\lambda)\in M\times\bR$. It will also be useful 
to consider the function $\overline{\nu}_{B}(\lambda)$,
that is right continuous in $\lambda$, defined by
$$
\overline{\nu}_{B}(\lambda)=\lim_{0<\varepsilon\to 0}
\nu_{B}(\lambda+\varepsilon).
\leqno(2.2)
$$ 
We then prove the following generalization of Corollary~1.7.
\medskip

{\statement Theorem 2.3. {\it When $k$ tends to~$+\infty$,
the number $N_{\Omega,k}(\lambda)$ of eigenvalues${}\leq\lambda$ of 
$Q_{\Omega,k}$ satisfies the asymptotic inequalities
$$
\int_{\Omega}\nu_{B}(V+\lambda)\,d\sigma\leq\lim\inf k^{-\frac{n}{2}}N_{\Omega,k}
(\lambda)
\leq\lim\sup k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)\leq\int_{\Omega}
\overline{\nu}_{B}(V+\lambda)\,d\sigma.
$$\vskip-\parskip}}

The function $\lambda\mapsto\int_{\Omega}\nu_{B}(V+\lambda)\,d\sigma$ is non 
decreasing and left continuous~; therefore it has at most a
countable set $\cD$ of points of discontinuity. The set $\cD$ is in
fact empty when $n$ is odd, since $\nu_B(\lambda)$ is then continuous.
From this, we immediately infer
\medskip

{\statement Corollary 2.4. {\it Assume that $\partial\Omega$ is 
of measure zero. Then
$$
\lim_{k\to+\infty}k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)=\int_{\Omega}\nu_{B}(V+
\lambda)\,d\sigma
$$ 
for all $\lambda\in\bR\ssm \cD$, and the spectral density measure
$k^{-\frac{n}{2}} \frac{d}{d\lambda}N_{\Omega,k}(\lambda)$ converges
weakly on~$\bR$ to $\frac{d}{d\lambda}\int_{\Omega}\nu_{B}(V+\lambda)\,
d\sigma$. If $n$ is odd, the limit measure has no atoms.\hfil\square}}
\medskip

The following lemma shows that the integrals involved in Theorem 2.3 are
well defined.
\medskip

{\statement Lemma 2.5.}{\it
{\parindent 6.5mm
\vskip2pt
\item{\rm(a)} We have inequalities 
$$\nu_{B}(\lambda)\leq\overline{\nu}_{B}(\lambda)
\leq\lambda_{+}^{n/2}.$$ 
\vskip2pt
\item{\rm(b)} $\nu_{B}(V)$ $($resp.\ $\overline{\nu}_{B}(V))$ is
lower $($resp.\ upper$\,)$  semi-continuous on~$M$.
\vskip2pt
\item{\rm(c)} At any point $x\in M$ where $s(x)< \frac{n}{2}$ we have 
$\nu_{B}(V)(x)=\overline{\nu}_{B}(V)(x)$, and the functions
$\nu_{B}(V)$, $\overline{\nu}_{B}(V)$ are continuous in~$x$.
\vskip2pt
\item{\rm(d)} If $n$ is odd, $\nu_{B}(V)=\overline{\nu}_{B}(V)$ is 
continuous on~$M$.\vskip-\parskip}}
\medskip

{\it Proof.~--} (a) We always have $\big(\lambda-\sum(2p_{j}+1)B_{j}
\big)^{\frac{n}{2}-s}_+\leq\lambda^{\frac{n}{2}-s}_+$, and the number of integers 
$p_{j}$ such that $\lambda-(2p_{j}+1)B_{j}$ is${}\geq 0$ is bounded above by
$\frac{\lambda_{+}}{B_{j}}$. As the resulting numerical factor occurring
in (1.5) is bounded by~$1$, inequality (a) follows.

(b, c) The rank $s=s(x)$ is a lower semi-continuous function
on~$M$, and the eigenvalues $B_{1},\,B_{2},\,\ldots\,$, extended
by $B_{j}(x)=0$ for $j>s(x)$, are continuous on~$M$. As the function
$t\mapsto t_{+}^{0}$ (resp.\ $t\mapsto(t+0)_{+}^{0})$ is lower
(resp.\ upper) semi-continuous, the semi-continuity of
$\nu_{B}(V)$ and $\overline{\nu}_{B}(V)$ is a problem only
at points $a\in M$ in the neighborhood of which $s(x)$ is not locally
constant. At such a point $a\in~M$, we necessarily have $s(a)<
\frac{n}{2}$, so $\nu_{B}(V)(a)=\overline{\nu}_{B}(V)(a)\;$; we are going to 
show that $\nu_{B}(V)$ and $\overline{\nu}_{B}(V)$ are then continuous
at~$a$. The continuity of $B_{j}$ gives $\lim_{x\to a}B_{j}(x)=0$ for
$j>s(a)$. When the integers $p_{1},\,\ldots\,,p_{s\langle a)}$ are fixed, 
the summation in (1.5) may be interpreted as a Riemann sum
for an integral over $\bR^{s(x)-s(a)}$, and we therefore get
the equivalent
$$
\eqalign{
\sum_{(p_{j};\;s(a)<j\leq s(x))}&\Big(V(x)-\sum(2p_{j}+1)B_{j}(x)
\Big)^{\frac{n}{2}-s(x)}_+\cr
&\sim\int_{t\in\bR^{s(x)-s(a)}}\bigg[V(a)-\sum_{j=1}^{s(a)}(2p_{j}+1)B_{j}(a)-
\sum_{j=s(a)+1}^{s(x)}2t_{j}B_{j}(x)\bigg]^{\frac{n}{2}-s(x)}_+dt\cr
&=\frac{2^{s(a)-s(x)}\Big(V(a)-\sum(2p_{j}+1)
B_{j}(a)\Big)^{\frac{n}{2}-s(a)}_+}{(\frac{n}{2}-s(x)+1)\cdots
(\frac{n}{2}-s(a))B_{s(a)+1}(x)\cdots B_{s(x)}(x)}~.\cr}
$$ 
Therefore, we find
$$
\lim_{x\to a}\nu_{B}(V)(x)=\nu_{B}(V)(a)=\lim_{x\to a}\overline{\nu}_{B}(V)(x).
$$ 
(d) Is a special case of (c).\hfil\square\medskip

The proof of theorem 2.3 is based essentially on two ingredients~:
firstly, an asymptotic localization principle of the eigenfunctions, which
is can be seen through a direct application of the minimax principle
(Proposition~2.6)~; secondly, an explicit evaluation of the
spectrum of the Schrödinger operator associated with a constant
magnetic field (see~\S1). Indeed, the localization principle
reduces the situation to the case of a constant field
by using a covering of $\Omega$ by small cubes.
\medskip

{\statement Proposition 2.6.\vskip3pt
\rm (a)} {\it If 
$\Omega_{1}, \cdots, \Omega_{N}\subset\Omega$ are pairwise disjoint
open subsets, then
$$
N_{\Omega,k}(\lambda)\geq\sum_{j=1}^{N}N_{\Omega_{j},k}(\lambda).
$$ 
{\parindent=6.5mm
\item{\rm(b)} Let $(\Omega_{j}')_{1\leq j\leq\bN}$ an open covering
of $\overline\Omega$ and let $(\psi_{j})_{1\leq j\leq\bN}$ be a system
of functions 
$\psi_j\in \cC^{\infty}(\bR^{n})$ with support in $\Omega_{j}'$, such
that $\sum\psi_{j}^{2}=1$ on~$\overline\Omega$. We set
$$
C(\psi)=\sup_{\Omega}\sum_{j=1}^{N}|d\psi_{j}|^{2}.
$$ 
Then
$$
N_{\Omega,k}(\lambda)\leq\sum_{j=1}^{N}N_{\Omega_{j}',k}\Big(\lambda+\frac{1}
{k}C(\psi)\Big).$$\vskip-\parskip}}


{\it Proof}.~-- (a) Let $\cF$ be the $\bC$-vector space
generated by the collection of all eigenfunctions of the
quadratic forms $Q_{\Omega_{j},k}$, $1\leq j\leq N$, corresponding to
eigenvalues${}\leq\lambda$. The space $\cF$ is of dimension
$$
\dim \cF=\sum_{j=1}^{N}N_{\Omega_{j},k}(\lambda)
$$ 
and for all $u\in \cF$, we have
$$
Q_{\Omega,k}(u)=\sum_{j=1}^{N}Q_{\Omega_{j},k}(u)\leq\sum_{j=1}^{N}\lambda\Vert u\Vert_{\Omega_{j}'}^{2}=\lambda\Vert u\Vert_{\Omega}^{2}.
$$ 
The minimax principle therefore shows that the eigenvalues of $Q_{\Omega,k}$ 
of index${}\leq\dim \cF$ are${}\leq\lambda$, whence inequality (a).

