# The heredity and bimeromorphic invariance of the $\partial\overline{\partial}$-lemma property

## TL;DR

This paper provides a new proof regarding the stability of the $ ext{dd}^c$-lemma under blow-ups, and discusses its inheritance and invariance under bimeromorphic transformations, using explicit cohomology formulas.

## Contribution

It introduces a simplified proof of the $ ext{dd}^c$-lemma's behavior under blow-ups and explores its heredity and invariance properties in complex geometry.

## Key findings

- The $ ext{dd}^c$-lemma property is preserved under blow-up transformations.
- Explicit formulas for Dolbeault cohomology facilitate understanding of the property.
- The $ ext{dd}^c$-lemma is hereditary and bimeromorphically invariant in certain contexts.

## Abstract

We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of spaces of forms and currents. At last, we discuss the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property.