Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

TL;DR

This paper establishes the existence of non-positively curved Kähler-Einstein metrics with cone singularities along normal crossing divisors and explores implications for holomorphic tensor fields, extending classical results in complex geometry.

Contribution

It proves existence results for Kähler-Einstein metrics with cone singularities under certain conditions and extends classical theorems on holomorphic tensor fields to this setting.

Findings

Existence of non-positively curved Kähler-Einstein metrics with cone singularities.

Extension of classical results on holomorphic tensor fields to singular metrics.

Discussion of positively-curved Kähler-Einstein metrics with cone singularities.

Abstract

We prove the existence of non-positively curved K\"ahler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact K\"ahler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved K\"ahler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.