Balanced Hermitian structures on almost abelian Lie algebras

TL;DR

This paper classifies six-dimensional almost abelian Lie algebras with balanced Hermitian structures, proves a conjecture relating balanced and SKT metrics on certain manifolds, and analyzes the behavior of specific geometric flows on these structures.

Contribution

It provides a classification of six-dimensional almost abelian Lie algebras with balanced structures and proves a conjecture linking balanced and SKT metrics on compact almost abelian solvmanifolds.

Findings

Classified six-dimensional almost abelian Lie algebras with balanced structures.

Proved the conjecture that balanced and SKT metrics imply Kähler on certain manifolds.

Showed that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

Abstract

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a K\"ahler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow by D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally K\"ahler metrics are fixed points.