A general Schwarz lemma for Hermitian manifolds

TL;DR

This paper advances the Schwarz lemma for Hermitian manifolds by introducing weaker curvature conditions, demonstrating invariance under connection changes, and extending results to Gauduchon connections, thereby improving geometric understanding.

Contribution

It introduces new intrinsic curvature constraints for Hermitian manifolds, broadening the applicability of the Schwarz lemma and extending it to Gauduchon connections.

Findings

Weaker curvature conditions than Ricci and bisectional curvatures.

Schwarz lemma invariance under Hermitian connection changes.

Extension of Schwarz lemma to Gauduchon connections.

Abstract

The Schwarz lemma for holomorphic maps between Hermitian manifolds is improved. New curvature constraints on the source and target manifolds are introduced and shown to be weaker than the Ricci and real bisectional curvature, respectively. The novel target curvature condition is intrinsic to the Hermitian structure and is controlled by the holomorphic sectional curvature if the metric is pluriclosed. This leads to significant improvements on the Wu--Yau theorem. Further, it is shown that the Schwarz lemma is largely invariant under a change of Hermitian connection, and the precise geometric quantity that varies with the connection is determined. This enables us to establish the Schwarz lemma for the Gauduchon connections.