A remark on the Bismut-Ricci form on 2-step nilmanifolds

TL;DR

This paper studies the Bismut-Ricci form on 2-step nilmanifolds with SKT structures, showing its seminegative definiteness and implications for invariant SKT static metrics and pluriclosed flow solutions.

Contribution

It provides a new observation about the Bismut-Ricci form's properties and simplifies proofs regarding SKT metrics and pluriclosed flow on 2-step nilmanifolds.

Findings

Bismut-Ricci form is seminegative definite on these manifolds.

Invariant SKT static metrics do not exist on 2-step nilmanifolds.

Long-time solutions to pluriclosed flow exist in this setting.

Abstract

In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds.