Bott-Chern cohomology of compact Vaisman manifolds

TL;DR

This paper explicitly describes the Bott-Chern cohomology of compact Vaisman manifolds, revealing relationships with basic cohomology, and explores their cohomological invariants and formality properties.

Contribution

It provides an explicit cohomology description for Vaisman manifolds and links Bott-Chern and Dolbeault numbers, also analyzing their invariants and formality.

Findings

Bott-Chern and Dolbeault numbers determine each other in Vaisman manifolds.

The invariants $ riangle^k$ are unbounded for Vaisman manifolds.

Cohomological criteria for formality are established.

Abstract

We give an explicit description of the Bott-Chern cohomology groups of a compact Vaisman manifold in terms of the basic cohomology. We infer that the Bott-Chern numbers and the Dolbeault numbers of a Vaisman manifold determine each other. On the other hand, we show that the cohomological invariants $\Delta^k$ introduced by Angella-Tomassini are unbounded for Vaisman manifolds. Finally, we give a cohomological characterization of the Dolbeault and Bott-Chern formality for Vaisman metrics.