Cohomological aspects on complex and symplectic manifolds

TL;DR

This paper explores how cohomological invariants like Bott-Chern and Aeppli groups reveal qualitative geometric properties of complex and symplectic manifolds, especially in non-Kähler contexts.

Contribution

It provides a comparative overview of cohomology group dimensions and their role in characterizing key geometric conditions such as the $ ext{dd}^c$-lemma and Hard-Lefschetz property.

Findings

Comparison of cohomology dimensions in different geometries

Characterization of the $ ext{dd}^c$-lemma via cohomology

Identification of the Hard-Lefschetz condition through cohomology

Abstract

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non K\"ahler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the $\partial\overline\partial$-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to [6] and [29].