${\overline{\partial }}$-cohomology and geometry of the boundary of pseudoconvex domains
Volume 91 / 2007
Annales Polonici Mathematici 91 (2007), 249-262
MSC: Primary 32V40; Secondary 53C40.
DOI: 10.4064/ap91-2-12
Abstract
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.
