The pluriclosed flow on nilmanifolds and Tamed symplectic forms

TL;DR

This paper investigates the pluriclosed flow on complex nilmanifolds, establishing long-term existence results for SKT structures and introducing a new flow for tamed symplectic forms based on Bismut Ricci curvature.

Contribution

It adapts Lauret's techniques to SKT structures on nilmanifolds, proving long-time existence and defining a novel flow for tamed symplectic forms.

Findings

Pluriclosed flow on SKT nilmanifolds is equivalent to a bracket flow.

Long-time existence of pluriclosed flow on nilpotent Lie groups.

Introduction of a flow for tamed symplectic forms via Bismut Ricci form.

Abstract

We study evolution of (strong K\"ahler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex manifold, by considering evolution of symplectic forms via the flow induced by the Bismut Ricci form.