Schwarz lemma for conical K\"ahler metrics with general cone angles

TL;DR

This paper extends the Schwarz lemma to conical Kähler metrics with arbitrary cone angles, including cases with singularities, broadening the scope of previous results in complex differential geometry.

Contribution

It generalizes Jeffres' Schwarz lemma for conical Kähler metrics to encompass all cone angles, even with metric blow-ups along divisors.

Findings

Extended Schwarz lemma to general cone angles

Included cases with metric blow-up along divisors

Broadened applicability of Schwarz lemma in complex geometry

Abstract

The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete Riemannian manifolds. Jeffres obtained Schwarz lemma for volume forms of conical K\"ahler metrics, based on a barrier function and the maximum principle argument. In this note, we generalize Jeffres' result to general cone angles including the case when the pullback of the metric would blows up along the divisors.