Pluriclosed and Strominger K\"ahler-like metrics compatible with abelian complex structures

TL;DR

This paper investigates conditions under which certain Hermitian metrics on unimodular Lie groups with abelian complex structures exist and remain stable under the pluriclosed flow, revealing structural constraints and flow behavior.

Contribution

It establishes that unimodular Lie groups with abelian complex structures admitting pluriclosed metrics must be 2-step nilpotent and shows the pluriclosed flow preserves Strominger Kähler-like conditions on such groups.

Findings

Unimodular Lie groups with abelian complex structures and pluriclosed metrics are 2-step nilpotent.

The pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.

Structural constraints on Lie groups admitting certain Hermitian metrics are identified.

Abstract

We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger K\"ahler-like condition.