Trèves, François A remark on the Poincaré lemma in analytic complexes with nondegenerate Levi form. (English) Zbl 0535.32005 Commun. Partial Differ. Equations 7, 1467-1482 (1982). 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 Cited in 8 Documents 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 Keywords:exterior differential form; nondegenerate Levi form; trivial cohomology; infinite dimensional cohomology space × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Baouendi M. S., Ann. of Math. 113 pp 421– (1981) · Zbl 0491.35036 · doi:10.2307/2006990 [2] Bony J. M., Asterisque 34 (35) pp 43– (1976) [3] Boutet de Monvel L., Comm. Pure Applied Math. pp 585– (1974) [4] Folland G. B., Ann. of Math, Studies 75, Princeton, N. J. (1972) [5] Geller D., Comm. in P.D.E. 5 pp 475– (1980) · Zbl 0488.22020 · doi:10.1080/0360530800882146 [6] Greiner P., Proc. Natl . Acad. Sci. USA 72 (9) pp 3287– (1975) · Zbl 0308.35017 · doi:10.1073/pnas.72.9.3287 [7] Kashiwara M., Springer Lecture Notes no 287 (1971) [8] Rothschild L. P., preprint [9] Treves F., Comm. P. D. E. 3 pp 475– (1978) · Zbl 0384.35055 · doi:10.1080/03605307808820074 [10] Treves F., Introduction to pseudodifferential and Fourier integral operators (1980) · doi:10.1007/978-1-4684-8780-0 [11] Treves F., Ecole Polytechnique Pal aiseau (France) May (1981) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.