Global Variants of Hartogs' Theorem

Abstract

Hartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.