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A remark on the Poincaré lemma in analytic complexes with nondegenerate Levi form. (English) Zbl 0535.32005

Let \(\Omega\) be a real analytic manifold, dim \(\Omega =m+n\), \(T\)’ an \(m\)-dimensional vector subbundle of \({\mathbb{C}}T^*\Omega\) locally generated by differentials of m analytic functions. Let \(\widetilde\bigwedge{}^{p,q}\) be the subbundle of \(\bigwedge^{p+q}{\mathbb{C}}T^*\) locally generated by exterior products of one-forms p of which are sections of \(T'\), \(\bigwedge{}^{p,q}=\widetilde\bigwedge{}^{p,q}/\widetilde\bigwedge{}^{p+1,q-1};\) then \(d'=d'\!_{p,q}:C^{\infty}(\Omega;\quad \bigwedge{}^{p,q})\to C^{\infty}(\Omega;\bigwedge{}^{p,q+1})\) form a sequence of differential operators, \(d'\!_{p,q}d'\!_{p,q-1}=0.\) Let \(T^{\prime\perp}\) be the \(n\)-dimensional subbundle of \({\mathbb{C}}T\Omega\), orthogonal to T’, \(T^ 0=T'\cap T^*\Omega;\) in general \(T^ 0\) is not a vector subbundle of \(T^*\Omega\); \(T^ 0\) is the characteristic set of \(T^{\prime\perp}\). For \((\omega_ o,\lesssim)\in T^ 0\) we can introduce a Hermitian form on \(T^{\prime\perp}: {\mathcal L}(\omega_ 0,\theta,v_ 1,v_ 2)=(1/2i)<d\phi,v_ 1\vee \bar v_ 2>\quad(\omega_ 0)=(1/2i)<\theta,[L_ 2,L_ 1]>(\omega_ 0)\) where [ , ] is commutation bracket, \(\phi\), \(L_ j\) are sections of T’, \(T^{\prime\perp}\) respectively such that \(\phi(\omega_ 0)=\theta\), \(L_ j(\omega_ 0)=v_ j\), \({\mathcal L}\) is the Levi form. We can introduce a Hilbert structure on \(L^ 2(\Omega;\bigwedge^{0,q})\) and define \(d^{\prime*}:C^{\infty}(\bigwedge^{0,q})\to C^{\infty}(\bigwedge{}^{0,q-1})\) and Kohn-Laplacian \(\square =d^{\prime*}d'+d'd^{\prime*}\). - It is known that if either \(\square\) is elliptic (i.e. \(T^ 0=0)\) or the Levi form is nondegenerate in every point of \(T^ 0\) then there exists a parametrix of \(\square\). - Let us consider the complex \[ (*)\quad d':C^{\infty}(\bigwedge{}^{p,q})\to C^{\infty}(\bigwedge{}^{p,q+1}). \] Let \(\nu\) (\(\omega\),\(\theta)\) be the number of negative eigenvalues of the Levi form. It is proved that if at every point \((0,\theta)\in T^ 0\backslash \{0\}\) the Levi form is nondegenerate and neither \(\nu =n-q\) nor \(\nu =q\) then the q-th cohomology of (*) is trivial. It is also proved that if at some point \((0,\theta)\in T^ 0\backslash \{0\}\) the Levi form is nondegenerate and either \(\nu =n-q\) or \(\nu =q\) then the q- th cohomology space is infinite dimensional.
Reviewer: V.Ja.Ivrii

MSC:

32C35 Analytic sheaves and cohomology groups
65H10 Numerical computation of solutions to systems of equations
58J10 Differential complexes
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
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References:

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