# Holomorphic Approximation via Dolbeault Cohomology

## TL;DR

This paper investigates holomorphic approximation in complex manifolds, extending classical theorems like Runge and Mergelyan, and characterizes these properties using Dolbeault cohomology and geometric conditions.

## Contribution

It extends classical approximation theorems to complex manifolds and links these properties to Dolbeault cohomology groups and geometric criteria.

## Key findings

- Characterization of Runge and Mergelyan properties via Dolbeault cohomology
- Extension of classical approximation theorems to complex manifolds
- Provision of geometric sufficient conditions for approximation properties

## Abstract

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.