Conic K\"{a}hler-Einstein metrics along simple normal crossing divisors on Fano manifolds

TL;DR

This paper proves the existence and uniqueness of conical Kähler-Einstein metrics with prescribed cone angles on certain Fano manifolds, and establishes new curvature and higher order estimates for these metrics.

Contribution

It introduces new existence and uniqueness results for conic Kähler-Einstein metrics on Fano manifolds with simple normal crossing divisors, extending previous estimates.

Findings

Existence and uniqueness of conical Kähler-Einstein metrics with prescribed cone angles.

A generalized curvature estimate for conic metrics along simple normal crossing divisors.

Derivation of high order estimates from the curvature estimate.

Abstract

We prove that on one K\"{a}hler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical K\"{a}hler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's estimate and derive high order estimates from this estimate.