Quantitative and qualitative cohomological properties for non-K\"ahler manifolds

TL;DR

This paper introduces a new qualitative property for Bott-Chern cohomology on non-K"ahler manifolds, characterizes the $ ext{dd}^c$-lemma, and provides bounds on cohomology dimensions with applications to symplectic cohomologies.

Contribution

It defines a new qualitative property for Bott-Chern cohomology that characterizes the $ ext{dd}^c$-lemma and establishes new bounds on cohomology dimensions, extending to symplectic contexts.

Findings

The qualitative property characterizes the $ ext{dd}^c$-lemma.

A new bound on Bott-Chern cohomology dimension is proven.

Generalizations apply to symplectic cohomologies.

Abstract

We introduce a "qualitative property" for Bott-Chern cohomology of complex non-K\"ahler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the $\partial\overline\partial$-Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.