Inequalities \`a la Fr\"olicher and cohomological decompositions

TL;DR

This paper investigates cohomological inequalities for vector spaces with two anti-commuting endomorphisms, characterizing when the bla bla-Lemma holds and applying results to various types of manifolds.

Contribution

It establishes a Frf6licher-type inequality relating Bott-Chern, Aeppli, and Dolbeault cohomologies and characterizes the bla bla-Lemma through equality conditions.

Findings

Proves an inequality relating cohomology dimensions.

Characterizes the bla bla-Lemma via equality in the inequality.

Applies results to complex, symplectic, and generalized complex manifolds.

Abstract

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality \`a la Fr\"olicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equality in such an inequality \`a la Fr\"olicher characterizes the validity of the so-called cohomological property of satisfying the $\partial\overline{\partial}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, generalized-complex manifolds.