\magnification=1200 \vsize=20cm\hsize=13.5cm\hoffset=-0.1mm \parindent=0mm \parskip=6pt plus 2pt minus 2pt \input macros_ocr.tex %% Further macros \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}} %% main tex \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\folio\hfil Annales de l'Institut Fourier 35, (1985) 189--229\hfil}\fi\fi} {\sevenrm\baselineskip=9pt 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~}: Inequalities of Morse -- d"-cohomology -- Fiber hermitian linear -- Shape of curvature -- Magnetic field -- Schrödinger operator -- Identity of Bochner-Kodaira-Nakano -- Me Space\v{s}ezon.\vskip-\parskip} \centerline{\bigbf 0. Introduction.} Let $X$ be a compact complex analytic manifold od dimension~$n$, $F$ a holomorphic fiber bundle of rank $r$ and $E$ a hermitian holomorphic line bundle of class $\cC^{\infty}$ on $X$. Let $D=D'+D''$ the canonical connection of $E$ and $c(E)=D^{2}=D'D''+D''D'$ the curvauture tensor of this connection. Let us designate by $X(q)$ , $0\leq q\leq n$, the open points of $X$ index $q$, i.e.\ the open subset of points $x\in X$ at which the form of curvature $ic(E)(x)$ has exactly $q$ negative eigenvalues and $n-q$ positive ones. We also set $$ X(\leq q)=X(0)\cup X(1)\cup\ldots\cup X(q). $$ We then prove the following Morse inequalities, which bound the dimension of the cohomology groups $H^{q}(X,E^{k}\otimes F)$ in terms of integral invariants of the curvature of $E$. \medskip {\statement Theorem 0.1.\pointir} {\it When $k$ tends towards $+\infty$, one has for all $q=0,1, \ldots, n$ the following asymptotic inequalities. {\parindent=6.5mm \vskip2pt \item{\rm (a)} Morse Inequalities~$:$ $$ \dim H^{q}(X,E^{k}\otimes F)\leq r\frac{k^{n}}{n!}\int_{X(q)}(-1)^{q} \Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}). $$ \vskip2pt \item{\rm(b)} Strong Morse inequalities~$:$ $$ \sum_{j=0}^{q}(-1)^{q-j}\dim H^{j}(X,E^{k}\otimes F)\leq r\frac{k^{n}}{n!} \int_{X(\leq q)}(-1)^{q}\Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}). $$ \vskip2pt \item{\rm(c)} Asymptotic Riemann-Roch Formula~$:$ $$ \sum_{q=0}^{n}(-1)^{q}\dim H^{q}(X,E^{k}\otimes F)=r\frac{k^{n}}{n!}\int_X \Big(\frac{i}{2\pi}c(E)\Big)^{n}+o(k^{n}) . $$}} \medskip Estimates 0.1 (a), (b) are new to our knowledge even in the case of projective varieties. Asymptotic equality 0.1~(c), quand à elle, est une version affai­blie of the theorem of Hirzebruch-Riemann-Roch, which is itself a special case of the Atiyah-Singer index theorem [1]. The latter theorem allows indeed to express the characteristic of Euler-Poincaré $$ \chi(X,E^{k}\otimes F)=\sum_{q=0}^{n}(-1)^{q}\dim H^{q}(X,E^{k}\otimes F) $$ in the form $$ \chi( X,E^{k}\otimes F)=r\frac{k^{n}}{n!}c_{1}(E)^{n}+P_{n-1}(k)~; \leqno(0.2) $$ $P_{n-1}(k)\in\bQ[k]$ here designates 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 ${i\over 2\pi}c(E)$ (cf.\ for example [16]). It can be observed that the numerical constant of the inequality 0.1~(a) is optimal, as shown by the example of the total tensor product fiber $X$q above $X=(\bP^{1}(\bC))^{n}$ . For this fiber, we have indeed $X(q)=X$ and $$ \eqalign{ &\dim H^{q}(X,E^{k})=(k+1)^{n-q}(k-1)^{q}, k\geq 1,\cr &\int_{X}\Big(\frac{i}{2\pi}c(E)\Big)^{n}=(-1)^{q}n!\,.\cr} $$ The existence of a 0.1(a) surcharge was speculated by Y. T. Siu, who has successively demonstrated the case where $ic(E)$ is${}>0$ in the complement of a particular set of measure zero [16], then the case where $ic(E)$${}\geq 0$ over $X$ [17]. We have borrowed from Siu a part of the techniques used here, especially at\S3 and\S5. The proof of theorem 0.1 is based on the method analytical approach recently introduced by E.~Witten [18], [19]. This method allows (among other things) to redemonstrate the classical Morse inequalities $b_{q}\leq m_{q}$ on a compact differentiable variety $M$ , where $b_{q}$ designates the $q$-th number of Betti and $m_{q}$ the number of critical points of index $q$ of any Morse function on~$M$ . In our situation, the role of the Morse function is held by the choice of the Hermitian metric on $E$ . We provide other part $X$ and F of arbitrary Hermitian metrics, which will intervene only in the $o(k^{n})$ terms of the estimates finals. Given a real $\lambda\geq 0$ , one can considers the $\cH_{k}^\bullet(\lambda)$ sub-complex of the Dolbeault $\cC_{0,\bullet}^{\infty}(X,E^ {k}\otimes F)$ des $(0,q)$ -forms of class $\cC^{\infty}$ on $X$ with values in $E^{k}\otimes F$ , generated by the Laplacian's own functions antiholomorph $\Delta''$ whose eigenvalues are${}\leq k\lambda$ . The cohomology groups of the $\cH_{k}^\bullet(\lambda)$ complex are then isomorphic to the $H^{q}(X,E^{k}\otimes F)$ groups (proposal 4.1), so that it is enough to know how to limit the dimension of the spaces $\cH_{k}^{q}(\lambda)$ . For this, two tools are essentially used. The first tool consists of a Weitzenböck type formula $$ \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) $$ demonstrated at\S3, and derived from the identity of Bochner- Kodaira-Nakano non-Kählerian~ [6]. $\nabla_{k}$ designates here the natural hermitian connection on the $\Lambda^{0.q}T^{*}X\otimes E^{k}\otimes F$ fiber, $\bC$y~ is a linear potential of order $0$ related to the curvature of the fiber $n$a , finally $S$ and $\Theta$ are operators of order $0$ from the twist of the Hermitian metric on $n$f and curvature of $F$ . The study of the spectrum of $n$g is is thus brought back to the study of the spectrum of the associated self-adhesive operator $\nabla_{k}^{*} \nabla_{k}$ to the actual connection $\nabla_{k}$ . The second tool fundamental consists precisely of a fundamental theorem very general spectrum relative to the operators of the type~$n$j . Let $n$k be a riemannian variety $n$l of real dimension $n$m , $n$n a fibrous in complex straight lines above~$X$ , provided with a connection hermitienne~$\nabla$ . If $n$q designates the connection induced by $\nabla$ on $E^{k}$ , we then study the spectrum of the quadratic form $$ 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^{k}) \leqno(0.4) $$ for the Dirichlet problem, where $\Omega$ is a relatively open-ended compact in~$M$ , and where $V$ is a continuous scalar potential on~$M$ . From a physical point of view, this is equivalent to studying the spectrum of the operator of Schrödinger $\frac{1}{k}(\nabla_{k}^{*}\nabla_{k}-kV)$ associated with the electric field $kV$ and the magnetic field $kB$ , where $B=-i\nabla^{2} $ is none other than the $2$ -form of curvature of the connection~$F$d . It is in the presence of this magnetic field that our main contribution to the method of E.~Witten [18], [19] (in the case of the cohomology of De Rham the magnetic field is always zero since $F$e ). At any point $x\in X$ , that is $2s=2s(x)\leq n$ the rank of $B(x)$ and $B_{1}(x) \geq\ldots\geq B_{s}(x)>0$ the non-zero eigenvalue modules of the associated antisymmetric endomorphism. We define a function $F$j of couple $F$k , continues at left in~$\lambda$ , by posing $$ v_{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) $$ with the convention $F$n . Finally, if $F$o denote the eigenvalues of $Q_{k}$ (counted with multiplicity), we consider the enumeration function $F$q , $F$r . \medskip {\statement Theorem 0.6.