On the curvature of the Bismut connection: Bismut Yamabe problem and Calabi-Yau with torsion metrics

TL;DR

This paper investigates the scalar and Ricci curvatures of the Bismut connection on Hermitian manifolds, establishing existence results for constant scalar curvature metrics and constructing explicit Calabi-Yau with torsion examples.

Contribution

It introduces an analog of the Yamabe problem for Bismut scalar curvature and constructs explicit Calabi-Yau with torsion Hermitian structures, providing new insights into their geometry.

Findings

Existence of metrics with constant Bismut scalar curvature on compact Hermitian manifolds.

Construction of explicit Calabi--Yau with torsion Hermitian structures.

Uniqueness results for these Hermitian structures.

Abstract

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis we construct explicit examples of Calabi--Yau with torsion Hermitian structures and prove a uniqueness result for them.