Bott-Chern blow-up formula and bimeromorphic invariance of the $\partial\bar{\partial}$-Lemma for threefolds

TL;DR

This paper establishes a blow-up formula for Bott-Chern cohomology and proves that the $ar{ ext{d}}ar{ ext{d}}$-Lemma is bimeromorphically invariant for threefolds, linking cohomological invariants to complex geometric properties.

Contribution

It introduces a Bott-Chern blow-up formula and demonstrates the bimeromorphic invariance of the $ar{ ext{d}}ar{ ext{d}}$-Lemma for threefolds, advancing understanding of complex manifold invariants.

Findings

Bott-Chern cohomology satisfies a blow-up formula.

Non-Kählerness degrees are bimeromorphic invariants for threefolds.

The $ar{ ext{d}}ar{ ext{d}}$-Lemma is bimeromorphically invariant on threefolds.

Abstract

The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-K\"{a}hlerness degrees, introduced by Angella-Tomassini [Invent. Math. 192, (2013), 71-81], are bimeromorphic invariants. Consequently, the $\partial\bar{\partial}$-Lemma on threefolds admits the bimeromorphic invariance.