\pointir}{\it If $\partial\Omega$ is of zero measurement, there is a countable set $0\leq q\leq n$d such as $$ \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 To demonstrate the 0.6 theorem, we start by studying the case of simple where $F$w with a constant magnetic field $F$x and with~$F$y . When $F$z is a cube, then we know how to explain the own functions by a partial Fourier transformation that brings the problem back to the classic oscillator problem harmonic in one variable. The idea of this calculation was strongly inspired by articles [3], [4] of Y.~Colin from Verdière. The extension of the result to the case of a field any kind of magnetic takes up an idea from [16], consisting of to use a paving of $\Omega$ by rather small cubes. Our method is nevertheless very different the one of Siu, since we work directly on the harmonic forms while Siu was reduced to the holomorphic cochains via the isomorphism of Dolbeault. We thus gain a lot of precision on the estimates sought. The side of the cubes must be chosen here with a order of magnitude intermediate between $k^{-\frac{1}{2}}$ and $k^{-\frac{1}{4}}$ , for example $k^{-\frac{1}{3}}$~: $k^{-\frac{1}{2}}$ is indeed the wavelength of the the first proper functions, so that the action of the field magnetic $r$f is not perceptible at a scale lower~; above $k^{-\frac{1}{4}}$ , the oscillation of $B$ is on the contrary too strong. We finally use the principle of minimax to compare the eigenvalues on $\Omega$ to the values clean on the cubes. In the ante'rieure method of [16] (as it was used in the is repeated in [7]), the size of the cubes was chosen equal to to $k^{-\frac{1}{2}}$~; one can easily see that this choice was critical to allow the effects of the magnetic field to be limited. independently of~$k$ , but the exact determination of the spectrum became impossible. The last paragraph is devoted to the study of geometrical characterizations of spaces of Me\v{s}ezon~ [13]. Recall that a compact analytical space irreducible $X$ is called ego space\v{s}ezon if the body $K(X)$ of the meromorphic functions on $X$ is degree of transcendence${}=n=\dim_{\bC}X$ . The conjecture of Grauert-Riemenschneider [10] states that $X$ is Me\v{s}ezon if and only if there is a quasi-positive beam $\cE$ Row 1 without torsion above~$X$ . By de-ingularization, we come back to the case where $X$ is smooth and where $\cE$ is the locally free bundle of sections of a fibrous material straight $E$ strictly positive on a dense open $X$ . Y.T.~Siu [17] has recently resolved the conjecture and strengthened it assuming only $ic(E)$ semi-positive and${}>0$ in at least one point. The use of theorem 0.1 (b) makes it possible to find the following conditions geometrically even weaker, which do not require the $r$y's semi-positive point activity, but only the positivity of an integral curvature oertain. For $q=1$ , the inequality 0.1~(b) indeed implies a minusing of the number holomorphic sections of $E^{k}$ , namely: $$ \dim H^{0}(X,E^{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) $$ On the other hand, we can show, using classical Siegel reasoning 15] formatted by [16] that $\dim H^{0}(X,E^{k})\leq{\rm cte}\cdot k^{n-1}$ if $X$ is not of Me\v{s}ezon (cf.\ theorem~5.1). From there it results in the\medskip {\statement Theorem 0.8.\pointir} {\it Let $X$ a variety $\bC$ -analytical compact related dimension~$n$ . For $E$h to be of Me\v{s}ezon, it is enough that $X$ has a holomorphic fiber in Hermitian straight lines verifying one of the hypotheses{\rm (a), (b), (c)} below. {\parindent=6.5mm \vskip2pt \item{\rm(a)} $\int_{X(\leq 1)}(ic(E))^{n}>0$. \vskip2pt \item{\rm(b)} $c_{1}(E)^{n}>0$, and the shape of curvature $E$l has no even index point$E$m . \vskip2pt \item{\rm(c)} $ic(E)$ is semi-positive at any point of $X$ and defined positive in at least one point of~$X$ .\vskip-\parskip}} \medskip This work was the subject of a note [8] of the same title, published in the Comptes Rendus. This article is a version improved from a previous memory [7], which was closer to Siu's initial techniques, and which demonstrated only the inequality 0.1~(a) to the constant to the nearest numerical value; therefore, the estimates 0.1~(b) and (c) remained inaccessible. The author would like to thank Mr. Gérard Besson, Mr. Alain Dufresnoy, Mr. Sylvestre Gallot and especially Yves Colin de Verdière, for de Stimulating conversations that contributed greatly to the fitness experience definitive ideas of this work, especially in the\S1. \bigskip \section{1}{Spectrum of Schrödinger operator\\ associated with a constant magnetic field.} Let $E$q a riemannian variety of class $E$r , of real dimension~$n$ , and $E\to M$ one fiber in complex straight lines above~$M$ , provided with a hermitian metric $E$v . Note $\cC_{q}^{\infty}(M,E)$ the space of the class sections $\cC^{\infty}$ of the fibered $\Lambda^{q}T^{*}M\otimes E$ , and $E$z the coupling canonical sesquilinear $$ \cC_{q}^{\infty}(M,E)\times \cC_{q}^{\infty}(M,E)\to \cC_{p+q}^{\infty}(M,\bC). $$ It is assumed to give a hermitian connection $\cC^{\infty}$b on $\cC^{\infty}$c, c'est-à-dire un opérateur dif­fé­order frame of order one $$ D:\cC_{q}^{\infty}(M,E)\to \cC_{q+1}^{\infty} (M,E),\qquad 0\leq q0$, $s \leq\frac{n}{2}$. You can then choose a trivialization of $E$ whose associated vector potential is $$ A = \sum_{j=1}^{s}B_{j}x_{j}\,dx_{j+s}. $$ The connection of $E$k is written as follows $$ D_{k}u=du+ikA\wedge u, $$ and the quadratic form $E$m 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 $E$o is the measure of Lebesgue on~$E$p . If one performs 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 side cubes $E$t . on the field~$E$u , 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 one poses by convention $E$x if $E$y and $\lambda_{+}^{0}=1$ si $c(E)=D^{2}=D'D''+D''D'$a . The function $c(E)=D^{2}=D'D''+D''D'$b is then increasing and continuous to the left on~$\bR$~; one will 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, of which the idea was suggested to us by Y.~Colin of Verdière [4].\medskip {\statement Theorem 1.6.\pointir}{\it Let $R$ be a real$c(E)=D^{2}=D'D''+D''D'$h , $$ P(R)=\Big\{x\in\bR^{n}\,;\;|x_{j}|<\frac{R}{2}\Big\} $$ the side paving stone $R$ , $Q_R$ the quadratic form $$ Q_{R}(u)=\int_{P(R)}\Bigg[\sum_{1\leq j\leq s}\Big(\Big| \frac{\partial u}{\partial x_{j}}\Big|^2 + \Big|\frac{\partial u}{\partial x_{j+s}}+iB_{j}x_{j}u\Big|^{2}\Big)+ \sum_{j>2s}\Big|\frac{\partial u}{\partial x_{j}}\Big|^{2}\Bigg]\,d\mu, $$ and ${}\leq\lambda$ the number of eigenvalues${}\leq\lambda$ of $Q_R$ for the Dirichlet problem. So for all $\lambda\in\bR$ we have $$ \lim_{R\to+\infty}R^{-n}N_{R}(\lambda)=\nu_{B}(\lambda). $$}\medskip When $s= \frac{n}{2}$ , $\nu_{B}$ is a step function. The eigenvalues of $Q_{R}$ are thus 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 eigen-state quantification. Returning to the initial problem of the quadratic form $c(E)=D^{2}=D'D''+D''D'$w , we get the \medskip {\statement Corollary 1.7.