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

Abstract

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-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.

arXiv comment36 pages, v3: added a section on the log-Fano case. To appear in Annales Scientifiques de l'ENS