On Gauduchon connections with Kähler-like curvature

Abstract

We study Hermitian metrics with a Gauduchon connection being "Kähler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate $6$-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermitian metrics. The results for this case give evidence for two conjectures that are expected to hold in more generality: first, if the Strominger-Bismut connection is Kähler-like, then the metric is pluriclosed; second, if another Gauduchon connection, different from Chern or Strominger-Bismut, is Kähler-like, then the metric is Kähler. As a further motivation, we show that the Kähler-like condition for the Levi-Civita connection assures that the Ricci flow preserves the Hermitian condition along analytic solutions.

to appear in Commun. Anal. Geom

References in corpus (4)

• Special Hermitian metrics on compact solvmanifolds
• Strominger connection and pluriclosed metrics
• Levi-Civita Ricci-flat metrics on compact complex manifolds
• Scalar curvatures in almost Hermitian geometry and some applications

Cited by in corpus (6)

• Strominger connection and pluriclosed metrics
• Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows
• Complex nilmanifolds and Kähler-like connections
• Curvatures of real connections on Hermitian manifolds
• Curvature properties of twistor spaces
• Convergence of Hermitian manifolds and the Type IIB flow