RC-positivity, Schwarz's lemma and comparison theorems

TL;DR

This paper extends Schwarz lemmas to holomorphic bundle maps between Hermitian vector bundles with positive curvature bounds, leading to new comparison theorems for complex manifolds with RC-positivity.

Contribution

It introduces Schwarz lemmas for vector bundles with various positive curvature bounds and applies them to derive new diameter and volume comparison theorems.

Findings

Schwarz lemmas established for holomorphic bundle maps with positive curvature bounds

New diameter comparison theorems for complex manifolds with RC-positivity

Volume comparison results derived from the Schwarz lemmas

Abstract

It is well-known that the classical Schwarz lemma yields an explicit comparison of two Hermitian metrics with uniform constant negative curvature bounds through holomorphic maps between complex manifolds. In this paper, we establish Schwarz lemmas for holomorphic bundle maps between abstract Hermitian holomorphic vector bundles with various positive curvature bounds. As applications, we prove Schwarz lemmas for holomorphic maps between complex manifolds whose curvature tensors are described by the notion ``RC-positivity''. In particular, new diameter and volume comparison theorems are obtained by using Schwarz lemmas.