\pointir} $\displaystyle \lim_{k\to+\infty}k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)= r^{n}\nu_{B}(\lambda)$ .\hfill$\square$\medskip {\it Demonstration of theorem 1.6. --} First we try to increase $N_{R}(\lambda)$ . To this end, being given $X(q)$a , one expresses $X(q)$b as a series of partial Fourier 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 notations $$ \eqalign{ &x'=(x_{1}, \ldots,x_{s}),\quad x''=(x_{s+1}, \ldots,x_{n}),\cr &\ell\cdot x''=\ell_{1}x_{s+1}+\cdots+\ell_{n-s}x_{n}.\cr} $$ The $u\in W_{0}^{1}(P(R))$ hypothesis implies that the series $$ \sum|\ell|^{2}|u_{\ell}(x')|^{2} $$ is in $L^{2}(\bR^{s})$ . Let's put $\ell'=(\ell_{1}, \ldots,\ell_{s})$ , $\ell''=(\ell_{s+1}, \ldots,\ell_{n-s})$ . $\Vert u\Vert_{P(R)}$ 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 therefore get a Dirichlet problem at "separate variables"\ on the cube $\bR^{s}\cap P(R)$ . By posing $X(q)$p , we are brought back to study the spectrum of the form quadratic of a variable $$ q(f)=\int_{R}\Big(\Big|\frac{df}{dt}\Big|^{2}+B_{j}^{2}t^{2}|f|^{2}\Big)\,dt, $$ with $f \in \smash{W_{0}^{1}\big(\,]-\frac{R}{2}\frac{R}{2}[{} +\frac{2\pi\ell_{j}}{RB_{j}}\big)}$ . So we come back to the problem classic harmonic oscillator (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 functions of $q$. are given by $\Phi_{m}(\sqrt{B_{j}}\,t)$ where $\Phi_{0}$ , $\Phi_{1},\ldots$ are the functions of Hermite~: $$ \Phi_{m}(t)=e^{t^{2}/2}\frac{d^{m}}{dt^{m}}(e^{-t^{2}}) . $$ For all $p_{j}\in\bN$ , note $\Psi_{p_{j},\ell_{j}}(x_{j})$ the $p_{j}$ the $p_{j}$ -third. proper function 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 $\lambda_{p_{j},\ell_{j}}$ the corresponding eigenvalue. We can then decompose each function $u_{\ell}$ as a series of own functions, which leads to write $u$ in 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 $$ u_{p,\ell}\in\bC,\qquad \Psi_{p,\ell'}(x')=\prod_{1\leq j\leq s}\Psi_{p_{j},\ell_{j}}(x_{j}). $$ We will take care of the fact that $\Psi_{p,\ell'}(x')\exp(\frac{2\pi i}{R}\ell \cdot x'')$ is not not a real proper function for the Dirichlet problem, because the term exponential takes non-zero values at the edge points $x_{j}= \pm\frac{R}{2}$ , $0\leq q\leq n$n . Therefore, the coefficients $(u_{p,\ell})$ are not arbitrary if $u\in W_{0}^{1}(P(R))\;$; they must check the cancellation conditions at the~ edge: $$ \sum_{t_{j}\in\bZ}(-1)^{\ell_{j}}u_{p,\ell}=0 \leqno(1.10) $$ for all $j=1, \ldots, n-s$ and all indices other than $\ell_{j}$ fixed~: $$ p\in \bN^{s},\quad \ell_{1},\ldots,\ell_{j-1},\;\ell_{j+1},\ldots,\ell_{n-s}\in\bZ. $$ With the writing (1.9), the norm $L^{2}$ and the quadratic form $Q_{R}$ are expressed in the form of $$ \Vert u\Vert_{P(R)}^{2}=\sum|u_{p,\ell}|^{2},\qquad Q_{R}(u)=\sum\Big(\lambda_{p,\ell'}+ \frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\Big)|u_{p,\ell}|^{2}, $$ where $\lambda_{p,\ell'}=\sum_{1\leq j\leq s}\lambda_{p_{j},\ell_{j}}$ . The principle of the minimax 1.20~(b) recalled below shows 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 adequate reduction of $\lambda_{p_{j},\ell_{j}}$ .\medskip {\statement Lemma 1.12.\pointir}{\it We have inequality $$ \lambda_{p_{j},\ell_{j}}\geq\max\bigg((2p_{j}+1)B_{j}~,~\frac{4\pi^{2}}{R^{2}} \Big[\Big(\frac{p_{j}+1}{2}\Big)^{2}+\Big(|\ell_{j}|-\frac{B_{j}R^{2}}{4\pi} \Big)_{+}^{2}\Big]\bigg), $$ and this one is strict if $\ell_{j}\neq 0$ or if $X$c .} \medskip The $\lambda_{p_{j},\ell_{j}}\geq(2p_{j}+1)B_{j}$ reduction results in fact of the minimax and the fact that the eigenvalues of $q(f)$ on $\bR$ are worth $X$g . To obtain the other inequality, we minimize (1.8) by the quadratic form $$ \widehat{q}(f)=\int_{x_{j}|\widehat{q}(f)$ for any ${}^{\scriptscriptstyle\circ}$on , and on the other hand $X$o cannot be proper function of $q$ on $]-R/2, R/2[{}+2\pi\ell_{j}/RB_{j}$ only if $$ \Phi_{p_{j}}\big(\pm R\sqrt{B_{j}}/2+2\pi t_{j}/R\sqrt{B_{j}}\big)=0. $$ Since the zeros of $\Phi_{p_{j}}$ are algebraic and $\pi$ is transcendent, this is only possible if $$\ell_{j}=0\quad\hbox{et}\quad\Phi_{p_{j}}(R\sqrt{B_{J}}/2)=0.\eqno\square$$ {\statement Lemma 1.13.\pointir} {\it Let $X$v the number of points of $\bZ^{n}$ located in the closed ball $\overline{B}(0,\rho) \subset\bR^{n}$ . Then $$ \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}.$$}\medskip Indeed, the meeting of the side cubes $1$ centered at the points $x\in\bZ^{n}$ such that $|x|\leq\rho$ is contained in the ball $\overline{B}(0,\rho+\frac{\sqrt{n}}{2})$ , and contains the ball $\overline{B}(0,\rho-\frac{\sqrt{n}}{2})$ si $q$e , because $q$f is the half-diagonal of the cube~; the integer $\tau_{n}(\rho)$ is thus framed by the volume of the balls $q$h .\hfill\square\medskip We now enhance $\lim\sup R^{-n}N_{R}(\lambda)$ using (1.11) and the lemmas 1.12, 1.13. For $q$j fixed, inequality $\lambda_{p,\ell'}+ \frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda$ implies $$ |\ell''| \leq\frac{R}{2\pi}\Big(\lambda-\sum(2p_{j}+1)B_{j}\Big)^{\frac{1}{2}}_+, \leqno(1.14) $$ and the inequality is strict for $q$m big enough. When $s0$ and $0$ if not, which is well in conformity to the convention we adopted for the rating~$\lambda_{+}^{0}$ . Inequality $\lambda_{p,\ell'} \leq\lambda$ implies on the other hand $$ |\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) $$ which asymptotically corresponds to a number of multi-indices $q$x equivalent to $$ \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) $$ The $q$z increase is obtained by then by performing the product of (1.15) by (1.17), and summing for all $p\in\bN^{s}$ (the sum is finite).\hfill\square\medskip For questions of convergence that will take place at the\S2, we will also need to know a field-independent $N_{R}(\lambda)$ increase magnetic~$B$ . One Such a uniform estimate is provided by the following proposal. \medskip {\statement Proposition 1.18.\pointir} $N_{R}(\lambda)\leq(R\sqrt{\lambda_{+}}+1)^{n}$ . \medskip {\it Demonstration}. -- For each index $j$ is increased the number of integers $x\in X$f and $x\in X$g such as inequality $$ \lambda_{p,\ell'}+\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda $$ takes place. The lemma 1.12 implies $$ \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, $$ while (1.16) results in $$ \card \{l_{i}\}\leq\frac{R}{\pi}\sqrt{\lambda_{+}}+\frac{B_{j}R^{2}}{2\pi}+1, \qquad 1\leq j\leq s. $$ We deduce therefore for $1\leq j\leq s$~: $$ \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} $$ For $x\in X$m , the inequality (1.14) gives on the other hand $$ |\ell_{j}|<\frac{R}{2\pi}\sqrt{\lambda_{+}}, $$ hence $\card \{l_{j}\}\leq\frac{R}{\pi}\sqrt{\lambda_{+}}+1$ . Proposition 1.18 follows.