(b) For every $u\in W_{0}^{1}(\Omega,E^{\otimes k})$, we have
$$
\sum_{j}|D_{k}(\psi_{j}u)|^{2}=\sum_{j}\big|\psi_{j}D_{k}u+(d\psi_{j})u\big|^{2}
=|D_{k}u|^{2}+\sum_{j}|d\psi_{j}|^{2}|u|^{2}
$$ 
since $2\sum\psi_{j}d\psi_{j}=d(\sum\psi_{j}^{2})=0$. Therefore we get
$$
\sum_{j=1}^{N}Q_{\Omega_{j}',k}(\psi_{j}u)=Q_{\Omega,k}(u)+\int_{\Omega}\frac{1}
{k}\sum_{j=1}^{N}|d\psi_{j}|^{2}|u|^{2}\,d\sigma
\leq Q_{\Omega,k}(u)+\frac{1}{k}C(\psi)\Vert u\Vert_{\Omega}^{2}.
$$ 
If each function $H^{q}(X,E^{\otimes k}\otimes F)$c is orthogonal
to the eigenfunctions of $Q_{\Omega_{j},k}$ associated with the
eigenvalues${}\leq\lambda+\frac{1}{k}C(\psi)$, we infer successively
$$
\eqalign{
Q_{\Omega_{j},k}(\psi_{j}u)&>\Big(\lambda+\frac{1}{k}C(\psi)\Big)\Vert \psi_{j}
u\Vert_{\Omega_{j}}^{2},\quad\hbox{if $\psi_{j}u\neq 0$},\cr
Q_{\Omega,k}(u)&>\lambda\Vert u\Vert_{\Omega}^{2},\quad\hbox{if $u\neq 0$}.\cr}
$$
The minimax principle 1.20 (b) then shows that $N_{\Omega,k}(\lambda)$ 
is bounded above by the number of linear constraints that we had to
impose on $u$, namely
$$
\sum_{j=1}^{N}N_{\Omega_{j},k}\Big(\lambda+\frac{1}{k}C(\psi)\Big).\eqno\square 
$$ 
Let $W_{1}, \ldots, W_{N}$ be a covering of $\Omega$ by open coordinate charts
of the manifold~$M$. For any $\varepsilon>0$, we can find open sets
$\Omega_{i}\subset\Omega_{j}'$, that are relatively compact in $W_{j}$, 
$1\leq j\leq N$, and such that
$$
\leqalignno{
\Omega&\supset\bigcup\Omega_{j}~~\hbox{(disjoint),~~~and~ $\Vol(\Omega)=
\sum\Vol(\Omega_{j})$},&(2.7)\cr
\overline\Omega&\subset\bigcup\Omega_{j}',\kern49pt\hbox{~~and~ 
$\sum \Vol(\overline\Omega_{j}')\leq\Vol(\overline\Omega)+\varepsilon$}.
&(2.8)\cr}
$$ 
Proposition 2.6 then reduces the proof of Theorem 2.3
to the case of the open sets $\Omega_{j}$ and $\Omega_{j}'$ (observe
for this that the function $\nu_{B}(V+\lambda)$ is bounded and that
the constant  $C(\psi)$ is independent of~$k$ ).

In the end, we can assume that $M=\bR^{n}$, with an arbitrary Riemannian metric
$g$. Since $M=\bR^{n}$ is contractible, the bundle $E$ is then trivial~;
let $A$ be a vector potential of the 
connection $D$ and $B=dA$ the corresponding magnetic field. 
We first prove the following local version of Theorem 2.3.
\medskip

{\statement Proposition 2.9.\pointir}{\it Let $a\in\bR^{n}$ be a given point, 
and $P_{k}$ a sequence of open cubes such that $P_{k}\ni a$.
Denote by $r_{k}$ the side length of $P_{k}$, and assume that
$$
r_{k}\leq 1,\qquad\lim k^{\frac{1}{2}}r_{k}=+\infty,\qquad
\lim k^{\frac{1}{4}}r_{k}=0.
$$ 
Then, when $k$ tends to $+\infty$, we have
$$
\eqalign{
&\liminf\frac{k^{-\frac{n}{2}}}{\Vol(P_{k})}N_{P_{k},k}(\lambda)\geq 
\nu_{B(a)}(V(a)+\lambda),\cr
&\limsup\frac{k^{-\frac{n}{2}}}{\Vol(P_{k})}N_{P_{k},k}(\lambda)\leq
\overline{\nu}_{B(a)}(V(a)+\lambda),\cr}
$$ 
and for every compact $K\subset\bR^{n}$, $N_{P_{k},k}(\lambda)$ admits the 
upper bound
$$
N_{P_{k},k}(\lambda)\leq C_{K}
\Big(1+r_{k}\sqrt{k\big(\lambda_{+}+\max_{K}V_{+}\big)}\Big)^{n}
$$ 
uniformly with respect to the point~$a$, as long as $P_{k}\subset K$.}
\medskip

{\it Proof}.~-- We proceed via a reduction to Theorem 1.6, after applying 
an homothety of ratio $\sqrt{k}$ to $P_{k}$ -- this is the reason
why we had to assume $\lim k^{\frac{1}{2}}r_{k}=+\infty$.
The following lemma measures how much the magnetic field
$B$ deviates from the field constant $B(a)$ on each cube~$P_{k}$.
\medskip

{\statement Lemma 2.10.\pointir} {\it On each cube $\overline P_{k}$,
one can choose a vector potential $\widetilde A_{k}$ of the constant
field $B(a)$ tel that for all $x\in\overline{P}_{k}$ we have
$$
|A_{k}(x)-A(x)|\leq C_{1}r_{k}^{2},
$$ 
where $C_{1}\geq 0$ is a constant independent of $k$ $($and 
independent of $a$ when $a$ runs over a compact set $K\subset\bR^{n})$.}
\medskip

The $\cC^{\infty}$ regularity of $B$ indeed leads to an upper bound
$$
|B(a)-B(x)|\leq C_{2}r_{k},\qquad x\in\overline P_{k}.
$$ 
Let $A_{k}'$ be a potential of the field $B(a)-B(x)$ on the cube
$\overline P_{k}$, calculated using the usual homotopy formula for
starlike open sets. We then get
$$
|A_{k}'(x)|\leq C_{3}r_{k}^{2},
$$ 
and it is sufficient to take $\widetilde A_{k}=A+A_{k}'$.\hfil\square\medskip

Let $(x_{1},\ldots,x_{n})$ be the standard coordinates on $\bR^{n}$.
Let us take new coordinates $(y_{1},\ldots,y_{n})$ depending linearly
on $x_{1}, \ldots, x_{n}$ such that $(dy_{1},\ldots,dy_{n})$ is an
orthonormal basis for the metric $g$ at point $a$, and such that
$B(a)$ admits in this basis a normalized form as in~(2.1)~:
$$
B(a)=\sum_{j=1}^{s}B_{j}(a)\,dy_{j}\wedge dy_{j+s}.
$$ 
Let $\widetilde{g}$ be the constant metric
$$
\widetilde{g}\equiv g(a)=\sum_{j=1}^{n}dy_{j}^{2}.
$$ 
Let us denote by $D_{k}=d+ikA\wedge{?}$,
$\widetilde D_{k}=d+ik\widetilde A_k\wedge{?}$ the 
connections on $E^{\otimes k}_{|P_{k}}$ associated with the potentials
$A$, ${\widetilde A}_{k}$, and by $Q_{k}=Q_{P_{k},k}$, ${\widetilde Q}_{k}$ 
the quadratic forms  associated respectively with the 
metrics $g$, $\widetilde{g}$, and the scalar potentials $V$, 
$\widetilde{V}\equiv V(a)$ (formula (1.3)).
\medskip

{\statement Lemma 2.11.\pointir}{\it There exists a sequence
$\varepsilon_{k}$ converging to $0$ $($ dépending on $r_{k}$,
but independent of $a$ when $a$ runs over a compact set $K\subset\bR^{n})$,
such that the global $L^{2}$ norms $\Vert~~\Vert_{g}$,
$\Vert~~\Vert_{\tilde g}$ associated with the metrics $g$ and
$\widetilde{g}$ satisfy
$$
\eqalign{
(1-\varepsilon_{k})\Vert u\Vert_{\tilde g}^{2}
&\leq\Vert u\Vert_{g}^{2}\leq(1+\varepsilon_{k})\Vert u\Vert_{\widetilde{g}}^{2},\cr
(1-\varepsilon_{k})\widetilde Q_{k}(u)-\varepsilon_{k}
\Vert u\Vert_{\tilde g}^{2}&\leq Q_{k}(u)\leq(1+\varepsilon_{k})
\widetilde Q_{k}(u)+\varepsilon_{k}\Vert u\Vert_{\tilde g}^{2}\cr}
$$ 
for all $u\in W_{0}^{1}(P_{k})$.}\medskip

Indeed, we have on $P_{k}$ inequalities
$$
(1-C_{4}r_{k})\,\widetilde{g}\leq g\leq(1+C_{4}r_{k})\,\widetilde{g},
$$ 
and this gives the first double inequality in 2.11. With the 
notation $A_{k}'=A_{k}-A$, we infer from there
$$
\eqalign{
Q_{k}(u)&=\int_{P_{k}}\Big(\frac{1}{k}|\widetilde D_{k}u-ikA_{k}'\wedge u|_{g}^{2}
-V|u|^{2}\Big)\,d\sigma\cr
&\leq(1+C_{5}r_{k})\int_{P_{k}}\Big(\frac{1}{k}|\widetilde D_{k}u-ikA_{k}'\wedge u
|^2_{\tilde g}-V(a)|u|^{2}\Big)\,d\widetilde{\sigma}+
\eta_{k}\Vert u\Vert^2_{\tilde g}\cr}
$$ 
with $\eta_{k}=\sup_{P_{k}}|V-V(a)|+C_{6}r_{k}$, a quantity that converges
to $0$ as $k$ tends to~$+\infty$. Using the inequality 
$(a+b)^{2}\leq(1+\alpha)(a^{2}+\alpha^{-1}b^{2})$, Lemma 2.10 implies 
on the other hand
$$
|\widetilde D_{k}u-ikA_{k}'\wedge u|^2_{\tilde g}\leq
(1+\alpha)\Big[\,|\widetilde D_{k}u|^{2}_{\tilde g}+
\alpha^{-1}C_{1}^{2}k^{2}r_{k}^{4}|u|^{2}\Big].
$$ 
Let us choose $\alpha=\alpha_{k}=C_{1}\sqrt{k}r_{k}^{2}$. The sequence
$\alpha_{k}$  tends to $0$ by the assumption $\lim k^{\frac{1}{4}}r_{k}=0$,
and we find
$$
\frac{1}{k}|\widetilde D_{k}u-ikA_{k}'\wedge u|^{2}_{\tilde g}\leq
(1+\alpha_{k})\Big[\frac{1}{k}|D_{k}u|^{2}_{\tilde g}+\alpha_{k}|u|^{2}\Big].
$$ 
This implies the upper bound for $Q_{k}$. The lower bound is obtained in
the same way by means of the inequality 
$(a+b)^{2}\geq(1-\alpha)(a^{2}-\alpha^{-1}b^{2})$.\hfil\square\medskip

