Poincaré lemma for tangential Cauchy Riemann complexes (Q1063753)

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 \textit{A. Andreotti} and \textit{H. Grauert} in Bull. Soc. Math. Fr. 90, 193-259 (1962; Zbl 0106.055).