Curvatures of real connections on Hermitian manifolds

TL;DR

This paper explores the geometry of real connections on Hermitian manifolds, focusing on the real Chern connection and its role in deriving Kähler-Einstein metrics, bridging real and Hermitian connection theories.

Contribution

It introduces the study of real connections compatible with both metric and complex structure, and links the real Chern connection to the construction of Kähler-Einstein metrics.

Findings

Analysis of the geometry of real Chern connections

Establishment of conditions for real Chern-Einstein metrics

Derivation of Kähler-Einstein metrics from real connection properties

Abstract

Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex structure $J\in\mathrm{End}(T_\mathbb{R} M)$ and $\mathcal{A}_{g,J}$ be the space of real connections on $T_\mathbb{R} M$ preserving both $g$ and $J$. In this paper, 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 $\nabla^{\mathrm{Ch,\mathrm{R}}}$ on $(M,g,J)$, and obtain K\"ahler-Einstein metrics by using real Chern-Einstein metrics.