# The $\partial\bar{\partial}$-lemma under surjective maps

## TL;DR

This paper investigates the behavior of the $ ext{d} ext{d}^c$-lemma in complex manifolds under surjective holomorphic maps and establishes a criterion for product manifolds to satisfy this lemma based on their components.

## Contribution

It proves that the $ ext{d} ext{d}^c$-lemma is preserved under surjective maps and characterizes product manifolds satisfying the lemma via their factors.

## Key findings

- The $ ext{d} ext{d}^c$-lemma is preserved under surjective holomorphic maps.
- A product manifold satisfies the $ ext{d} ext{d}^c$-lemma iff all its factors do.
- The result leverages Deligne-Griffiths-Morgan-Sullivan's theorem.

## Abstract

We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if so do all its components.