Nacinovich, M. Poincaré lemma for tangential Cauchy Riemann complexes. (English) Zbl 0574.32045 Math. Ann. 268, 449-471 (1984). The author studies the tangential Cauchy-Riemann complex of a generic real submanifold S of a complex manifold X. The Levi form of S at \(x_ 0\in S\) is a quadratic form on the analytic tangent space \(H_{x_ 0}S\) with values in \(T_{x_ 0}S/H_{x_ 0}S\). For \(x_ 0\in S\), the set \(E(S,x_ 0)\) is defined with the help of the Levi form: \(E(S,x_ 0)\) is a set of pairs of non-negative integers (p,q) with \(p+q\leq \dim_{{\mathbb{C}}}H_{x_ 0}S.\) \(E(S,x_ 0)\) is a local pseudoconformal invariant on S. The author shows: If \(p_ 0=\min imum p: (p,q)\in E(s,x_ 0)\) then the tangential Cauchy-Riemann complex admits the Poincaré lemma at j wherever \(1\leq j<p_ 0\) and \(j>\dim_{{\mathbb{C}}}H_{x_ 0}-p_ 0.\) The approach in the proof follows ideas of A. Andreotti and H. Grauert in Bull. Soc. Math. Fr. 90, 193-259 (1962; Zbl 0106.055). Reviewer: A.Aeppli Cited in 5 ReviewsCited in 18 Documents MSC: 32L05 Holomorphic bundles and generalizations 32C35 Analytic sheaves and cohomology groups 32C36 Local cohomology of analytic spaces 58J10 Differential complexes 32V40 Real submanifolds in complex manifolds Keywords:sheaves of germs of differential forms; tangential Cauchy-Riemann complex; real submanifold; complex manifold; Levi form; Poincaré lemma Citations:Zbl 0106.055 PDFBibTeX XMLCite \textit{M. Nacinovich}, Math. Ann. 268, 449--471 (1984; Zbl 0574.32045) Full Text: DOI EuDML References: [1] Andreotti, A.: Complexes of partial differential operators. Chicago: Yale University Press 1975 · Zbl 0309.58020 [2] Andreotti, A.: Complessi di operatori differenziali. Boll. Un. Mat. Ital. A13, 273-281 (1976) · Zbl 0355.58013 [3] Andreotti, A.: E. E. Levi convexity and the H. Lewy problem. Actes du Congr?s Intern. des Math?m. Nice (1970), Tome 2, pp. 607-611. 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