\hfill\square\medskip {\it End of the demonstration of the theorem} 1.6 (minus $x\in X$p ). To minimize $x\in X$q , it is sufficient to construct a vector space of finite dimension on which $Q_{R}(u)\leq\lambda\Vert u\Vert_{P(R)}^{2}$ . We consider for this vector space $\cF_{\lambda}$ of linear combinations of "own functions"\ of the type (1.9), subject to the conditions of cancellation on board (1.10), and summoned on the $(p,\ell)\in\bN^{s}\times\bZ^{n-s}$ indices such as $$ \lambda_{p,\ell'}+\frac{4\pi^{2}}{R^{2}}|\ell''|^{2}\leq\lambda. $$ Based on the reasoning in Proposition 1.18, the number of conditions (1.10) to be carried out is increased by $$ \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_{s0$ . We deduce therefore: $$ \eqalign{ \widetilde{q}(\chi_{R}f) &\leq\widetilde{q}(f)+\int_{|t|>\sqrt{R}/4}\bigg(\frac{10}{\sqrt{R}}\Big|f \frac{df}{dt}\Big|+\frac{25}{R}|f|^{2}\bigg)\,dt\cr &\leq\widetilde{q}(f)+\int_{|t|>\sqrt{R}/4}\bigg(\frac{1}{R}\Big|\frac{df}{dt}\Big|^{2} +25\Big(1+\frac{1}{R}\Big)|f|^{2}\bigg)\,dt\cr &\leq\Big(1+\frac{C_{2}}{R}\Big)\widetilde{q}(f) \leq\Big(1+\frac{C_{2}}{R}\Big)(2p_{j}+1)B_{j}\,\Vert f\Vert^{2}\cr &\leq\Big(1+\frac{C}{R}\Big)(2p_{j}+1)B_{j}\,\Vert \chi_{R}f\Vert^{2}\cr} $$ This gives $q$a .\hfil\square \medskip For the reader's convenience, we now state the principle of minimax in the form in which it has served us. \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 space of Hilbert $\cH$. We assume that $Q$ is bounded from below, i.e.\ $Q(f)\geq-C\Vert f\Vert^{2}$ if $f\in D(Q)$ , that $D(Q)$ is complete for the $\Vert f\Vert_{Q}=[Q(f)+ (C+1)\Vert f\Vert^{2}]^{\frac{1}{2}}$ norm, and finally that injection $(D(Q), \Vert~~\Vert_{Q})\hookrightarrow(\cH,\Vert~~\Vert)$ is compact. Then $Q$ has a discrete spectrum $\lambda_{1}\leq\lambda_{2}\leq\ldots~$, and we have the 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 $F$ describes 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 $F$ describes all the $Q$-closed subspaces of codimension $p$ of~$D(Q)$ .\vskip-\parskip} \bigskip \section{2}{Asymptotic distribution of the spectrum\\ (case of a variable field).} Again, we place ourselves within the general framework described above. at the beginning of the\S1. Our objective is to study the spectrum of the form quadratic $Q_{\Omega,k}$ (see (1.3)) in the case of a magnetic field $B$ and any $V$ electric field. For any point $q$z , or $$ B(a)=\sum_{j=1}^{s}B_{j}(a)\,dx_{j}\wedge dx_{j+s} \leqno(2.1) $$ reduced writing of $B(a)$ in a suitable orthonormal base $(dx_{1},\ldots,dx_{n})$ of ${}<0$d , where ${}<0$e is the row of $B(a)$ , and where ${}<0$g are the non-zero eigenvalue modules of endomorphism associated antisymmetric. Equality of definition 1.5 allows you to view $\nu_{B}(\lambda)$ as a function of the torque $(a,\lambda)\in M\times\bR$ . We will need also to consider the function ${}<0$j , continuous right in ${}<0$k , defined by~: $$ \overline{\nu}_{B}(\lambda)=\lim_{0<\varepsilon\to 0} \nu_{B}(\lambda+\varepsilon). \leqno(2.2) $$ We then demonstrate the following generalization of the corollary 1.7. \medskip {\statement Theorem 2.3.\\\it When $k$ tends towards $+\infty$ , the number $N_{\Omega,k}(\lambda)$ of eigenvalues${}\leq\lambda$ of $Q_{\Omega,k}$ checks the asymptotic framing $$ \int_{\Omega}\nu_{B}(V+\lambda)\,d\sigma\leq\lim\inf k^{-\frac{n}{2}}N_{\Omega,k} (\lambda) \leq\lim\sup k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)\leq\int_{\Omega} \overline{\nu}_{B}(V+\lambda)\,d\sigma. $$\vskip-\parskip} The $\lambda\mapsto\int_{\Omega}\nu_{B}(V+\lambda)\,d\sigma$ function is increasing and continuous to the left~; therefore it has at most one set $\cD$ countable points of discontinuity. The ${}<0$u set is also empty if ${}<0$v is odd, because ${}<0$w is then continuous. From this, we immediately deduce the \medskip {\statement Corollary 2.4.\\\it It is assumed that $\partial\Omega$ is zero measurement. So $$ \lim_{k\to+\infty}k^{-\frac{n}{2}}N_{\Omega,k}(\lambda)=\int_{\Omega}\nu_{B}(V+ \lambda)\,d\sigma $$ for all ${}<0$z , and density measurement spectral $k^{-\frac{n}{2}} \frac{d}{d\lambda}N_{\Omega,k}(\lambda)$ converge weakly on $\bR$ to $\frac{d}{d\lambda}\int_{\Omega}\nu_{B}(V+\lambda)\, d\sigma$ . If $n$ is odd, the limit measurement is diffuse.\hfil\square} \medskip The following lemma shows that the integrals of theorem 2.3 have well a sense. \medskip {\statement Lemma 2.5.}{\it {\parindent 6.5mm \vskip2pt \item{\rm(a)} We have the 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 semi-continuous lower $($ resp.\ upper$)$ on~$M$ . \vskip2pt \item{\rm(c)} At any point $(n-q)$l where $s(x)< \frac{n}{2}$ we have $(n-q)$n and $\nu_{B}(V),\overline{\nu}_{B}(V)$ are continuous in~$x$ . \vskip2pt \item{\rm(d)} If $n$ is odd, $\nu_{B}(V)=\overline{\nu}_{B}(V)$ is continues on~$M$ .\vskip-\parskip} \medskip {\it Demonstration. --} (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 $\lambda-(2p_{j}+1)B_{j}$ or${}\geq 0$ is increased by $\frac{\lambda_{+}}{B_{j}}$ . As the numerical quantity shown in (1.5) is increased by~$1$ , the inequality (a) follows. (b, c) Rank $s=s(x)$ is a semi-continuous function below on~${}^{\scriptscriptstyle\circ}$va , and the eigenvalues ${}>0$b , prolonged by $B_{j}(x)=0$ for $j>s(x)$ , are continuous on~${}>0$e . As the function $t\mapsto t_{+}^{0}$ (resp.\ $t\mapsto(t+0)_{+}^{0})$ is semi-continuous lower (resp. upper), the semi-continuity of $\nu_{B}(V)$ and $\overline{\nu}_{B}(V)$ is a problem only at points $a\in M$ in the vicinity of which $s(x)$ is not local constant. At such a point $a\in~M$ , we necessarily have $s(a)< \frac{n}{2}$ , so ${}>0$n ; we are going to then show that $\nu_{B}(V)$ and $\overline{\nu}_{B}(V)$ are continuous in~${}>0$q . The continuity of $B_{j}$ gives ${}>0$s for $j>s(a)$ . If the integers $p_{1},\,\ldots\,,p_{s\langle a)}$ are fixed, the summation in (1.5) may be construed as a sum of of Riemann's complete works on $\bR^{s(x)-s(a)}$ , and we have therefore the equivalent~: $$ \eqalign{ \sum_{(p_{j};\;s(a)0$ . To solve this difficulty, you just have to pave every parallelepiped $P_{k}$ by cubes $P_{k,\alpha}$ of which the edges are parallel to the coordinate axes $(y_{1}, \ldots,y_{n})$ . let's choose $\varepsilon\in{}]0,1[$ . For all $\alpha\in\bZ^{n}$ , are $(P_{k,\alpha})$ , $(P_{k,\alpha}')$ the cubes open sides $\varepsilon r_{k}$ , $\varepsilon(1+\varepsilon)r_{k}$ , and of common center $\varepsilon r_{k}\alpha$ . We will limit ourselves to considering the $P_{k,\alpha}$ cubes contained in $P_{k}$ and $P_{k,\alpha}'$ cubes meeting~$P_{k}$ . Then we have $$ \leqalignno{ &P_{k}\supset\bigcup_{\alpha}P_{k,\alpha}~~\hbox{(disjointe),~~~et}~~~ \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{et}~~~ \frac{\sum_\alpha\Vol(P_{k,\alpha}')}{\Vol(P_{k})} \leq 1+C_{7}\varepsilon,&(2.13)\cr} $$ where $C_{7}$ is a constant independent of $k$ (and also of $a$ , if $a$ describes a compact). The number of cubes $P_{k,\alpha}$ , $P_{k,\alpha}'$ which are listed in (2.12) or (2.13) is increased by $C_{8}\varepsilon^{-n}$ . As the $P_{k,\alpha}'$ cubes overlap two by two on a length $\varepsilon^{2}r_{k}$ when they are contiguous, a partition of the unit can be constructed $\sum \psi_{k,\alpha}^{2}=1$ on $P_{k}$ , with \hbox{Supp$\,\psi_{k,\alpha}\subset P_{k,\alpha}'$} and $$ \sup_{P_{k}}\sum_{\alpha}|d\psi_{k,\alpha}|^{2}=C(\psi_{k})\leq C_{9} (\varepsilon^{2}r_{k})^{-2}. $$ The $\lim k^{\frac{1}{2}}r_{k}=+\infty$ hypothesis leads well to $\lim\frac{1}{k}C(\psi_{k})=0$ , which allows to apply 2.6~(b). On the cubes $P_{k\alpha}$ , $P_{k,\alpha}'$ we are maintenant dans la situation du théo­rth~1.6~: after ratio homothety $\sqrt{k}$ , the side of the cube homothetic $\sqrt{k}\,P_{k,\alpha}$ is worth $R_{k}=\varepsilon r_{k}\sqrt{k}$ and tends well to $+\infty$ by hypothesis. The uniform mark-up of $N_{P_{k},k}(\lambda)$ results from Proposition 1.18 and the fact that all our $C_{1}, \ldots, C_{9}$ constants were uniforms. Proposition 2.9 is demonstrated.