Abstract
LetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If\(q< \tfrac{1}{2}n\), then we prove thatH pn−q (X) is Hausdorff for allp. This is not true in general if\(q \geqslant \tfrac{1}{2}n\) (Rossi’s example withn=2 andq=1). If all ends areq-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for\(q \geqslant \tfrac{1}{2}n\)). Moreover the result was already known in the case when theq-concave ends can be ‘filled in’ (again also for\(q \geqslant \tfrac{1}{2}n\)). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries,\(q< \tfrac{1}{2}n\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andreotti, A. andGrauert, H., Théorèmes de finitude pour la cohomologie des espaces complexes,Bull. Soc. Math. France 90 (1962), 193–259.
Andreotti, A. andVesentini, E., Carleman estimates for the Laplace-Beltrami equation on complex manifolds,Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81–130.
Chirka, E. M. andStout, E. L., Removable singularities in the boundary, inContributions to Complex Analysis and Analytic Geometry (Skoda, H. and Trépreau, J.-M., eds.), Aspects of Math.E26, pp. 43–104, Viehweg, Braunschweig, 1994.
Dieudonné, J. andSchwartz, L., La dualité dans les espaces\((\mathcal{F})\) et\((\mathcal{L}F)\),Ann. Inst. Fourier (Grenoble) 1 (1949), 61–101.
Fischer, W. andLieb, I., Lokale Kerne und beschränkte Lösungen für den\(\bar \partial \)-Operator aufq-konvexen Gebieten,Math. Ann. 208 (1974), 249–265.
Henkin, G. M. andLeiterer, J.,Andreotti-Grauert Theory by Integral Formulas, Progress in Math.74, Birkhäuser, Basel, 1988.
Kohn, J. J. andRossi, H., On the extension of holomorphic functions from the boundary of a complex manifold,Ann. of Math. 81 (1965), 451–472.
Laufer, H. B., On Serre duality and envelopes of holomorphy,Trans. Amer. Math. Soc. 128 (1967), 414–436.
Laurent-ThiéBaut, C. andLeiterer, J., The Andreotti-Vesentini separation theorem withC k estimates and extension ofCR-forms, inSeveral Complex Variables, Proceedings of the Mittag-Leffler Institute, 1987–1988 (Fornæss, J. E., ed.), Mathematical Notes38, pp. 416–439, Princeton Univ. Press, Princeton, N. J., 1993.
Laurent-Thiébaut, C. andLeiterer, J., The Andreotti-Vesentini separation theorem and global homotopy representation,Math. Z. 227 (1998), 711–727.
Laurent-Thiébaut, C. andLeiterer, J., On Serre duality,Bull. Sci. Math. 124 (2000), 93–106.
Ramis, J.-P., Théorèmes de séparation et de finitude pour l’homologie et la cohomologie des espaces (p, q)-convexes-concaves,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27 (1973), 933–997.
Rossi, H., Attaching analytic spaces to an analytic space along a pseudoconcave boundary, inProceedings of the Conference on Complex Analysis (Minneapolis, 1964) (Aeppli, A., Calabi, E. and Röhrl, H., eds.), pp. 242–256, Springer-Verlag, Berlin, 1965.
Serre, J. P., Un théorème de dualité,Comment. Math. Helv. 29 (1955), 9–26.
Trèves, F.,Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
Author information
Authors and Affiliations
Additional information
This research was partially supported by TMR Research Network ERBFMRXCT 98063.
Rights and permissions
About this article
Cite this article
Laurent-Thiébaut, C., Leiterer, J. A separation theorem and Serre duality for the Dolbeault cohomology. Ark. Mat. 40, 301–321 (2002). https://doi.org/10.1007/BF02384538
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384538