Degree of non-K\"ahlerianity for 6-dimensional nilmanifolds

TL;DR

This paper uses Bott-Chern cohomology to quantify how much 6-dimensional nilmanifolds with invariant complex structures deviate from being Kähler, and explores the existence of special Hermitian metrics like pluriclosed metrics.

Contribution

It introduces a method to measure non-Kählerianity using Bott-Chern cohomology on classified nilmanifolds and examines the link to pluriclosed metrics.

Findings

Quantifies non-Kählerianity for classified 6-dimensional nilmanifolds.

Identifies conditions for the existence of pluriclosed metrics.

Provides new insights into the geometry of nilmanifolds with invariant complex structures.

Abstract

We use Bott-Chern cohomology to measure the non-K\"ahlerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Fr\"olicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.