\hfil\square\medskip {\it Demonstration of theorem 2.3.} -- According to the remark preceding the proposition 2.9, we can assume that $M=\bR^{n}$ and $\Omega$ is a open bounded of $\bR^{n}$ . The idea of the reasoning is to combine the proposals 2.6 and 2.9 using a paving of $\Omega$ by cubes of side $r_{k}=k^{-\frac{1}{3}}$ . The actual implementation requires nevertheless a little care because of the difficulties related to the possible non-uniformity of the $\limsup$ and $\liminf$ . Designate by $\Pi_{k,\alpha}$ , $\Pi_{k,\alpha}'$ , $\alpha\in \bZ^{n}$ , the cubes open on their respective sides $$ k^{-\frac{1}{3}},\qquad k^{-\frac{1}{3}}(1+ k^{-\frac{1}{8}})= k^{-\frac{1}{3}}+k^{-\frac{11}{24}} $$ and common center $k^{-\frac{1}{3}}\alpha$ . Either $I(k)$ (resp.\ $I'(k)$ ) the set of $\alpha\in\bZ^{n}$ indices such as $\Pi_{k.\alpha}\subset\Omega$ (resp.\ $\overline\Pi_{k,\alpha}' \cap\overline\Omega\neq\emptyset$ ). As in the reasoning of proposal 2.9, there is a partition of the unit $\sum_{\alpha\in I'(k)}\psi_{k,\alpha}^{2}=1$ on $\Omega$ , with \hbox{Supp$\,\psi_{k,\alpha}\subset\Pi_{k,\alpha}'$} and $$ C(\psi_{k})=\sup_{\Omega}\sum_{\alpha\in I'(k)}|d\psi_{k,\alpha}|^{2} \leq C_{10}k^{\frac{11}{12}}, $$ hence $\lim\frac{1}{k}C(\psi_{k})=0$ . We pose $$ \Omega_{k}=\bigcup_{\alpha\in I(k)}\Pi_{k,\alpha},\qquad \Omega_{k}'=\bigcup_{\alpha\in I'(k)}\Pi_{k,\alpha}' $$ and one considers for every $\lambda\in\bR$ fixed, the functions on $\bR^{n}$ defined by $$ \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} $$ where $\bOne_{\Pi_{k,\alpha}}$ designates the characteristic function of $\Pi_{k,\alpha}$ . La proposition 2.6 implique l'enca­hard $$ \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) $$ Either $x\in\bR^{n}$ a fixed point not belonging to the set negligible $$ Z=\bigcup_{k\in\bN,\,\alpha\in\bZ^n}\partial\Pi_{k,\alpha}. $$ There is then a unique $\alpha(k)\in\bZ^{n}$ index sequence such as $x\in\Pi_{k,\alpha(k)}$ . Proposition 2.9 applied to $P_{k}=\Pi_{k,\alpha(k)}$ cubes (resp.\ $P_{k}'=\Pi_{k,\alpha(k)}'$ ) with $\Vol(P_{k})\sim \Vol P_{k}'$ shows that the punctual sequences $$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), $$ are such that $$\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) $$ The uniform increase in proposal 2.9 also results in the existence of constants $C_{11}$ , $C_{12}$ independent of $k$ , $x$ and $\lambda$ such as $$ f_{k}(x)\leq f_{k}'(x)\leq C_{11}\big(1+\sqrt{\lambda_{+}+C_{12}}\,\big)^{n}. $$ The theorem 2.3 then results from (2.14), (2.15) and the lemma of Fatou.\hfil\square\medskip For applications with complex geometry, we will require d'une légère généra­lisation of theorem 2.3. We give ourselves a $F$ hermitian fiber of rank $r$ and class $\cC^{\infty}$ above of~$M$ , provided with a hermitian connection $\nabla$ , and continuous sections $S$ of the fiber $\Lambda_{R}^{1}T^{*}X\otimes_{R}\Hom_{\bC}(F,F)$ and $V$ of the fibered $\Herm(F)$ of~$F$ hermitian endomorphisms. Either $\nabla_{k}$ the hermitian connection on $E^{k}\otimes F$ induced by the connections $D$ and $\nabla$ . To shorten the notations, one will still designate by $S$ and $V$ the endomorphisms $\Id_{E^{k}}\otimes S$ and $\Id_{E^{k}}\otimes V$ operating on~$E^{k}\otimes F$ . Given an open $\Omega$ relatively compact in~$M$ , we consider the quadratic form $$ Q_{\Omega,k}(u)=\int_{\Omega}\Big(\frac{1}{k}|\nabla_{k}u+Su|^{2}-\langle Vu, u\rangle\Big)\,d\sigma, $$ where $u\in W_{0}^{1}(\Omega,E^{k}\otimes F)$. Let $V_{1}(x)\leq V_{2}(x) \leq\cdots\leq V_{r}(x)$ be the eigenvalues of $V(x)$ at any point $x\in M$ . We then have the following result.\medskip {\statement Theorem 2.16.\pointir}{\it The counting function $N_{\Omega,k}(\lambda)$ of the eigenvalues of $Q_{\Omega,k}$ admits for all $\lambda\in\bR$ asymptotic estimates $$ \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} $$ where $B$ is the magnetic field associated with the connection $D$ on~$E$ .} \medskip {\it Demonstration}. -- The localization principle 2.6 is still valid in the present situation. It is therefore sufficient to demonstrate the inequalities 2.16 when $\Omega$ is small enough. Let $a\in M$ a fixed point and $(e_{1},\ldots,e_{r})$ an orthonormal marker $\cC^{\infty}$ of $F$ above a neighborhood $W$ of $a$ , such as $(e_{1}(a),\ldots,e_{r}(a))$ or a clean base for $V(a)$ . Let's write $u$ as $$ u=\sum_{j=1}^{r}u_{j}\otimes e_{j} $$ where $u_{j}$ is a section of $E^{k}$ . For any $\varepsilon>0$ , there is exists a neighborhood $W_{\varepsilon}'\subset W$ of $a$ on which $$ \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}, $$ and the term $u_{j}\otimes\nabla e_{j}$ can be absorbed into $Su$ (which actually brings us back to the case where the $\nabla$ connection is flat). Coaching $$ (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} $$ shows that the term $Su$ only changes $Q_{\Omega,k}$ by a factor of multiplicative $1\pm\varepsilon$ and by an additive factor $\pm\varepsilon\Vert u\Vert^{2}$ . For all $\varepsilon>0$ , there are so a neighborhood $W_{\varepsilon}$ of $a$ and an integer $k_{0}(\varepsilon)$ such as $$ (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} $$ as soon as $k\geq k_{0}(\varepsilon)$ and $\Omega\subset W_{\varepsilon}$ , where $\widetilde Q_{\Omega,k}$ designates the quadratic form $$ \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. $$ As $\widetilde Q_{\Omega,k}$ is a direct sum of $r$ forms quadratic, the spectrum of $\widetilde Q_{\Omega.k}$ is the meeting (counted with multiplicities) of the spectra of each of the terms of the sum. The theorem 2.16 follows.\hfil\square\bigskip \section{3}{Identity of Bochner-Kodaira-Nakano\\\ in hermitian geometry.} The purpose of the following paragraphs is to draw the consequences of the théorème de répar­spectral analysis 2.16 for the study of the $d''$ -cohomology of holomorphic vectorial fibrils hermitians. For this purpose, we will need to connect the laplacian antiholomorph $\Delta''$ to Schrödinger's ope'rateur of a connection adequate real one. This is done by means of a formula Weitzenböck type, known in geometry and design for the complex under the identity of Bochner-Kodaira-Nakano. Either $X$ a compact complex analytical variety of dimension $n$ and $F$ a hermitian holomorphic vectorial fiber of rank $r$ above of~$X$ . It is known that there is a unique hermitian connection $D=D'+D''$ on $F$ whose $D''$ component of type $(0,1)$ coincides with the operator $\overline\partial$ of the fibrous (such a connection is called holomorphic). Let $c(F)=D^{2}=D'D''+D''D'$ the bending shape of~$F$ . Let's provide $X$ with an arbitrary hermitian me'trique $\omega$ of type $(1,1)$ and class $\cC^{\infty}$ . $\cC_{p,q}^{\infty}(X,F)$ space sections of class $\Lambda^{p,q}T^{*}X\otimes F$ fiber $\Lambda^{p,q}T^{*}X\otimes F$ se trouve alors muni d'une structure préhil­ber­natural hold. We note $\delta=\delta'+\delta''$ the formal assistant to $D$ considered as a differential operator on $\cC^{\infty}(X,F)$ , and $\Lambda$ the operator's assistant $L:u\mapsto\omega\wedge u$ . We will use the identity of Bochner-Kodaira-Nakano in the form of demonstrated in [6], although one can in fact make a general statement in be content, as Y.T. does.