Lemma 2.11 reduces the proof of Proposition~2.9 to the case where the metric
$g$ and the magnetic field $B$ are constant~:
$$
g=\sum_{j=1}^{n}dy_{j}^{2},\qquad B=\sum_{j=1}^{n}B_{j}\,dy_{j}\wedge dy_{j+s}.
$$ 
We can assume moreover that $V \equiv 0$ by performing a translation 
$\lambda\mapsto\lambda+V(a)$. The only remaining difficulty for 
applying directly Theorem 1.6 comes from the fact that the cubes
$P_{k}$ become oblique parallelepipeds in the coordinates
$(y_{1}, \ldots,y_{n})\;$; the angles 
between the different edges of each $P_{k}$ and the ratios of 
their side lengths remain however bounded by positive constants.
In order to solve this difficulty, it is sufficient to cover each
parallelepiped $P_{k}$ by cubes $P_{k,\alpha}$ whose edges are parallel
to the coordinate axes $(y_{1}, \ldots,y_{n})$. Let us fix
$\varepsilon\in{}]0,1[$. For all $\alpha\in\bZ^{n}$, let
$(P_{k,\alpha})$, $(P_{k,\alpha}')$ be the open cubes of
side lengths $\varepsilon r_{k}$, $\varepsilon(1+\varepsilon)r_{k}$,
and the common center $\varepsilon r_{k}\alpha$. We can limit
ourselves to the consideration of the cubes  $P_{k,\alpha}$ contained in
$P_{k}$ and of the cubes $P_{k,\alpha}'$ meeting~$P_{k}$. Then we get
$$
\leqalignno{
&P_{k}\supset\bigcup_{\alpha}P_{k,\alpha}~~\hbox{(disjointe),~~~and}~~~
\frac{\sum_\alpha\Vol(P_{k,\alpha})}{\Vol(P_{k})}\geq 1-C_{7}\varepsilon,
&(2.12)\cr
&P_{k}\subset\bigcup_{\alpha}P_{k,\alpha}',\kern60pt\hbox{and}~~~
\frac{\sum_\alpha\Vol(P_{k,\alpha}')}{\Vol(P_{k})}
\leq 1+C_{7}\varepsilon,&(2.13)\cr}
$$ 
where $C_{7}$ is a constant independent of $k$ (and also of $a$, 
when $a$ runs over a compact set). The number of cubes $P_{k,\alpha}$, 
$P_{k,\alpha}'$ which are involved in (2.12) or (2.13) is bounded above by 
$C_{8}\varepsilon^{-n}$. As the cubes $P_{k,\alpha}'$ overlap each other
on an interval of length${}\sim\varepsilon^{2}r_{k}$ 
when they are contiguous, one can construct a partition of unity
$\sum \psi_{k,\alpha}^{2}=1$ on $P_{k}$, such that
\hbox{Supp$\,\psi_{k,\alpha}\subset P_{k,\alpha}'$} and
$$
\sup_{P_{k}}\sum_{\alpha}|d\psi_{k,\alpha}|^{2}=C(\psi_{k})\leq C_{9}
(\varepsilon^{2}r_{k})^{-2}.
$$ 
The hypothesis $\lim k^{\frac{1}{2}}r_{k}=+\infty$ actually implies
$\lim\frac{1}{k}C(\psi_{k})=0$, and this allows to apply 2.6~(b). 
On the cubes $P_{k\alpha}$, $P_{k,\alpha}'$ we are
now in the situation of Theorem~1.6~:
after applying a homothety of ratio $\sqrt{k}$, the side of the
homothetic cube $\sqrt{k}\,P_{k,\alpha}$ becomes
$R_{k}=\varepsilon r_{k}\sqrt{k}$ and this value indeed tends
to $+\infty$ by hypothesis. The uniform upper bound of 
$N_{P_{k},k}(\lambda)$ follows from Proposition 1.18 and from the fact that 
all our constants $C_{1}, \ldots, C_{9}$ are uniform. Proposition 2.9
is proved.\hfil\square\medskip

{\it Proof of Theorem 2.3.}~-- According to the remark 
preceding Proposition 2.9, we can assume $M=\bR^{n}$ and $\Omega$ to be a
bounded open subset of $\bR^{n}$. The main idea of our argument is to 
combine Propositions 2.6 and 2.9 with a covering of $\Omega$ by cubes of 
side $r_{k}=k^{-\frac{1}{3}}$. The actual implementation requires
nevertheless some care, in view of the difficulties related to
the possible non-uniformity of the $\limsup$ and $\liminf$ involved.

Let us denote by $\Pi_{k,\alpha}$, $\Pi_{k,\alpha}'$, $\alpha\in \bZ^{n}$, 
the open cubes of respective side lengths
$$
k^{-\frac{1}{3}},\qquad k^{-\frac{1}{3}}(1+ k^{-\frac{1}{8}})=
k^{-\frac{1}{3}}+k^{-\frac{11}{24}}
$$ 
and common center $k^{-\frac{1}{3}}\alpha$. Let $I(k)$ (resp.\ 
$I'(k)$) be the set of indices $\alpha\in\bZ^{n}$ such that
$\Pi_{k.\alpha}\subset\Omega$ (resp.\ $\overline\Pi_{k,\alpha}'
\cap\overline\Omega\neq\emptyset$). As in the
reasoning of Proposition 2.9, there exists a partition of unity
$\sum_{\alpha\in I'(k)}\psi_{k,\alpha}^{2}=1$ on $\Omega$, such that 
\hbox{Supp$\,\psi_{k,\alpha}\subset\Pi_{k,\alpha}'$} and
$$
C(\psi_{k})=\sup_{\Omega}\sum_{\alpha\in I'(k)}|d\psi_{k,\alpha}|^{2}
\leq C_{10}k^{\frac{11}{12}},
$$ 
hence $\lim\frac{1}{k}C(\psi_{k})=0$. We set
$$
\Omega_{k}=\bigcup_{\alpha\in I(k)}\Pi_{k,\alpha},\qquad
\Omega_{k}'=\bigcup_{\alpha\in I'(k)}\Pi_{k,\alpha}'
$$ 
and consider for every given $\lambda\in\bR$, the functions 
on $\bR^{n}$ defined by
$$
\eqalign{
f_{k}&=k^{-\frac{n}{2}}\sum_{\alpha\in I(k)}N_{\Pi_{k,\alpha},k}(\lambda)
\frac{1}{\Vol(\Pi_{k,\alpha})}\bOne_{\Pi_{k,\alpha}},\cr
f_{k}'&=k^{-\frac{n}{2}} \sum_{\alpha\in I'(k)}N_{\Pi'_{k,\alpha},k}
\Big(\lambda+\frac{1}{k}C(\psi_{k})\Big)
\frac{1}{\Vol(\Pi_{k,\alpha})}\bOne_{\Pi_{k,\alpha}}
\cr}
$$ 
where $\bOne_{\Pi_{k,\alpha}}$ denotes the characteristic function of 
$\Pi_{k,\alpha}$. Proposition~2.6 implies the inequalities
$$
\int_{\bR^n}f_{k}\,d\sigma\leq k^{-\frac{n}{2}}N_{\Omega,k}
(\lambda)\leq\int_{\bR^n}f_{k}'\,d\sigma.
\leqno(2.14)
$$ 
Let $x\in\bR^{n}$ be a point taken in the complement of the 
negligible set
$$
Z=\bigcup_{k\in\bN,\,\alpha\in\bZ^n}\partial\Pi_{k,\alpha}.
$$ 
Then there exists a unique sequence of indices $\alpha(k)\in\bZ^{n}$ such 
that $x\in\Pi_{k,\alpha(k)}$. Proposition 2.9 applied to the cubes
$P_{k}=\Pi_{k,\alpha(k)}$ (resp.\ $P_{k}'=\Pi_{k,\alpha(k)}'$) 
with $\Vol(P_{k})\sim \Vol P_{k}'$ shows that the pointwise sequences 
$$
f_{k}(x)= \frac{k^{-\frac{n}{2}}}{\Vol(P_{k})}N_{P_{k},k}(\lambda)
\bOne_{\Omega_{k}}(x),\qquad f_{k}'(x)= \frac{k^{-\frac{n}{2}}}
{\Vol(P_{k})}N_{P_{k}',k}(\lambda)\bOne_{\Omega_{k}'}(x),
$$ 
satisfy
$$\cases{
\liminf f_{k}(x)\kern3pt{}\geq \nu_{B(x)}(V(x)+\lambda)\,\bOne_{\Omega}(x),\cr
\noalign{\vskip5pt}
\limsup f_{k}'(x)\leq\overline{\nu}_{B(x)}(V(x)+\lambda)\,
\bOne_{\overline\Omega}(x).\cr}
\leqno(2.15)
$$ 
On the other hand, the uniform upper bound of Proposition~2.9 implies
the existence of constants $C_{11}$, $C_{12}$ independent of 
$k$, $x$ and $\lambda$ such that
$$
f_{k}(x)\leq f_{k}'(x)\leq C_{11}\big(1+\sqrt{\lambda_{+}+C_{12}}\,\big)^{n}.
$$ 
Theorem 2.3 then follows from (2.14), (2.15) and the Fatou 
lemma.\hfil\square\medskip

