Non-K\"ahler Calabi-Yau geometry and pluriclosed flow

TL;DR

This paper explores non-K"ahler Calabi-Yau geometry through pluriclosed flow, introduces Bismut Hermitian-Einstein metrics, and establishes new existence, convergence, and classification results for complex manifolds.

Contribution

It reinterprets pluriclosed metrics via Courant algebroids, introduces Bismut Hermitian-Einstein metrics, and proves global existence and convergence results for pluriclosed flow on non-K"ahler manifolds.

Findings

Existence of infinitely many non-K"ahler Calabi-Yau manifolds without Bismut Hermitian-Einstein metrics.

Global existence of pluriclosed flow on non-K"ahler surfaces of nonnegative Kodaira dimension.

Convergence of pluriclosed flow to Bismut-flat metrics on certain complex manifolds.

Abstract

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-K\"ahler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-K\"ahler surfaces of nonnegative Kodaira dimension. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized K\"ahler structures on these spaces.