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Abstract

Given a locally pseudoconvex bounded domain $\varOmega $, in a complex manifold $M$, the Hartogs type extension theorem is said to hold on $\varOmega $ if there exists an arbitrarily large compact subset $K$ of $\varOmega $ such that every holomorphic function on $\varOmega -K$ is extendible to a holomorphic function on $\varOmega $. It will be reported, based on still unpublished papers of the author, that the Hartogs type extension theorem holds in the following two cases: 1) $M$ is Kähler and $\partial \varOmega $ is $C^2$-smooth and not Levi flat; 2) $M$ is compact Kähler and $\partial \varOmega $ is the support of a divisor whose normal bundle is nonflatly semipositive.