In view of applications to complex geometry, we will need below a slight
generalization of Theorem~2.3. Consider a Hermitian vector bundle $F$
of rank $r$ and of class $\cC^{\infty}$ over~$M$, equipped with a 
Hermitian connection $\nabla$, and continuous sections 
$S$ of the fiber $\Lambda_{\bR}^{1}T^{*}X\otimes_{\bR}\Hom_{\bC}(F,F)$ 
and $V$ of the bundle $\Herm(F)$ of Hermitian endomorphisms of~$F$. Let
$\nabla_{k}$ be the Hermitian connection on $E^{\otimes k}\otimes F$ induced 
by the connections $D$ and $\nabla$. For the simplicity of 
notation, we will still denote by $S$ and $V$ the endomorphisms 
$\Id_{E^{\otimes k}}\otimes S$ and $\Id_{E^{\otimes k}}\otimes V$ 
acting on~$E^{\otimes k}\otimes F$. Given a relatively compact 
open set $\Omega$ in~$M$, we consider the quadratic form
$$
Q_{\Omega,k}(u)=\int_{\Omega}\Big(\frac{1}{k}|\nabla_{k}u+Su|^{2}-\langle Vu,
u\rangle\Big)\,d\sigma,
$$ 
where $u\in W_{0}^{1}(\Omega,E^{\otimes k}\otimes F)$. Let $V_{1}(x)\leq V_{2}(x)
\leq\cdots\leq V_{r}(x)$ be the eigenvalues of $V(x)$ at any point
$x\in M$. In this setting, the following result holds.\medskip

{\statement Theorem 2.16.\pointir}{\it The counting function 
$N_{\Omega,k}(\lambda)$ of the eigenvalues of $Q_{\Omega,k}$ admits for 
all $\lambda\in\bR$ asymptotic estimates
$$
\eqalign{
\liminf_{k\to+\infty}~k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)
&\geq\sum_{j=1}
^{r}\int_{\Omega}\nu_{B}(V_{j}+\lambda)\,d\sigma,\cr
\limsup_{k\to+\infty}~k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)
&\leq\sum_{j=1}^{r}\int_{\Omega}\overline{\nu}_{B}(V_{j}+\lambda)\,d\sigma,\cr}
$$ 
where $B$ is the magnetic field associated with the connection $D$ 
on~$E$.}
\medskip

{\it Proof}.~-- The localization principle 2.6 is still 
valid in the 
present situation. It is therefore sufficient to prove the inequalities 
of proposition 2.16 when $\Omega$ is small enough. Let $a\in M$ be a fixed 
point and $(e_{1},\ldots,e_{r})$ a $\cC^{\infty}$ orthonormal frame of $F$ 
over a neighborhood $W$ of $a$, such that $(e_{1}(a),\ldots,e_{r}(a))$ 
diagonalizes the endomorphism $V(a)$. Let us express $u$ under the form
$$
u=\sum_{j=1}^{r}u_{j}\otimes e_{j}
$$ 
where $u_{j}$ is a section of $E^{\otimes k}$. For every $\varepsilon>0$, 
there is  exists a neighborhood $W_{\varepsilon}'\subset W$ of $a$ on which
$$
\sum_{j=1}^{r}(V_{j}(a)-\varepsilon)|u_{j}|^{2}\leq\langle Vu,u\rangle\leq\sum_{j=
1}^{r}(V_{j}(a)+\varepsilon)|u_{j}|^{2}
$$ 
On the other hand, we have
$$
\nabla_{k}u=\sum_{j=1}^{r}D_{k}u_{j}\otimes e_{j}+u_{j}\otimes\nabla e_{j},
$$ 
and the term $u_{j}\otimes\nabla e_{j}$ can be absorbed into $Su$ 
(this actually brings us back to the case where the connection $\nabla$ 
is flat). The inequalities
$$
(1-k^{-\frac{1}{2}})|\nabla_{k}u|^{2}+(1-k^{\frac{1}{2}})|Su|^{2}
\leq|\nabla_{k}u+Su|^{2}\leq
(1+k^{-\frac{1}{2}})|\nabla_{k}u|^{2}+(1+k^{\frac{1}{2}})|Su|^{2}
$$ 
shows that the term $Su$ only changes $Q_{\Omega,k}$ by a 
multiplicative factor $1\pm\varepsilon$ and by an additive factor 
$\pm\varepsilon\Vert u\Vert^{2}$. As a consequence, for every 
$\varepsilon>0$, there exist a neighborhood $W_{\varepsilon}$ of $a$ 
and an integer $k_{0}(\varepsilon)$ such that
$$
(1-\varepsilon)\widetilde Q_{\Omega.k}(u)-\varepsilon\Vert u\Vert^{2}
\leq Q_{\Omega,k}(u)\leq(1+\varepsilon)
\widetilde Q_{\Omega,k}(u)+\varepsilon\Vert u\Vert^{2},
$$ 
as soon as $k\geq k_{0}(\varepsilon)$ and $\Omega\subset W_{\varepsilon}$, 
where $\widetilde Q_{\Omega,k}$ denotes the quadratic form
$$
\widetilde Q_{\Omega.k}(u)=
\sum_{j=1}^{r}\int_{\Omega}\Big(\frac{1}{k}|D_{k}u_{j}|^{2}-V_{j}(a)
|u_{j}|^{2}\Big)\,d\sigma.
$$ 
Since $\widetilde Q_{\Omega,k}$ is a direct sum of $r$ quadratic
forms, the spectrum of $\widetilde Q_{\Omega.k}$ is the union
(counted with multiplicities) of the spectra of the terms
in the sum. Theorem 2.16 follows.\hfil\square\bigskip

\section{3}{Bochner-Kodaira-Nakano identity in Hermitian geometry.}

The goal of the forthcoming sections is to draw consequences of the 
spectral theorem~2.16 in the study of $d''$-cohomology of Hermitian
holomorphic vector bundles. In this direction, we need to relate the
conjugate-holomorphic Laplace-Beltrami operator $\Delta''$ with the
Schrödinger operator of a suitable real connection. This is done by
means of a Weitzenböck type formula, known in complex differential
geometry as the Bochner-Kodaira-Nakano identity.

Let $X$ be a compact complex analytic manifold of dimension $n$ 
and $F$ a Hermitian holomorphic vector bundle of rank $r$ over~$X$.
As is well known, there exists a unique Hermitian connection
$D=D'+D''$ on $F$ whose component $D''$ of type $(0,1)$ coincides
with the $\overline\partial$ operator of the bundle (this 
connection is sometimes termed ``holomorphic'' and called the
Chern connection). Let $c(F)=D^{2}=D'D''+D''D'$ be the curvature form of
$D$ on~$F$. Let us equip $X$ with an arbitrary Hermitian metric 
$\omega$ of type $(1,1)$ and class $\cC^{\infty}$. The space 
$\cC_{p,q}^{\infty}(X,F)$ of $\cC^\infty$ sections of the
bundle $\Lambda^{p,q}T^{*}X\otimes F$ is then equipped with a natural
prehilbertian structure. 
We let $\delta=\delta'+\delta''$ denote the formal adjoint
of~$D$, considered as a differential operator on 
$\cC^{\infty}(X,F)$, and $\Lambda$ the adjoint of the Lefschetz 
operator $L:u\mapsto\omega\wedge u$.

We will use the Hermitian Bochner-Kodaira-Nakano identity in the form it
was established in [6], although one could in fact get by with the less
precise formula given by P.~Griffiths, as Y.T.~Siu does in [16], [17].
For differential operators  $A$, $B$ on $\cC^{\infty}(X,F)$, one defines
their graded bracket $[A,B]$ by the formula
$$
[A,B]=AB-(-1)^{ab}BA
$$ 
where $a$, $b$ are the respective degrees of $A$ and~$B$. The 
Laplace-Beltrami operators $\Delta'$ and $\Delta''$ are then
classically given by
$$
\Delta'=[D',\delta']=D'\delta'+\delta'D',\qquad \Delta''=[D'', \delta''].
$$ 
With the torsion tensor $d'\omega$, we associate the wedge multiplication
operator $u\mapsto d'\omega\wedge u$ on 
$\cC^{\infty}(X,F)$, type $(2,1)$, which we still denote $d'\omega$, 
and the operator $\tau$ of type $(1,0)$ defined by
$\tau=[\Lambda,d'\omega]$. We finally put
$$
D_{\tau}'=D'+\tau,\qquad 
\delta_{\tau}' =(D_{\tau}')^{*}=\delta'+\tau^{*},\qquad
\Delta_{\tau}'=[D_{\tau}',\delta_{\tau}'].
$$ 
Then the following identity holds. The reader is referred to [6]
for a proof.\medskip

{\statement Proposition 3.1. {\it We have
$$
\Delta''=\Delta_{\tau}'+[ic(F),\Lambda]+T_{\omega}
$$
where $T_{\omega}$ is the operator of order $0$ and of type $(0,0)$ defined by
$$
T_{\omega}=\Big[\Lambda,\Big[\Lambda,\frac{i}{2}d'd''\omega\Big]\Big]-
[d'\omega,(d'\omega)^{*}].
$$}\vskip-\parskip}

