Curvatures of real connections on Hermitian manifolds

Jun Wang and Xiaokui Yang

Vol. 337 (2025), No. 2, 365–391

DOI: 10.2140/pjm.2025.337.365

Abstract

Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex structure $J\in \mathrm{End}<!--FUNCTION\; APPLICATION-->(T_{ℝ}M)$ and ${\mathcal{𝒜}}_{g,J}$ be the space of real connections on $T_{ℝ}M$ preserving both $g$ and $J$. We investigate the relationship between the geometry of real connections in ${\mathcal{𝒜}}_{g,J}$ and that of Hermitian connections on $T^{1,0}M$. In particular, we study the geometry of the real Chern connection ${\nabla <!--FUNCTION\; APPLICATION-->}^{\mathrm{Ch}<!--FUNCTION\; APPLICATION-->,ℝ}$ on $(M,g,J)$, and obtain Kähler–Einstein metrics by using real Chern–Einstein metrics.

Keywords

real Chern curvature, Chern–Einstein metrics, scalar curvature

Mathematical Subject Classification

Primary: 53C55

Milestones

Received: 26 December 2023

Revised: 22 March 2025

Accepted: 19 April 2025

Published: 3 June 2025