~Siu [16], [17], of the formula minus precise given by P.~Griffiths. If $A$ , $B$ are differential operators on $\cC^{\infty}(X,F)$ , one defines their anti-switch $[A,B]$ by the formula $$ [A,B]=AB-(-1)^{ab}BA $$ where $a$ , $b$ are the respective degrees of $A$ and~$B$ . The Laplace-Beltrami operators $\Delta'$j and $\Delta''$ are then classically given by $$ \Delta'=[D',\delta']=D'\delta'+\delta'D',\qquad \Delta''=[D'', \delta''] $$ To the torsion shape $d'\omega$ , we associate the operator of the outward multiplication $u\mapsto d'\omega\wedge u$ on $\cC^{\infty}(X,F)$ , type $(2,1)$ , simply noted $d'\omega$ , and the operator $\tau$ of type $(1,0)$ defined by $\tau=[\Lambda,d'\omega]$ . We finally pose $$ D_{\tau}'=D'+\tau,\qquad \delta_{\tau}' =(D_{\tau}')^{*}=\delta'+\tau^{*},\qquad \Delta_{\tau}'=[D_{\tau}',\delta_{\tau}']. $$ We then have the following identity, for a demonstration of which the reader will refer to [6].\medskip {\statement Proposition 3.1.\\\it We have $\Delta''=\Delta_{\tau}'+[ic(F),\Lambda]+ T_{\omega}$ where $T_{\omega}$ is the operator of order $0$ and of type $(0,0)$ defined by $$ T_{\omega}=\Big[\Lambda,\Big[\Lambda,\frac{i}{2}d'd''\omega\Big]\Big]- [d'\omega,(d'\omega)^{*}]. $$}\vskip-\parskip According to Hodge-De Rham's theory, the cohomology group $H^{q}(X,F)$ identifies itself with the space of $(0,q)$ -forms $\Delta''$ -harmonics with values in~$F$ . Or $u\in \cC_{p.q}^{\infty}(X,F)$ . Proposition 3.1 gives us equality $$ \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) $$ where integrals are calculated relative to l'élément de volume $d\sigma=\frac{\omega^{n}}{n!}$ . En parti­culier, if $u$ is of bidegré $(0,q)$ , we have $\delta_{\tau}'u=0$ by reason of bidegré, from where $$ \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) $$ One can also consider $u$ as a $(n,q)$ -form to values in fiber $$ \widetilde F:=F\otimes\Lambda^{n}TX~; $$ we will note $\widetilde D=\widetilde D'+\widetilde D''$ the hermitian connection holomorph of $\widetilde F$ and $\widetilde{u}$ the canonical image of $u$ in $\cC_{n,q}^{\infty}(X,F)$ . \medskip {\statement Lemma 3.4.\\\it We have switching diagrams $$ \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} $$ where the vertical arrows are the isometrics $u\mapsto\widetilde{u}$ .} {\it Demonstration}. -- Switching of the left diagram results from the fact that $\Lambda^{n}TX$ is a holomorphic fiber (be careful that the corresponding result for $D'$ and $\widetilde D'$ is wrong). So we have a diagram analogue switch for assistants $\delta''$ , $\widetilde\delta''$ and for $\Delta''$ , $\widetilde \Delta''$ .\hfil\square\medskip The lemma 3.4 and the identity (3.2) give us $$ \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) $$ We will now slightly transform the writing of (3.3) and (3.5). The holomorphic hermitian connection of the fibroid $\Lambda^{q}T^{*}X$ induced on conjugated fiber $\Lambda^{0,q}T^{*}X$ a connection with a component of type $(1,0)$ coincides with the operator~$d'$ . From this we deduce then a natural hermitian connection $\nabla$ on the fibered product produces tensorial $\Lambda^{0,q}T^{*}X\otimes F$ (we will observe that this fibrous vector is not generally holomorphic if $q\neq 0$ ). Let $\nabla'$~et $\nabla''$ the components of $\nabla$ of type $(1,0)$ and $(0,1)$ . \medskip {\statement Proposition 3.6.\\\it We have $$ \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} $$ where the vertical arrows are isometries, that of left being given by $u\mapsto\widetilde{u}$ .} \medskip {\it Demonstration}. -- The equality $\nabla' =D'$ comes from the makes the $(1,0)$ component of the connection from $\Lambda^{0,q}T^{*}X$ coincides with~$d'$ . For the diagram, we start by defining the vertical arrow~$\Psi$ . Either $$ \{?|?\} : (\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 $$ the canonical sesquilinear coupling induced by the metric on the ~$F$ fibers, and $$ *{}: \Lambda^{p,q}T^{*}X\otimes\widetilde F\longrightarrow \Lambda^{n-q,n-p}T^{*}X\otimes\widetilde F $$ the operator of Hodge-De Rham-Poincaré defined by $$ \{v|*w\}=\langle v,w\rangle\,d\sigma,\qquad v,~w\in \Lambda^{p,q}T^{*}X\otimes\widetilde F. $$ We deduce from this by composition an isometry $$ \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 $$ and the arrow $E^{k}\otimes F$ is obtained by definition by tensorizing $-i^{-n^{2}}\Psi_{0}$ by $\Lambda^{0,q}T^{*}X$ . To demonstrate the commutativity, we first assume $q=0$ . Let $u\in \cC^{\infty}(F)$ . We have classically $$ \widetilde{\delta}'\widetilde{u}=-*\widetilde D''*\widetilde{u}, $$ and like $\widetilde{u}\in \cC_{n,0}^{\infty}(X,F)$ , he comes $*\widetilde{u}=i^{-n^{2}}\widetilde{u}$ , hence $$ \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). $$ In case $q$ is arbitrary, you just have to trivialize $\Lambda^{0,q}T^{*}X$ in the vicinity of an arbitrary point $x$, in choosing an orthonormal marker $(e_{1}, \ldots,e_{N})$ of this fiber, such as\hbox{$\nabla e_{1}(x)=\cdots=\nabla e_{N}(x)=0$}. \hfil\square\medskip We now consider the morphisms of fibrils $$ \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} $$ where $S'=\tau=[\Lambda,d'\omega]$ , and where $S''$ is the reading by the $\sim$ and $\Psi$ isometries of the morphism $$ \tau^{*}=[(d'\omega)^{*},L]:\Lambda^{n,q}T^{*}X\otimes\widetilde F \to\Lambda^{n-1.q}T^{*}X\otimes\widetilde F. $$ According to proposal 3.6, we have $$ |D_{\tau}'u|=|\nabla'u+S'u|,\qquad |\widetilde\delta'_{\tau}\widetilde{u}|=|\nabla''u+S''u|. $$ If $S=S'\oplus S''$ is set, identities (3.3) and (3.5) imply by addition $$ \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} $$ for all $u\in \cC_{0,q}^{\infty}(X,F)$ . Let now be $E$ a hermitian holomorphic fiber of rank~$1$ above of~$X$ . For the whole~$k$ , we note $D_{k}$ and $\nabla_{k}$ the natural hermitian connections on the fibrils $F_{k}=E^{k}\otimes F$ and $\Lambda^{0,q}T^{*}X\otimes F_{k}$ , and on pose $\Delta_{k}''=[D_{k}'',\delta_{k}'']$ . The curvature of $F_{k}$ (resp.