According to the Hodge-De Rham theory, the sheaf cohomology group
$H^{q}(X,F)$ is isomorphic to the space of $\Delta''$-harmonic
$(0,q)$-forms with values in~$F$. Let $u\in \cC_{p.q}^{\infty}(X,F)$.
Proposition 3.1 gives us an equality
$$
\int_{X}|D''u|^{2}+|\delta''u|^{2}=\int_{X}\langle\Delta''u,u\rangle
=\int_{X}|D_{\tau}'u|^{2}+|\delta_{\tau}'u|^{2}+\langle[ic(F),\Lambda]u,u\rangle
+\langle T_{\omega}u,u\rangle,
\leqno(3.2)  
$$ 
where integrals are calculated relatively to 
the volume element $d\sigma=\frac{\omega^{n}}{n!}$. In particular, 
if $u$ is of bidegré $(0,q)$, we have $\delta_{\tau}'u=0$ by a bidegree
consideration. Therefore
$$
\int_{X}\langle\Delta''u,u\rangle=\int_{X}|D_{\tau}'u|^{2}+\langle[ic(F),
\Lambda]u,u\rangle+\langle T_{\omega}u,u\rangle.
\leqno(3.3)  
$$ 
One can also consider $u$ as a $(n,q)$-form with
values in the vector bundle
$$
\widetilde F:=F\otimes\Lambda^{n}TX.
$$ 
We will denote by $\widetilde D=\widetilde D'+\widetilde D''$ the
Hermitian holomorphic connection on $\widetilde F$, and by 
$\widetilde{u}$ the canonical image of $u$ in $\cC_{n,q}^{\infty}(X,F)$.
\medskip

{\statement Lemma 3.4. {\it We have commutative diagrams
$$
\matrix{
\cC_{0.q}^{\infty}(X,F)&\mathop{\longrightarrow}\limits^{\textstyle D''}
&\cC_{0,q+1}^{\infty}(X,F)\cr
\noalign{\vskip5pt}
\sim\big\downarrow&\phantom{{}\longrightarrow{}}&\big\downarrow\sim\cr
\cC_{n,q}^{\infty}(X,\widetilde F)&
\mathop{\longrightarrow}\limits^{\textstyle\widetilde D''}&
\cC_{n,q+1}^{\infty}(X,\widetilde F),\cr}\kern40pt
\matrix{
\cC_{0,q}^{\infty}(X,F)&\mathop{\longrightarrow}\limits^{\textstyle 
\Delta''}&\cC_{0,q}^{\infty}(X,F)\cr
\noalign{\vskip5pt}
\sim\big\downarrow&\phantom{{}\longrightarrow{}}&\big\downarrow\sim\cr
\cC_{n,q}^{\infty}(X,\widetilde F)&
\mathop{\longrightarrow}\limits^{\textstyle\widetilde\Delta''}&
\cC_{n,q}^{\infty}(X,\widetilde F),\cr}
$$ 
where the vertical arrows are the metric isomorphisms 
$u\mapsto\widetilde{u}$.}}

{\it Proof}.~-- The commutativity of the left diagram 
is a consequence of the fact that the line bundle $\Lambda^{n}TX$ is 
holomorphic (notice however that the corresponding result for 
$D'$ and $\widetilde D'$ is wrong). Therefore, there is a 
corresponding commutative diagram for the adjoint operators
$\delta''$, $\widetilde\delta''$ 
and for $\Delta''$, $\widetilde \Delta''$.\hfil\square\medskip

Lemma 3.4 and identity (3.2) imply
$$
\int_{X} \langle\Delta''u,u\rangle=
\int_{X}\langle\widetilde\Delta''\widetilde{u},\widetilde{u}\rangle
=\int_{X}|\widetilde{\delta}'_\tau\widetilde{u}|^{2}+
\langle[ic(\widetilde F),\Lambda]\widetilde{u},\widetilde
{u}\rangle+\langle T_{\omega}\widetilde{u},\widetilde{u}\,\rangle.
\leqno(3.5)
$$ 
We now slightly transform the expression of (3.3)
and (3.5). The holomorphic Hermitian connection on the bundle
$\Lambda^{q}T^{*}X$ induces on conjugate bundle
$\Lambda^{0,q}T^{*}X$ a connection whose component of type $(1,0)$ 
coincides with the operator~$d'$. From this we get
a natural Hermitian connection $\nabla$ on the tensor product bundle
$\Lambda^{0,q}T^{*}X\otimes F$ (observe however that this vector 
bundle is in general non holomorphic for $q\neq 0$). 
Let $\nabla'$ and $\nabla''$ be the components of $\nabla$ of type 
$(1,0)$ and $(0,1)$.
\medskip

{\statement Proposition 3.6. {\it We have
$$
\nabla'=D' : \cC^{\infty}(\Lambda^{0,q}T^{*}X\otimes F)\to \cC
_{1,0}^{\infty}(\Lambda^{0,q}T^{*}X\otimes F) ,
$$ 
and there is a commutative diagram
$$
\matrix{
\cC^{\infty}(X,\Lambda^{0.q}T^{*}X\otimes F)&
\mathop{\longrightarrow}\limits^{\textstyle\nabla''}
&\cC^{\infty}_{0,1}(X,\Lambda^{0.q}T^{*}X\otimes F)\cr
\noalign{\vskip5pt}
\sim\big\downarrow&\phantom{{}\longrightarrow{}}&\big\downarrow\Psi\cr
\cC_{n,q}^{\infty}(X,\widetilde F)&
\mathop{\longrightarrow}\limits^{\textstyle\widetilde\delta ''}&
\cC_{n-1,q}^{\infty}(X,\widetilde F),\cr}
$$ 
where the vertical arrows are isometries, the one on the
left being given by $u\mapsto\widetilde{u}$.}}
\medskip

{\it Proof}.~-- The equality $\nabla'=D'$ comes from the 
fact that the $(1,0)$ component of the connection prescribed above
on $\Lambda^{0,q}T^{*}X$ coincides with~$d'$. The diagram requires
the vertical arrow~$\Psi$ to be defined. Let
$$
\{?|?\} : (\Lambda^{p_{1},q_{1}}T^{*}X\otimes\widetilde F)\times
(\Lambda^{p_{2},q_{2}}T^{*}X\otimes\widetilde F)
\longrightarrow\Lambda^{p_{1}+q_{2},q_{1}+p_{2}}T^{*}X
$$ 
be the sesquilinear canonical pairing induced by the metric on the 
fibers of~$F$, and let
$$
*{}: \Lambda^{p,q}T^{*}X\otimes\widetilde F\longrightarrow
\Lambda^{n-q,n-p}T^{*}X\otimes\widetilde F
$$ 
be the Hodge-De Rham-Poincaré operator defined by
$$
\{v|*w\}=\langle v,w\rangle\,d\sigma,\qquad v,~w\in
\Lambda^{p,q}T^{*}X\otimes\widetilde F.
$$ 
By taking the composition, we derive from this an isometry
$$
\Psi_{0} : \Lambda^{0,1}T^{*}X\otimes F\mathop{\longrightarrow}\limits^{
\displaystyle\sim}\Lambda^{n,1}T^{*}X\otimes\widetilde F
\mathop{\longrightarrow}\limits^{\displaystyle *}\Lambda^{n-1,0}T^{*}X\otimes 
\widetilde F
$$ 
and the arrow $\Psi$ is obtained by definition by tensorizing 
$-i^{-n^{2}}\Psi_{0}$ with $\Lambda^{0,q}T^{*}X$. To prove 
the commutativity, we first assume $q=0$. Let $u\in \cC^{\infty}(F)$.
We have classically
$$
\widetilde{\delta}'\widetilde{u}=-*\widetilde D''*\widetilde{u}.
$$ 
Since $\widetilde{u}\in \cC_{n,0}^{\infty}(X,F)$, we get
$*\widetilde{u}=i^{-n^{2}}\widetilde{u}$, hence
$$
\widetilde{\delta}'\widetilde{u}=-i^{-n^{2}}*D''\widetilde{u}=
-i^{-n^{2}}*\sim D''u=-i^{-n^{2}}\Psi_{0}(D''u)=\Psi(\nabla''u).
$$ 
In case $q$ is arbitrary, one just needs trivializing
$\Lambda^{0,q}T^{*}X$ in a neighborhood of an arbitrary point $x$
and choosing an orthonormal frame $(e_{1}, \ldots,e_{N})$ 
of this bundle, such that\hbox{$\nabla e_{1}(x)=\cdots=\nabla e_{N}(x)=0$}.
\hfil\square\medskip

We now consider the bundle morphisms
$$
\eqalign{
S' &: \Lambda^{0,q}T^{*}X\otimes F\to\Lambda^{1,0}T^{*}X\otimes\Lambda^{0,q}T^{
*}X\otimes F\cr
S''&:\Lambda^{0,q}T^{*}X\otimes F\to\Lambda^{0,1}T^{*}X\otimes\Lambda^{0,q}T^{*}
X\otimes F\cr}
$$ 
where $S'=\tau=[\Lambda,d'\omega]$, and where $S''$ is the lifting by
the isometries $\sim$ and $\Psi$ of the morphism
$$
\tau^{*}=[(d'\omega)^{*},L]:\Lambda^{n,q}T^{*}X\otimes\widetilde F
\to\Lambda^{n-1.q}T^{*}X\otimes\widetilde F.
$$ 
According to Proposition~3.6, we have
$$
|D_{\tau}'u|=|\nabla'u+S'u|,\qquad
|\widetilde\delta'_{\tau}\widetilde{u}|=|\nabla''u+S''u|.
$$ 
Putting $S=S'\oplus S''$, the addition of identities (3.3) and (3.5) imply
$$
\leqalignno{
2\int_{X}\langle\Delta''u,u\rangle
=\int_{x}|\nabla u&+Su|^{2}+\int_{X}\langle[ic(F),\Lambda]u,u\rangle\cr
&+\int_{X}\langle[ic(\widetilde{F}),\Lambda]\widetilde{u},\widetilde{u}\rangle+
\langle T_{\omega}u,u\rangle+\langle T_{\omega}\widetilde{u},\widetilde{u}\rangle
&(3.7)\cr}
$$ 
for all $u\in \cC_{0,q}^{\infty}(X,F)$.

