On the Bott-Chern cohomology and balanced Hermitian nilmanifolds

TL;DR

This paper investigates the Bott-Chern cohomology of 6-dimensional nilmanifolds with invariant complex structures, focusing on balanced and strongly Gauduchon Hermitian metrics, and explores their deformation properties and related invariants.

Contribution

It analyzes the behavior of complex invariants related to Bott-Chern cohomology on nilmanifolds and characterizes deformation spaces in the balanced case within type IIB supergravity.

Findings

Vanishing of certain invariants is not stable under holomorphic deformations.

Deformation spaces in the balanced case are described using Bott-Chern cohomology.

The study links complex invariants to geometric and physical deformation parameters.

Abstract

The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\partial \bar\partial$-lemma condition and defined in terms of the Bott-Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in type IIB supergravity described by Tseng and Yau in terms of the Bott-Chern cohomology group of bidegree (2,2).