\ $\widetilde F_{k}$ ) is given by $$ 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) $$ Let us recall, although it is useless for the rest, that $$ c(\widetilde F)= c(F)+c(\Lambda^{n}TX)\otimes\Id_{F}=c(F)+{\rm Ricci}(\omega)\otimes\Id_{F}. $$ We will therefore need to evaluate the terms $[ic(E),\Lambda]$ . For all point $x\in X$ , are $\alpha_{1}(x)$ , $\alpha_{2}(x), \ldots, \alpha_{n}(x)$ the eigenvalues of $ic(E)(x)$ relative to the metrics hermitienne $\omega$ on $X$ . So there is a system of local coordinates $(z_{1}, \ldots,z_{n})$ centered in $x$ such that $(\frac{\partial}{\partial z_{1}},\ldots,\frac{\partial} {\partial z_{n}})$ is an orthonormal base of $T_{X}X$ , and such as $$ \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} $$ Either $(e_{1}, \ldots,e_{r})$ an orthonormal marker of the fiber $E_{x}^{k}\otimes F_{x}$ . For $v\in\Lambda^{p.q}T^{*}X\otimes F_{k}$ , you can write $$ 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} $$ An elementary calculation, as explained for example in [6], gives the formula $$ \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) $$ with $\alpha_{I}=\sum_{j\in I}\alpha_{j}$ . Either $u\in\Lambda^{0,q}T^{*}X\otimes F_{k}$ . Let's post $$ u= \sum_{J,\ell}u_{J,\ell}\,d\overline{z}_{J}\otimes e_{\ell}. $$ According to (3.9), it comes from $$ \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} $$ Either $V$ the hermitian endomorphism of $\Lambda^{0,q}T^{*}X\otimes F_{k}$ defined by $$ \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) $$ The eigenvalues of $V$ are therefore the coefficients $\alpha_{\complement J}-\alpha_{J}$ , counted with multiplicity $r=\rank(F)$ . Or finally $\Theta$ the hermitian endomorphism defined by $$ \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) $$ where the operators $S$ , $V$ , $\Theta$ act only on the component $\Lambda^{0,q}T^{*}X\otimes F$ of $\Lambda^{0,q}T^{*}X\otimes F_{k}$ . So we're going to be able to use theorem 2.16 to determine the spectral distribution asymptotic of $\Delta_{k}''$ , because the term $\frac{1}{k}\langle\Theta u,u\rangle$ tends towards $0$ as norm. Let $h_{k}^{q}(\lambda)$ be the number of eigenvalues${}\leq k\lambda$ of $\Delta_{k}''$ operating on $\cC_{0,q}^{\infty}(E^{k}\otimes F)$ . The magnetic field $B$ is here given by $$ B=-ic(E)=-\sum_{j=1}^{n}\alpha_{j}\,dx_{j}\wedge dy_{j},\qquad z_{j}=x_{j}+iy_{j}.\leqno(3.13) $$ Given that $\dim_{\bR}X=2n$ , the theorem 2.16 is transcribed as follows. \medskip {\statement Theorem 3.14.\\\it There is a set of countable $\cD$ such as for any $q=0,1, \ldots, n$ and all $\lambda\in\bR\ssm \cD$ we have $$ 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}) $$ when $k$ tends towards $+\infty$ .} \bigskip \section{4}{ Witten's complex and Morse inequalities.} E.~Witten [18], [19] has recently introduced a new method analytique pour démon­to overcome the inequalities of Morse in de Rham's cohomology. We adapt here his method to the study of $d''$ -cohomology. The main difference lies in the fact that the magnetic field is always null. in the case of de Rham's cohomology (we have indeed $d^{2}=0$~!), and it is the electric field which intervenes alone in this case. With the\S3 notations, that is $\cH_{k}^{q}(\lambda)\subset \cC_{0,q}^{\infty} (X,E^{k}\otimes F)$ the direct sum of the clean subspaces of $\Delta_{k}''$ attached to eigenvalues${}\leq k\lambda$ . $\cH_{k}^{q}(\lambda)$ is therefore a vector space of finite dimension. $$ h_{k}^{q}(\lambda)=\dim_{\bC}\cH_{k}^{q}(\lambda) . $$ Hodge's theory gives an isomorphism $$ H^{q}(X,E^{k}\otimes F)\simeq \cH_{k}^{q}(0) . $$ We will pose to shorten $$ h_{k}^{q}=\dim H^{q}(X,E^{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^{k}\otimes F). $$ In addition, the $\cH_{k}^{\bullet}(\lambda)\subset \cC_{0,\bullet}^{\infty} (X,E^{k}\otimes F)$ inclusion and the orthogonal projection $$ P_{\lambda}:\cC_{0,\bullet}^{\infty}(X,E^{k}\otimes F)\to \cH_{k}^{\bullet} (\lambda) $$ induce in cohomology inverse isomorphisms of each other.} \medskip {\it Demonstration}. -- The fact that $\cH^{\bullet}_{k}(\lambda)$ is a sub-complex of $\cC_{0,\bullet}^{\infty}(X,E^{k}\otimes F)$ comes from the switching property of the operators $D_{k}''$ and $\Delta_{k}''$ . Either now $$ G=\int_{\lambda>0}\frac{1}{\lambda}dP_{1} $$ the Green operator of the laplacian $\Delta_{k}''$ . Like $[P_{\lambda},\Delta_{k}'']=0$ , we have the relations $[G,\Delta_{k}'']=0$ and $$ \Delta_{k}''G+P_{0}=\Id. $$ In addition, $[P_{\lambda},D_{k}'']=[G,D_{k}'']=0$ . We can therefore deduce $$ \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} $$ so that the operator $\delta_{k}''G(Id-P_{\lambda})$ is a homotopy between $\Id$ and $P_{\lambda}$ .\hfil\square\medskip We now use a simple classical lemma of homological algebra. \medskip {\statement Lemma 4.2.\pointir} {\it Either $$ 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 $$ a complex of vector spaces of finite dimensions $c^{0}$ , $c^{1}$ , $\ldots$q, $c^{n}$ on a body~$\bK$ . Let $h^{q}=\dim_{\bK}H^{q}(C^{\bullet})$ . So we have the following inequalities: {\parindent=6.5mm \vskip2pt \item{\rm(a)} Morse code Inequalities~$:$~ $h^{q}\leq c^{q}$ ,~ $0\leq q\leq n$ . \vskip2pt \item{\rm(b)} Equality of characteristics of Euler-Poincaré~ $\chi(H^{\bullet}(C^{\bullet}))=\chi(C^{\bullet})~:$ $$ h^{0}-h^{1}+\cdots+(-1)^{n}h^{n}=c^{0}-c^{1}+\cdots+(-1)^{n}c^{n}. $$ \item{\rm(c)} Strong Morse code inequalities~$:$ for all $q$ , $0\leq q\leq n$ , $$ h^{q}-h^{q-1}+\cdots+(-1)^{q}h^{0}\leq c^{q}-c^{q-1}+\cdots+(-1)^{q}c^{0}. $$}}\vskip0pt {\it Demonstration}. -- If $Z^{q}=\Ker d^{q}$ and $B^{q}=\Im d^{q-1}$ are for dimensions $z^{q}$ and $b^{q}$ , the equality (b) results from the formulas $$ c^{q}=z^{q}+b^{q+1},\qquad h^{q}=z^{q}-b^{q}, $$ while (c) results from (b) applied to the complex $$ 0\to C^{0}\to C^{1}\to\cdots \to C^{q-1}\to Z^{q}\to 0.\eqno\square $$ If $F$ is a holomorphic vectorial fiber on $X$ , we define its Euler-Poincaré characteristic by $$ \chi(X,F)=\sum_{q=0}^{n}(-1)^{q}\dim H^{q} (X,F). $$ Combining proposition 4.1 and lemma 4.2, we get for everything $\lambda\geq 0$ and all~$q$ , $0\leq q\leq n$ , inequality $$ 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). $$ Let us now evaluate $h_{k}^{q}(\lambda)$ using the 3.14 theorem and let's stretch $\lambda\in\bR\ssm \cD$ to $0$ by values${}>0$ . It follows~: \medskip {\statement Corollary 4.3.\pointir} {\it We have asymptotic inequalities {\parindent=6.5mm \vskip2pt \item{\rm(a)} $h_{k}^{q}\leq k^{n}I^{q}+o(k^{n}),$ \vskip2pt \item{\rm(b)} $\chi(X,E^{k}\otimes F)=k^{n}(I^{0}-I^{1}+\cdots+(-1)^{n}I^{n})+o(k^{n}),$ \vskip2pt \item{\rm(c)} $h_{k}^{q}-h_{k}^{q-1}+\cdots+(-1)^{q}h_{k}^{0}\leq k^{n} (I^{q}-I^{q-1}+\cdots+(-1)^{q}I^{0})+o(k^{n})$ , \vskip4pt where $I^{q}$ is the integral of curvature $$ I^{q}=r\sum_{|J|=q}\int_{X}\overline{\nu}_{B}(\alpha_{\complement J} -\alpha_{J})d\sigma. $$}}\vskip-\parskip According to (3.13), the eigenvalue modules of the magnetic field $B$ are the $|\alpha_{j}|$ , $1\leq j\leq n$ . For any point $x\in X$ , let us arranke these eigenvalues so that $$ |\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}| $$ is always${}\leq 0$ , $\overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})$ can be non-null only if $s=n$ . In the latter case $\alpha_{\complement J}-\alpha_{J}-\sum(2p_{j}+1)|\alpha_{j}|=0$ if and only if $ p_{1}=\cdots=p_{n}=0$ and $\alpha_{j}<0$ for $j\in J$ , $\alpha_{j}>0$ for $j\in{}{\complement}J$ . This results in the form $ic(E)$ is non-degenerated~$q$ . For $x\in X(q)$ (see~notations