Now, let $E$ be a Hermitian holomorphic fiber of rank~$1$ over~$X$. 
For any integer~$k$, we denote by $D_{k}$ and $\nabla_{k}$ the 
natural Hermitian connections on the bundles
$F_{k}=E^{\otimes k}\otimes F$ and $\Lambda^{0,q}T^{*}X\otimes F_{k}$, and we
put $\Delta_{k}''=[D_{k}'',\delta_{k}'']$. The curvature form of $F_{k}$ 
(resp.\ $\widetilde F_{k}$) is given by
$$
c(F_{k})=c(F)+kc(E)\otimes\Id_{F},\quad\hbox{resp.}\quad
c(\widetilde F_{k})=c(\widetilde F)+kc(E)\otimes\Id_{\widetilde F}.
\leqno(3.8) 
$$ 
Recall, although this will not be used in the sequel, that
$$
c(\widetilde F)=
c(F)+c(\Lambda^{n}TX)\otimes\Id_{F}=c(F)+{\rm Ricci}(\omega)\otimes\Id_{F}.
$$ 
We thus have to evaluate the terms $[ic(E),\Lambda]$. At any 
point $x\in X$, let $\alpha_{1}(x)$, $\alpha_{2}(x)$, $\ldots$~, 
$\alpha_{n}(x)$  be the eigenvalues of $ic(E)(x)$ relatively to the 
Hermitian metric $\omega$ on~$X$. There exists a local coordinate system
$(z_{1}, \ldots,z_{n})$ centered at $x$ 
such that $(\frac{\partial}{\partial z_{1}},\ldots,\frac{\partial}
{\partial z_{n}})$ is an orthonormal basis of $(T_xX,\omega(x))$,
and such that
$$
\eqalign{
\omega(x)&=\frac{i}{2}\sum_{j=1}^{n}dz_{j}\wedge d\overline{z}_{j},\cr
ic(E)(x)&=\frac{i}{2}\sum_{j=1}^{n}\alpha_{j}(x)\,dz_{j}\wedge d\overline{z}_{j}.
\cr}
$$ 
Let $(e_{1},\ldots,e_{r})$ be an orthonormal frame of the fiber 
$E_{x}^{k}\otimes F_{x}$. For $v\in\Lambda^{p.q}T^{*}X\otimes F_{k}$, 
we can write
$$
v=\sum_{|I|=p,|J|=q,\,\ell}v_{I,J,\ell}\,dz_{I}\wedge d\overline{z}_{J}
\otimes e_{\ell},\qquad|v|^{2}=2^{p+q}\sum_{I,J,\ell}|v_{I,J,\ell}|^{2}.
$$ 
An elementary calculation, explained for example in [6], gives the 
formula
$$
\langle[ic(E),\Lambda]v,v\rangle=2^{p+q}\sum_{I,J,\ell}(\alpha_{I}+\alpha_{J}-
\sum_{j=1}^{n}\alpha_{j})|v_{I,J,\ell}|^{2}
\leqno(3.9)
$$ 
with $\alpha_{I}=\sum_{j\in I}\alpha_{j}$. Let
$u\in\Lambda^{0,q}T^{*}X\otimes F_{k}$. Put
$$
u= \sum_{J,\ell}u_{J,\ell}\,d\overline{z}_{J}\otimes e_{\ell}.
$$ 
According to (3.9), we have
$$
\eqalign{
\langle[ic(E),\Lambda]u,u\rangle&=2^{q}\sum_{J,\ell}
-\alpha_{\complement J}|u_{J,\ell}|^{2},\cr
\langle[ic(E\rangle,\Lambda]\widetilde{u},\widetilde{u}\rangle&=
2^{q}\sum_{J,\ell}\alpha_{J}|u_{J,\ell}|^{2}.\cr}
$$ 
Let $V$ be the Hermitian endomorphism of $\Lambda^{0,q}T^{*}X\otimes F_{k}$ 
defined by
$$
\langle Vu,u\rangle=-\langle[ic(E),\Lambda]u,u\rangle-\langle[ic(E),
\Lambda]\widetilde{u},\widetilde{u}\rangle
=2^{q}\sum_{J,\ell}(\alpha_{\complement J}-\alpha_{J})|u_{J,\ell}|^{2}.
\leqno(3.10) 
$$ 
The eigenvalues of $V$ are therefore the coefficients 
$\alpha_{\complement J}-\alpha_{J}$, counted with multiplicity 
$r=\rank(F)$. Finally, let $\Theta$ be the Hermitian endomorphism defined by
$$
\langle\Theta u,u\rangle=\langle[ic(F),\Lambda]u,u\rangle+\langle[
ic(\widetilde{F}),\Lambda]\widetilde{u},\widetilde{u}\rangle
+\langle T_{\omega}u,u\rangle+
\langle T_{\omega}\widetilde{u},\widetilde{u}\rangle.
\leqno(3.11) 
$$ 
The identities (3.7-11) then imply
$$
\frac{2}{k}\int_{X}\langle\Delta_{k}''u,u\rangle=\int_{X}\frac{1}{k}
|\nabla_{k}u+Su|^{2}-\langle Vu,u\rangle+\frac{1}{k}\langle\Theta u,u\rangle
\leqno(3.12)  
$$ 
where the operators $S$, $V$, $\Theta$ act merely on the 
component $\Lambda^{0,q}T^{*}X\otimes F$ of 
$\Lambda^{0,q}T^{*}X\otimes F_{k}$. We can thus use
Theorem~2.16 to determine the asymptotic spectral distribution 
of $\Delta_{k}''$, because the term 
$\frac{1}{k}\langle\Theta u,u\rangle$ tends to $0$ in norm.

Let $h_{k}^{q}(\lambda)$ be the number of eigenvalues${}\leq k\lambda$ 
of $\Delta_{k}''$ acting on $\cC_{0,q}^{\infty}(E^{\otimes k}\otimes F)$. 
The magnetic field $B$ is given here by
$$
 B=-ic(E)=-\sum_{j=1}^{n}\alpha_{j}\,dx_{j}\wedge dy_{j},\qquad
z_{j}=x_{j}+iy_{j}.\leqno(3.13)
$$ 
Since $\dim_{\bR}X=2n$, Theorem 2.16 can be reinterpreted
as follows.
\medskip

\strut\vbox{\statement Theorem 3.14. {\it There exists a countable set
$\cD$ such that for every $q=0,1, \ldots, n$ and 
$\lambda\in\bR\ssm \cD$ the asymptotic estimate
$$
h_{k}^{q}(\lambda)=rk^{n}\sum_{|J|=q}\int_{X}\nu_{B}(2\lambda+\alpha_{\complement J}
-\alpha_{J})\,d\sigma+o(k^{n})
$$ 
holds when $k$ tends to $+\infty$.}}
\bigskip

\section{4}{ Witten complex and Morse inequalities.}

E.~Witten [18], [19] recently introduced a new analytic method
that reproves the standard Morse inequalities for
de Rham cohomology. We adapt here his method to
the study of $d''$-cohomology. The main difference
lies in the fact that the magnetic field is always zero
in the case of the de Rham cohomology (since $d^{2}=0$~!),
and, in this case, the only thing that plays a role is
the ``electric field''.

With the notation of~\S3, let $\cH_{k}^{q}(\lambda)\subset \cC_{0,q}^{\infty}
(X,E^{\otimes k}\otimes F)$ be the direct sum of the eigensubspaces of 
$\Delta_{k}''$ attached to eigenvalues${}\leq k\lambda$. The vector
space $\cH_{k}^{q}(\lambda)$ has therefore a finite dimension
$$
h_{k}^{q}(\lambda)=\dim_{\bC}\cH_{k}^{q}(\lambda) .
$$ 
Hodge theory gives us an isomorphism
$$
H^{q}(X,E^{\otimes k}\otimes F)\simeq \cH_{k}^{q}(0) .
$$ 
For brevity of notation, we put
$$
h_{k}^{q}=\dim H^{q}(X,E^{\otimes k}\otimes F)=h_{k}^{q}(0) .
$$ 

{\statement Proposition 4.1.\pointir}{\it $\cH_{k}^{\bullet}(\lambda)$ is 
a sub-complex of the Dolbeault complex
$$
D_{k}'':\cC_{0,\bullet}^{\infty}(X,E^{\otimes k}\otimes F).
$$ 
Moreover, the inclusion $\cH_{k}^{\bullet}(\lambda)\subset \cC_{0,\bullet}^{\infty}
(X,E^{\otimes k}\otimes F)$ and the orthogonal projection
$$
P_{\lambda}:\cC_{0,\bullet}^{\infty}(X,E^{\otimes k}\otimes F)\to \cH_{k}^{\bullet}
(\lambda)
$$ 
induce inverse isomorphisms in cohomology.}
\medskip

