Cross theorems with singularities

TL;DR

This paper proves extension theorems for separately holomorphic functions with singularities on complex manifolds, broadening the understanding of holomorphic extension in the presence of pluripolar and thin singular sets.

Contribution

It introduces new extension theorems for separately holomorphic mappings with singularities on complex manifolds, considering locally pluripolar and thin sets.

Findings

Extension theorems established for functions with singularities

Applicable to complex manifolds with the Hartogs extension property

Handles singularities that are locally pluripolar or thin

Abstract

We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.