## Journal of Differential Geometry

### Conic singularities metrics with prescribed Ricci curvature: 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 concerning the Monge-Ampère equations on $(X,D)$ by establishing Laplacian and $\mathscr{C}^{2,\alpha,\beta}$ estimates for the solution of these equations regardless of the size of the coefficients $0 \lt \beta_k \lt 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kähler–Einstein metrics with conic singularities along a normal crossing divisor.

#### Article information

Source
J. Differential Geom., Volume 103, Number 1 (2016), 15-57.

Dates
First available in Project Euclid: 12 April 2016

https://projecteuclid.org/euclid.jdg/1460463562

Digital Object Identifier
doi:10.4310/jdg/1460463562

Mathematical Reviews number (MathSciNet)
MR3488129

Zentralblatt MATH identifier
1344.53053

#### Citation

Guenancia, Henri; Paun, Mihai. Conic singularities metrics with prescribed Ricci curvature: General cone angles along normal crossing divisors. J. Differential Geom. 103 (2016), no. 1, 15--57. doi:10.4310/jdg/1460463562. https://projecteuclid.org/euclid.jdg/1460463562