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∂̅-cohomology and geometry of the boundary of pseudoconvex domains

Takeo Ohsawa (2007)

Annales Polonici Mathematici

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 ∂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...

Łojasiewicz-Siciak condition for the pluricomplex Green function

Marta Kosek (2011)

Banach Center Publications

A compact set K N satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that V K ( z ) B ( d i s t ( z , K ) ) β if dist(z,K) ≤ 1. (LS) Here V K denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

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