The extension of holomorphic functions on a non-pluriharmonic locus

Abstract

Let $n\ge 4$ and let $\Omega$ be a bounded hyperconvex domain in ${\mathbb{C}}^n$. Let $\phi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. We show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial }\phi {)}^{n-3}$ can be extended to the holomorphic function on $\Omega$.

12pages, mistakes in Lemma 1 are removed, the assumptions of continuity of the plurisubharmonic functions are added in Corollary 1 and 3, minor corrections are added, a reference is added