Paper detail

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

Let $X$ be a non-singular compact Kähler manifold, endowed with an effective divisor $D= \sum (1-β_k) Y_k$ having simple normal crossing support, and satisfying $β_k \in (0,1)$. 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-Ampère equations on $(X, D)$ by establishing Laplacian and ${\mathscr C}^{2,α, β}$ estimates for the solution of this equations regardless to the size of the coefficients $0<β_k< 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kähler-Einstein metrics with conic singularities along a normal crossing divisor.