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Tangential Cauchy-Riemann complexes on distributions. (English) Zbl 0631.58024

The results of this paper are quite interesting. It is a continuation of the paper of the first author [Math. Ann. 268, 449-471 (1984; Zbl 0574.32045)], where the same problems are considered in the case of \(C^{\infty}\) category. The paper is organized in seven sections as follows: Let X be a smooth \((C^{\infty})\) differentiable manifold and E be a smooth vector bundle over X. In section 1 are studied Mayer Vietoris’s exact sequences for distributions \(f\in D'(U,E)\), where D’(U,E) is the space of distribution sections of E over the open subset U of X. In section 2 systems of the linear partial differential operators with constant coefficients are considered. If X is a complex manifold with a finite dimension and S a generic submanifold of X, in section 3 is introduced a complex of differential operators of the first order on S, which is termed a tangential Cauchy-Riemann complex on S. A dual formal Cauchy-Kowalewski theorem is obtained. In the next section, by means of the mathematical technique designed, vanishing theorems for some local Dolbeault cohomology at edge points are obtained. In section 5 the Poincaré lemma for the tangential Cauchy-Riemann complex on distributions is considered. In the next section the problem of extension of Cauchy-Riemann distributions defined on S is studied. Let us note theorem B: “If S is \(q_ 0\)-pseudoconcave at \(x_ 0\) with \(q_ 0\geq 1\), then every germ of Cauchy-Riemann p-distribution on S at \(x_ 0\) (i.e. every element of \(H^ 0(x_ 0,^ 0\tau_ S^{p,k+*},{\bar \partial}_*))\) is the restriction to S of a germ of holomorphic p-form at \(x_ 0.''\) In the last, seventh section the tangential Cauchy-Riemann complex on compact submanifolds and the Dolbeault complex on an open domain in the complex manifold with piecewise smooth boundaries are discussed. The finiteness theorems for some cohomology groups of these complexes are proved.
{For future applications in other categories of functions or of generalized functions more detailed data can be found in the references cited by the authors.}
Reviewer: M.S.Marinov

MSC:

58J10 Differential complexes
32C05 Real-analytic manifolds, real-analytic spaces
32C35 Analytic sheaves and cohomology groups
32V40 Real submanifolds in complex manifolds

Citations:

Zbl 0574.32045
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References:

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