On the deformed Bott-Chern cohomology

TL;DR

This paper introduces a deformation of Bott-Chern cohomology on complex manifolds using Beltrami differentials, establishing a deformation theory, and computes examples for specific manifolds, linking the $ar{ar{ ext{d}}}$-lemma to formality.

Contribution

It develops a deformation theory for Bott-Chern cohomology under Beltrami differentials and computes the deformed cohomology for specific complex manifolds.

Findings

Deformation theory for Bott-Chern cohomology established

Explicit computations for Iwasawa and Nakamura manifolds

$ar{ar{ ext{d}}}$-lemma implies formality

Abstract

Given a compact complex manifold $X$ and a integrable Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$, we introduce a double complex structure on $A^{\bullet,\bullet}(X)$ naturally determined by $\phi$ and study its Bott-Chern cohomology. In particular, we establish a deformation theory for Bott-Chern cohomology and use it to compute the deformed Bott-Chern cohomology for the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold. The $\partial\bar{\partial}_{\phi}$-lemma is studied and we show a compact complex manifold satisfying $\partial\bar{\partial}_{\phi}$-lemma is formal.