${\overline{\partial }}$-cohomology and geometry of the boundary of pseudoconvex domains

Takeo Ohsawa¹

¹ Graduate School of Mathematics Nagoya University Chikusaku Furocho 464-8602 Nagoya, Japan

Annales Polonici Mathematici, Tome 91 (2007) no. 2-3, pp. 249-262

Résumé

In 1958, H. Grauert proved: If $D$ is a strongly pseudoconvex domain in a complex manifold, then $D$ is holomorphically convex. In contrast, various cases occur if the Levi form of the boundary of $D$ is everywhere zero, i.e. if $\partial D$ is Levi flat. A review is given of the results on the domains with Levi flat boundaries in recent decades. Related results on the domains with divisorial boundaries and generically strongly pseudoconvex domains are also presented. As for the methods, it is explained how Hartogs type extension theorems and $L^2$ finiteness theorem for the $\overline I$-cohomology are applied.

Détail

DOI : 10.4064/ap91-2-12

Keywords: grauert proved strongly pseudoconvex domain complex manifold holomorphically convex contrast various cases occur levi form boundary everywhere zero partial levi flat review given results domains levi flat boundaries recent decades related results domains divisorial boundaries generically strongly pseudoconvex domains presented methods explained hartogs type extension theorems finiteness theorem overline i cohomology applied