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

Abstract

Let $X$ be a non-singular compact Kähler 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-Ampère equations on $(X,D)$ by establishing Laplacian and ${\mathcal{C}}^{2,\alpha ,\beta }$ 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 Kähler-Einstein metrics with conic singularities along a normal crossing divisor.

34 pages; v2: details provided in the last section; v3: this is a major revision. In section 2, we give a new general laplacian estimate, in section 6 we add the mixed cone and cusp case, and more substantially, we added section 7 where we prove Hölder regularity for the second derivatives; v4: published version - corrected a few typos, updated references

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