Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors

TL;DR

This paper advances the understanding of conic singularity metrics with prescribed Ricci curvature on complex manifolds, establishing regularity and existence results for Kähler-Einstein metrics with general cone angles along normal crossing divisors.

Contribution

It extends previous work by providing Laplacian and ${ m C}^{2,eta}$ estimates for Monge-Ampère equations with arbitrary cone angles, ensuring existence and regularity of conic Kähler-Einstein metrics.

Findings

Established Laplacian and ${ m C}^{2,eta}$ estimates for solutions regardless of cone angle size.

Proved existence and regularity of Kähler-Einstein metrics with general cone angles.

Extended previous results to more general cone angle configurations along normal crossing divisors.

Abstract

Let $X$ be a non-singular compact K\"ahler manifold, endowed with an effective divisor $D= \sum (1-\beta_k) Y_k$ having simple normal crossing support, and satisfying $\beta_k \in (0,1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X, D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work \cite{CGP} concerning the Monge-Amp\`ere equations on $(X, D)$ by establishing Laplacian and ${\mathscr C}^{2,\alpha, \beta}$ estimates for the solution of this equations regardless to the size of the coefficients $0<\beta_k< 1$. In particular, we obtain a general theorem concerning the existence and regularity of K\"ahler-Einstein metrics with conic singularities along a normal crossing divisor.