Function theory on q-pseudoconvex domains

点击翻译按钮获取中文摘要

The results are the following:(1)The intersection of finite number of q-complete domains is cohomologically (q^^〜1)-complete if q does not devide n. This is best and shaper than that obtained by the approximation theorem of Diederich-Fornaess and the vanishing theorem of Andreotti-Grauert.(2)The complements of Levi flat real hypersurfaces in the complex tori of dimension 2 are Stein if they contain a local complex hypersurface which is not linear. This is the essential step to approach to the conjecture that there exists no Levi flat real hypersurfaces in complex tori except restricted types.(3)For a given complex submanifold M in C^n, we expressed the Hermitian matrix determined by the Levi form of the function -log d by using the defing function of M, where d denotes the distance to M. As its application, the degeneracy condition of the Levi form is equivalent to that of Gauss map.(4)For a given real submanifold M in C^2,we find a necessary and sufficient condition that the Levi form of distance to M degenerates. By applying this result, the Levi problem for complex tori can affimatively solved except the examples pointed out by Grauert.