×

zbMATH — the first resource for mathematics

Microlocal pseudoconvexity and “edge of the wedge” theorem. (English) Zbl 0722.32012
At first, the authors prove that, for a bounded pseudoconvex domain \(\Omega\) in \({\mathbb{C}}^ n\) with real analytic boundary \(\partial \Omega\) and an open subset \(\omega\) of \(\partial \Omega\), \(\Omega\cup \omega\) has a fundamental system of pseudoconvex neighbourhoods. They consider a complex manifold X, its real analytic generic submanifold of codimension d, an open subset U of the conormal bundle \(T_ N*X\), the Levi form of N having r negative eigenvalues, and the Sato’s microlocalization \(\mu_ N({\mathcal O}_ X)\). They prove the flabbyness of the sheaf \(H^{d+r}(\mu_ N({\mathcal O}_ X))\) under the assumption that U contains no germ of complex curve.

MSC:
32T99 Pseudoconvex domains
32C35 Analytic sheaves and cohomology groups
58J15 Relations of PDEs on manifolds with hyperfunctions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. S. Baouendi, C. H. Chang, and F. Treves, Microlocal hypo-analyticity and extension of CR functions , J. Differential Geom. 18 (1983), no. 3, 331-391. · Zbl 0575.32019
[2] E. Bedford and J. E. Fornaess, Complex manifolds in pseudoconvex boundaries , Duke Math. J. 48 (1981), no. 1, 279-288. · Zbl 0472.32007
[3] 1 K. Diederich and J. E. Fornaess, Pseudoconvex domains: existence of Stein neighborhoods , Duke Math. J. 44 (1977), no. 3, 641-662. · Zbl 0381.32014
[4] 2 K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary , Ann. Math. (2) 107 (1978), no. 2, 371-384. · Zbl 0378.32014
[5] 3 K. Diederich and J. E. Fornaess, Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary , Ann. of Math. (2) 110 (1979), no. 3, 575-592. JSTOR: · Zbl 0394.32012
[6] 1 M. Kashiwara and T. Kawai, On the boundary value problem for elliptic system of linear differential equations. I , Proc. Japan Acad. 48 (1972), 712-715. · Zbl 0271.35028
[7] 2 M. Kashiwara and T. Kawai, On the boundary value problem for elliptic system of linear differential equations. II , Proc. Japan Acad. 49 (1973), 164-168. · Zbl 0279.35037
[8] 1 M. Kashiwara and P. Schapira, Microlocal study of sheaves , Astérisque (1985), no. 128, 235, S.M.F., or: Sheaves on manifolds, Grundlehren der Math. Wiss. 292, Springer, (1990). · Zbl 0589.32019
[9] 2 M. Kashiwara and P. Schapira, A vanishing theorem for a class of systems with simple characteristics , Invent. Math. 82 (1985), no. 3, 579-592. · Zbl 0626.58028
[10] A. Martineau, Théorèmes sur le prolongement analytique du type “edge of the Wedge” , Sem. Bourbaki, 340, 1967-68. · Zbl 0209.14802
[11] M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations , Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Springer, Berlin, 1973, 265-529. Lecture Notes in Math., Vol. 287. · Zbl 0277.46039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.