Rigidity of Pseudoconvex Neighborhoods and Ohsawa--Takegoshi Extension

Abstract

Let $Y$ be a holomorphic section of a holomorphic submersion with one-dimensional fibers over a connected compact complex manifold. Assuming that its normal bundle is unitary flat, we prove that $Y$ admits a fundamental system of locally pseudoconvex neighborhoods if and only if it admits a holomorphic tubular neighborhood. The proof uses the sharp Ohsawa--Takegoshi extension theorem. As an application, we rule out such neighborhood systems in Ueda's example of class $(\gamma )$.

arXiv comment14 pages