Strominger connection and pluriclosed metrics

Abstract

In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold, then the metric must be pluriclosed. Actually, we show that Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian metric plus the parallelness of the torsion, even without the compactness assumption