of the introduction) and $|J|=q$ , so we have $$ \overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})=(2\pi)^{-n} |\alpha_{1}\ldots\alpha_{n}|>0 $$ if $J$ is the multi-index $J(x)=\{j\,;\;\alpha_{j}(x)<0\}$ and $\overline{\nu}_{B}(\alpha_{\complement J}-\alpha_{J})=0$ si $J\neq J(x)$ . This results in $$ 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}. $$ The fundamental theorem 0.1 is then only a reformulation of the corollary 4.3. The above reasoning shows that the harmonic forms of $H^{q}(X,E^{k}\otimes F)$ focus asymptotically on~$X(q)$ , and that by each point of $X(q)$ their direction tends to align with the $q$ -sub-space of $TX$ corresponding to the negative part of~$ic(E)$ . Moreover, only the minimum eigenvalue of energy $p_{1}=\cdots=p_{n}=0$ of the harmonic oscillator intervenes for these forms. For $q=1$ , the inequality of Morse code 4.3~(c) is written as follows $$ h_{k}^{1}-h_{k}^{0}\leq k^{n}(I^{1}-I^{0})+o(k^{n}) , $$ hence in particular an asymptotic reduction of the number of sections. holomorphs of the fiber $E^{k}\otimes F$ . \medskip {\statement Theorem 4.4.\pointir}{\it On a $$ \dim H^{0}(X,E^{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}). $$}\vskip-\parskip More generally, the addition of inequality 4.3 (c) for the indexes $q+1$ and $q-2$ leads to $$ 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}) , $$ hence the reduction $$ \dim H^{q}(X,E^{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) $$ \medskip \section{5}{ Characterization of the varieties of Ego\v{s}ezon.} Let $X$ a variety $\bC$ -analytical compact related of dimension $n$ . We call algebraic dimension of $X$ , noted $a(X)$ , the degree of transcendence on $\bC$ of the body $K(X)$ of the meromorphic functions on~$X$ . According to a well-known theorem by Siegel [15], the dimension algebraic of $X$ always checks for inequality $0\leq a(X)\leq n$ . When $a(X)=n$ , it is said that $X$ is a space of Me\v{s}ezon. As we will see, the algebraic dimension of $X$ asymptotically imposes strong constraints on the dimension of the section spaces of a holomorphic vectorial fiber. \medskip {\statement Theorem 5.1.\pointir}{\it Let $a$ the dimension algebraic of $X$ , $F$ a fibrous one holomorphic vector of rank $r$ and $E$ a linear fiber on~$X$ . Then, there is a constant $C_{E}\geq 0$ that does not depend on than $E$ such as $$ \dim H^{0}(X,E^{k}\otimes F)\leq C_{E}rk^{a}+o(k^{a}) . $$}\vskip-\parskip {\it Demonstration}. -- We essentially take the arguments of Y.T.~Siu [16]. Let $\{W_{\ell}\}$ be an overlay of $X$ by open of coordinates $W_{\ell}\subset\bC^{n}$ , and $B_{j}=B(a_{j},R_{j})$ , $1\leq j\leq m$ , a family of relatively compact balls in the open $W_{\ell}$ , such as concentric balls $B_{j}'=B(a_{j},\frac{1}{7}R_{j})$ cover~$X$ . Provide $E$ , $F$ of hermitian metrics, and either $\exp(-\varphi_{j})$ the weight representing the metric of $E$ in a trivialization of $E$ in the vicinity of $\overline{B}_{j}$ . Let $s\in H^{0}(X,E^{k}\otimes F)$ be a holomorphic section that cancels itself out to the order $p$ in one point $x_{j}\in B_{j}'$ . Inclusions $$ B_{j}'\subset B(x_{j},\frac{2}{7}R_{j})\subset B(x_{j},\frac{6}{7}R_{j})\subset B_{j} $$ and the Schwarz lemma applied to the two intermediate balls lead to inequality $$ \sup_{B_{j}'}|s|\leq\exp(Ak+C_{F})3^{-p}\sup_{B_{j}}|s|, \leqno(5.2) $$ where $A = \max_{1\leq j\leq m}\diam \varphi_{j}(B_{j})$ depends only on of~$E$ , and where $C_{F}$ is a constant${}\geq 0$ which depends on the metric of~$F$ . Let $\rho\leq r=\rank(F)$ be the maximum for $x\in X$ of the dimension of the subspace of the $F_{x}$ fiber generated by the $s(x)$ vectors when $s$ describes $\bigcup_{k\in\bN}H^{0}(X,E^{k}\otimes F)$ . If $\rho=0$ , then $H^{0}(X,E^{k}\otimes F)=0$ for all~$k$ . Let us now distinguish two cases according to whether $\rho=1$ or~$\rho>1$ . (a){\it Assume $\rho=1$}. Let $h_{k}=\dim H^{0}(X,E^{k}\otimes F)$ , assumed${}>0$ . Under the hypothesis $\rho=1$ , the global sections of $E^{k}\otimes F$ define a holomorphic application $$ \Phi_{k}:X\ssm Z_{k}\to\bP^{h_{k}-1}(\bC) $$ where $Z_{k}\subset X$ is the analytical subset of their zeros common. Let $d$ the maximum rank of the differential $\Phi_{k}'$ on $X\ssm Z_{k}$ . We necessarily have $d\leq a$ , otherwise the body of the rational fractions of $\bP^{h_{k}-1}(\bC)$ would induce a body of functions metamorphs on~$X$ of degree of transcendence${}\geq d>a$ , which is absurd. Let's choose for all $j=1, \ldots, m$ a point $x_{j}\in B_{j}' \ssm Z_{k}$ such that $\Phi_{k}'$ is of maximum rank${}=d$ in $x_{j}$ , and either $s_{0}\in H^{0}(X,E^{k}\otimes F)$ a section that does not cancel in any way point~$x_{j}$ . For all $s\in H^{0}(X,E^{k}\otimes F)$ , the quotient $s/s_{0}$ is well defined as a meromorphic function on~$X$ , and moreover $s/s_{0}$ is a holomorphic function in the neighborhood of~$x_{j}$ , constant along the fibers of~$\Phi_{k}$ . As $\Phi_{k}$ is a submersion in the vicinity of each point~$x_{j}$ , one can choose a submergence sub-variety $M_{j}$ of dimension $d$ passing through $x_{j}$ and transverse to the fiber $\Phi_{k}^{-1}(\Phi_{k}(x_{j}))$ . The section $s$ will cancel in the order $p$ at each point $x_{j}$ , $1\leq j\leq m$ , if and only if partial derivatives of order${}1$}. There are then sections $s_{t}\in H^{0}(X,E^{k_{t}}\otimes F)$ , $1\leq t\leq\rho$ , and a point $x_{0}\in X$ such as the vectors $s_{1}(x_{0}), \ldots, s_{\rho}(x_{0})$ are linear independent. By construction, for any $k\in\bN$ and any section $s\in H^{0}(X,E^{k}\otimes F)$ , the right $\bC\cdot s(x)$ is contained in the subspace generated by $(s_{1}(x), \ldots,s_{\rho}(x))$ , except perhaps above the analytical subset $\{x\in X;s_{1} \wedge\ldots\wedge s_{\rho}(x)\}=0$ . So we have an injectable morphism $$ H^{0}(X,E^{k}\otimes F)\to\bigoplus_{1\leq t\leq\rho}H^{0}(X,E^{k+k_{\hat{t}}} \otimes\Lambda^{p}F) $$ where $k_{\hat t}=(k_{1}+\cdots+k_{\rho})-k_{t}$ , whose index component $t$ is given by the morphism $s\to s_{1}\wedge\cdots\wedge\widehat{s}_{t}\wedge \cdots\wedge s_{\rho}\wedge s$ . The image of $H^{0}(X,E^{k}\otimes F)$ on each component is formed collinear sections at almost any point in $s_{1}\wedge\cdots\wedge s_{\rho}$ . So we meet again in a situation analogous to (a), where $F$ is replaced by by $E^{k_{\hat{t}}}\otimes\Lambda^{\rho}F$~; thereafter~: $$ \dim H^{0}(X,E^{k}\otimes F)\leq C_{E}\rho k^{a}+o(k^{a}),\qquad\rho\leq r. \eqno\square $$ Let's choose in particular for $F$ the trivial fiber $X\times\bC$ . By comparing the theorems 4.4 and 5.1, we obtain the characterization geometrically following varieties of ego\v{s}ezon. \medskip {\statement Theorem 5.2.\pointir} {\it For a variety $\bC$ -analytical compact related $X$ of dimension $n$ either of Me\v{s}ezon, it is enough that there is a fiber in straight lines hermitian holomorph $E$ above $X$ such as $$\int_{X(\leq 1)}(ic(E))^{n}>0.\eqno\square $$}\vskip-\parskip This theorem in turn leads to the 0.8 theorem. since 0.8 (c)${}\Rightarrow{}$ 0.8(b)${}\Rightarrow{}$ 0.8(a). 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