{\it Proof}.~-- The fact that $\cH^{\bullet}_{k}(\lambda)$ is a 
sub-complex of $\cC_{0,\bullet}^{\infty}(X,E^{\otimes k}\otimes F)$ comes
from the commutativity of operators $D_{k}''$ and $\Delta_{k}''$. Now, let
$$
G=\int_{\lambda>0}\frac{1}{\lambda}dP_{1}
$$ 
be the Green operator of the Laplacian $\Delta_{k}''$. Since
$[P_{\lambda},\Delta_{k}'']=0$, we have equalities $[G,\Delta_{k}'']=0$ and
$$
\Delta_{k}''G+P_{0}=\Id.
$$ 
Moreover, $[P_{\lambda},D_{k}'']=[G,D_{k}'']=0$. Therefore we get
$$
\eqalign{
\Id-P_{\lambda}&=
\Delta_{k}''G(\Id-P_{\lambda})+P_{0}(\Id-P_{\lambda})=
\Delta_{k}''G(\Id-P_{\lambda})\cr
&=D_{k}''\big(\delta_{k}''G(\Id-P_{\lambda})\big)+
\big(\delta_{k}''G(\Id-P_{\lambda})\big)D_{k}'',\cr}
$$ 
hence the operator $\delta_{k}''G(Id-P_{\lambda})$ is a homotopy 
between $\Id$ and $P_{\lambda}$.\hfil\square\medskip

We now use a simple and classical lemma of homological algebra.
\medskip

{\statement Lemma 4.2.\pointir} {\it Let
$$
0\longrightarrow C^{0}\mathop{\longrightarrow}\limits^{d^{0}}C^{1}\mathop{\longrightarrow}\limits^{d^{1}}~\cdots~\mathop{\longrightarrow}\limits^{d^{n-1}} C^{n}
\longrightarrow  0
$$ 
be a complex of vector spaces of finite dimensions $c^{0}$, $c^{1}$, 
$\ldots$, $c^{n}$ over a field~$\bK$. Let $h^{q}=\dim_{\bK}H^{q}(C^{\bullet})$. 
Then we have the following inequalities.
{\parindent=6.5mm
\vskip2pt
\item{\rm(a)} Morse inequalities~$:$~ $h^{q}\leq c^{q}$,~
$0\leq q\leq n$.
\vskip2pt
\item{\rm(b)} Equality of Euler-Poincaré characteristics~
$\chi(H^{\bullet}(C^{\bullet}))=\chi(C^{\bullet})~:$ 
$$
h^{0}-h^{1}+\cdots+(-1)^{n}h^{n}=c^{0}-c^{1}+\cdots+(-1)^{n}c^{n}.
$$ 
\item{\rm(c)} Strong Morse inequalities~$:$ for all $q$, 
$0\leq q\leq n$,
$$
h^{q}-h^{q-1}+\cdots+(-1)^{q}h^{0}\leq c^{q}-c^{q-1}+\cdots+(-1)^{q}c^{0}.
$$}}\vskip0pt

{\it Proof}.~-- Let $Z^{q}=\Ker d^{q}$ and $B^{q}=\Im d^{q-1}$ have 
dimensions $z^{q}$ and $b^{q}$. Equality (b) follows from the trivial
formulas
$$
c^{q}=z^{q}+b^{q+1},\qquad h^{q}=z^{q}-b^{q},
$$ 
while (c) is a consequence of (b) applied to the complex
$$
0\to C^{0}\to C^{1}\to\cdots \to C^{q-1}\to Z^{q}\to 0.\eqno\square 
$$ 

If $F$ is a holomorphic vector bundle on $X$, we define 
its Euler-Poincaré characteristic by
$$
\chi(X,F)=\sum_{q=0}^{n}(-1)^{q}\dim H^{q} (X,F).
$$ 
Combining proposition 4.1 and lemma 4.2, we get for all
$\lambda\geq 0$ and $q\in\bN$, $0\leq q\leq n$, the inequality
$$
h_{k}^{q}-h_{k}^{q-1}+\cdots+(-1)^{q}h_{k}^{0}\leq h_{k}^{q}(\lambda)-h_{k}^{q-
1}(\lambda)+\cdots+(-1)^{q}h_{k}^{0}(\lambda).
$$ 
Now, we evaluate $h_{k}^{q}(\lambda)$ by means of Theorem 3.14
and we let $\lambda\in\bR\ssm \cD$ tend to $0+$. We then get
the following conclusion.
\medskip

{\statement Corollary 4.3.\pointir} {\it There are asymptotic inequalities
{\parindent=6.5mm
\vskip2pt
\item{\rm(a)} $h_{k}^{q}\leq k^{n}I^{q}+o(k^{n}),$ 
\vskip2pt
\item{\rm(b)} $\chi(X,E^{\otimes k}\otimes F)=k^{n}(I^{0}-I^{1}+\cdots+(-1)^{n}I^{n})+o(k^{n}),$ 
\vskip2pt
\item{\rm(c)} $h_{k}^{q}-h_{k}^{q-1}+\cdots+(-1)^{q}h_{k}^{0}\leq k^{n}
(I^{q}-I^{q-1}+\cdots+(-1)^{q}I^{0})+o(k^{n})$,
\vskip4pt
where $I^{q}$ is the curvature integral
$$
I^{q}=r\sum_{|J|=q}\int_{X}\overline{\nu}_{B}(\alpha_{\complement J}
-\alpha_{J})d\sigma.
$$}}\vskip-\parskip

According to (3.13), the non negative eigenvalues of the magnetic field 
$B$ are the $|\alpha_{j}|$, $1\leq j\leq n$. For any point $x\in X$,
let us arrange these eigenvalues so that
$$
|\alpha_{1}\geq|\alpha_{2}|\geq\cdots\geq|\alpha_{s}|>0=|\alpha_{s+1}|=
\cdots=|\alpha_{n}|,\qquad s=s(x).
$$ 
Formula (1.5) gives
$$
\overline{\nu}_{B}(a_{\complement J}-\alpha_{J})=
\frac{2^{s-2n}\pi^{-n}}{\Gamma(n-s+1)}|\alpha_{1}\ldots\alpha_{s}|
\sum_{(p_{1},\ldots,p_{s})}
\Big\{\alpha_{\complement J}-\alpha_{J}-\sum(2p_{j}+1)
|\alpha_{j}|\,\Big\}_{+}^{n-s}
$$ 
with the notation $\{\lambda\}_{+}^{0}=0$ if $\lambda<0$ and 
$\{\lambda\}_{+}^{0}=1$ if $\lambda\geq 0$. As the quantity 
$$
\alpha_{\complement J}-\alpha_{J}-\sum(2p_{j}+1)|\alpha_{j}|
$$ 
is always${}\leq 0$, $\overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})$ 
can be non zero only for $s=n$. In the latter case 
$$
\alpha_{\complement J}-\alpha_{J}-\sum(2p_{j}+1)|\alpha_{j}|=0
$$
if and only if $ p_{1}=\cdots=p_{n}=0$ and $\alpha_{j}<0$ for 
$j\in J$, $\alpha_{j}>0$ for $j\in{}{\complement}J$. This requires the 
form $ic(E)$ to be non degenerate of index~$q$. For 
$x\in X(q)$ and $|J|=q$ (see notation from the introduction), we thus get
$$
\overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})=(2\pi)^{-n}
|\alpha_{1}\ldots\alpha_{n}|>0
$$ 
where $J$ is the multi-index $J(x)=\{j\,;\;\alpha_{j}(x)<0\}$ and 
$\overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})=0$ si $J\neq J(x)$. 
This implies
$$
I^{q}=r\int_{X(q)}(2\pi)^{-n}(-1)^{q}\alpha_{1}\ldots\alpha_{n}\,d\sigma=
\frac{r}{n!}\int_{X(q)}(-1)^{q}\Big(\frac{i}{2\pi}c(E)\Big)^{n}.
$$ 
Our fundamental theorem 0.1 is then just a reformulation of
Corollary 4.3. The above arguments show that the harmonic forms associated
with $H^{q}(X,E^{\otimes k}\otimes F)$ concentrate asymptotically on~$X(q)$,
and that, at each point of $X(q)$, their direction tends to be aligned with the 
$q$-subspace of $TX$ corresponding to the negative part of~$ic(E)$.
Moreover, the eigenvalue of minimal energy
$p_{1}=\cdots=p_{n}=0$ of the harmonic oscillator is the only one that
is involved in those harmonic forms. For $q=1$, the Morse inequality
4.3~(c) can be written
$$
h_{k}^{1}-h_{k}^{0}\leq k^{n}(I^{1}-I^{0})+o(k^{n}).
$$ 
In particular, we find an asymptotic lower bound of the number of
holomorphic sections of the vector bundle $E^{\otimes k}\otimes F$.
\medskip

{\statement Theorem 4.4.\pointir}{\it We have
$$
\dim H^{0}(X,E^{\otimes k}\otimes F)\geq r\frac{k^{n}}{n!}
\int_{X(\leq 1)}\Big(\frac{i}{2\pi}c(E)\Big)^{n}-o(k^{n}).
$$}\vskip-\parskip

More generally, the addition of the two instances of inequality 4.3 (c) for
indices $q+1$ and $q-2$ leads to
$$
h_{k}^{q+1}-h_{k}^{q}+h_{k}^{q-1}\leq k^{n}(I^{q+1}-I^{q}+I^{q-1})+o(k^{n}) ,
$$ 
hence the lower bound
$$
\dim H^{q}(X,E^{\otimes k}\otimes F)\geq r\frac{k^{n}}{n!}\sum_{j=0,\pm 1}(-1)
^{q}\int_{X(q+j)}\Big(\frac{i}{2\pi}c(E)\Big)^{n}-o(k^{n}).
\leqno(4.5) 
$$ 
\medskip

\section{5}{ Characterization of Moi\v{s}ezon  varieties.}

Let $X$ be a compact $\bC$-analytic variety of dimension $n$.
The algebraic dimension of $X$, denoted $a(X)$,
is by definition the transcendence degree over $\bC$ of the field
$K(X)$ of meromorphic functions on~$X$. 
According to a well-known theorem of Siegel [15], the 
algebraic dimension of $X$ always satisfies the inequality
$0\leq a(X)\leq n$. When $a(X)=n$, the variety $X$ is called a
Moi\v{s}ezon space. As we recall below, the algebraic dimension of
$X$ imposes strong asymptotic bounds for the dimension of spaces
of sections of tensor powers of any holomorphic line or vector bundle on $X$.
\medskip

{\statement Theorem 5.1.\pointir}{\it Let $X$ be a compact complex manifold,
$a$ the algebraic dimension of~$X$, $F$ a holomorphic vector bundle of rank $r$,
and $E$ a holomorphic line bundle over~$X$. Then, there exists a constant
$C_{E}\geq 0$ that depends only on $E$, such that
$$
\dim H^{0}(X,E^{\otimes k}\otimes F)\leq C_{E}rk^{a}+o(k^{a}) .
$$}\vskip-\parskip

{\it Proof}.~-- We essentially repeat the arguments detailed by 
Y.T.~Siu in~[16]. Let $\{W_{\ell}\}$ be a covering of $X$ by open 
coordinate charts $W_{\ell}\subset\bC^{n}$, and $B_{j}=B(a_{j},R_{j})$, 
$1\leq j\leq m$, a family of relatively compact balls in the
open sets $W_{\ell}$, such that the concentric balls 
$B_{j}'=B(a_{j},\frac{1}{7}R_{j})$ still cover~$X$. Let us provide $E$, $F$ 
with Hermitian metrics, and let $\exp(-\varphi_{j})$ be the weight 
representing the metric of $E$ in a trivialization of $E$ 
in a neighborhood of~$\overline{B}_{j}$.

Let $s\in H^{0}(X,E^{\otimes k}\otimes F)$ be a holomorphic section
that vanishes at order $p$ at a point $x_{j}\in B_{j}'$. The inclusions
$$
B_{j}'\subset B(x_{j},\frac{2}{7}R_{j})\subset B(x_{j},\frac{6}{7}R_{j})\subset 
B_{j}
$$ 
and the Schwarz lemma applied to the two intermediate balls 
lead to the inequality
$$
\sup_{B_{j}'}|s|\leq\exp(Ak+C_{F})3^{-p}\sup_{B_{j}}|s|,
\leqno(5.2)
$$ 
where $A = \max_{1\leq j\leq m}\diam \varphi_{j}(B_{j})$ depends only on
of~$E$, and where $C_{F}\geq 0$ is a constant that depends on 
the metric of~$F$.

Let $\rho\leq r=\rank(F)$ be the maximum for all $x\in X$ of the dimension
of the  subspace of the fiber $F_{x}$ generated by elements $s(x)$ 
when $s$ runs over $\bigcup_{k\in\bN}H^{0}(X,E^{\otimes k}\otimes F)$. 
If~$\rho=0$, then $H^{0}(X,E^{\otimes k}\otimes F)=0$ for all~$k$. 
Let us now distinguish two cases according to whether $\rho=1$ or~$\rho>1$.

(a) {\it Assume $\rho=1$}.

Let $h_{k}=\dim H^{0}(X,E^{\otimes k}\otimes F)$, and suppose that $h_{k}>0$. 
Under the hypothesis $\rho=1$, the global sections of $E^{\otimes k}\otimes F$ 
define a holomorphic map
$$
\Phi_{k}:X\ssm Z_{k}\to\bP^{h_{k}-1}(\bC)
$$ 
where $Z_{k}\subset X$ is the analytic subset of common zeros.
Let $d$ the maximum rank of the differential $\Phi_{k}'$ on
$X\ssm Z_{k}$. We necessarily have $d\leq a$, otherwise the field of
rational functions of $\bP^{h_{k}-1}(\bC)$ would induce a field of 
meromorphic functions on~$X$ of transcendence degree${}\geq d>a$, which 
is absurd. Let us choose for all $j=1, \ldots, m$ a point $x_{j}\in B_{j}'
\ssm Z_{k}$ such that $\Phi_{k}'$ is of maximum rank${}=d$ at $x_{j}$,
and let $s_{0}\in H^{0}(X,E^{\otimes k}\otimes F)$ be a section that
does not vanish at any point~$x_{j}$. For every
$s\in H^{0}(X,E^{\otimes k}\otimes F)$, the quotient 
$s/s_{0}$ is well defined as a meromorphic function on~$X$, 
and moreover $s/s_{0}$ is a holomorphic function
in a neighborhood of~$x_{j}$, that is constant along the
fibers of~$\Phi_{k}$. As $\Phi_{k}$ is a 
subimmersion in the neighborhood of each point~$x_{j}$, we
can choose a subvariety $M_{j}$ of dimension $d$ passing through $x_{j}$ and 
transverse to the fiber $\Phi_{k}^{-1}(\Phi_{k}(x_{j}))$. The section $s$ 
will vanish at order $p$ at each point $x_{j}$, 
$1\leq j\leq m$, if and only if partial derivatives 
of order${}<p$ of $s/s_{0}$ along $M_{j}$ vanish at~$x_{j}$.
In total, this corresponds to the vanishing of
$$
m{p+d-1\choose d}
$$ 
derivatives. If we choose $p=[Ak+C_{F}]+1$, then 
inequality (5.2) implies
$$
\sup_{X}|s|\leq\Big(\frac{e}{3}\Big)^{p}\sup_{X}|s|,
$$ 
hence $s=0$. As $d\leq a$, we therefore get
$$
\dim H^{0}(X,E^{\otimes k}\otimes F)\leq m{p+a-1\choose a}
\leq C_{E}k^{a}+o(k^{a})
$$ 
with $C_{E}=mA^{a}/a!$~.

(b) {\it Assume $\rho>1$}.

Then there are sections $s_{t}\in H^{0}(X,E^{k_{t}}\otimes F)$, 
$1\leq t\leq\rho$, and a point $x_{0}\in X$ such that the vectors 
$s_{1}(x_{0}), \ldots, s_{\rho}(x_{0})$ are linearly
independent. By construction, for any $k\in\bN$ and any 
section $s\in H^{0}(X,E^{\otimes k}\otimes F)$, the line $\bC\cdot s(x)$ is 
contained in the subspace generated by 
$(s_{1}(x), \ldots,s_{\rho}(x))$, except perhaps above the 
analytic subset $\{x\in X;s_{1}\wedge\ldots\wedge s_{\rho}(x)\}=0$.
Therefore, we get an injective morphism
$$
H^{0}(X,E^{\otimes k}\otimes F)\to\bigoplus_{1\leq t\leq\rho}H^{0}(X,E^{k+k_{\hat{t}}}
\otimes\Lambda^{p}F)
$$ 
where $k_{\hat t}=(k_{1}+\cdots+k_{\rho})-k_{t}$, whose component of index $t$ 
is given by the morphism 
$s\to s_{1}\wedge\cdots\wedge\widehat{s}_{t}\wedge
\cdots\wedge s_{\rho}\wedge s$. The image of 
$H^{0}(X,E^{\otimes k}\otimes F)$ on each component consists
of sections that are colinear at almost any point
with $s_{1}\wedge\cdots\wedge s_{\rho}$. We are led to a situation
analogous to the one considered in (a), where $F$ is replaced by
$E^{k_{\hat{t}}}\otimes\Lambda^{\rho}F$. Therefore we infer
$$
\dim H^{0}(X,E^{\otimes k}\otimes F)\leq C_{E}\rho k^{a}+o(k^{a}),\qquad\rho\leq r.
\eqno\square 
$$ 
Let us choose in particular for $F$ the trivial line bundle $X\times\bC$.
The combination of Theorems 4.4 and 5.1 leads to the following characterization 
of Moi\v{s}ezon varieties.
\medskip

{\statement Theorem 5.2.\pointir} {\it Let $X$ be a 
compact $\bC$-analytic manifold of dimension $n$. Then $X$ is
Moi\v{s}ezon as soon as there exists a Hermitian holomorphic line bundle
$E$ over $X$ such that
$$\int_{X(\leq 1)}(ic(E))^{n}>0.\eqno\square $$}\vskip-\parskip

This theorem in turn leads to Theorem~0.8, since
0.8~(c)${}\Rightarrow{}$ 0.8(b)${}\Rightarrow{}$ 0.8(a). This
strengthens Y.T.~Siu's results [17], [18], and gives
in particular a new proof of the Grauert-Riemenschneider conjecture~[10].
\bigskip

{\bigbf References}
{\parindent 7.5mm
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\item{[15]} {\petcap C.\ L.\ Siegel}, {\it Meromorphe Funktionen auf
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\item{[16]} {\petcap Y.T.\ Siu}, {\it A vanishing theorem for semipositive
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\item{[17]} {\petcap Y.T.\ Siu}, {\it Some recent results in complex 
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\item{[18]} {\petcap E.\ Witten}, {\it Supersymmetry and Morse theory}~; 
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\item{[19]} {\petcap E.\ Witten}, {\it Holomorphic Morse inequalities}~; 
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Ed.\ G.\ M.\ Rassias (1984), 318--333.\par}
\bigskip

Manuscript received May 30, 1985.
\medskip

Jean-Pierre Demailly\\
Université de Grenoble I\\
Institut Fourier\\
Laboratoire de Mathématiques, B.P.\ 74\\
38402 St-Martin d'Hères Cedex

\end
Q