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

By Henri Guenancia, Mihai Puaun. 2013

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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.

Subfields: Applied Mathematics, Geometry and Topology

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A variational approach to the Yau–Tian–Donaldson conjecture (2015, result): “honest Kahler form on any open set on which is smooth. In case is the integration current on an eective Q-divisor with ( X;) klt, !further has cone singularities along the normal crossing part of [GP16]. When <0 (resp. = 0 and klt), the variational approach of [BBGZ13] provides a unique solution ”
Existence and deformations of Kähler–Einstein metrics on smoothable $\mathbb{Q}$-Fano varieties (2014, background): “al Hermitian metric induced from the one on K−1 X . We should remark that in the case when (X,D) is a smooth pair, the weak conical K¨ahlerEinstein metric is actually genuinely conical by the work of [10, 24, 25, 31, 44, 16]. Recall the various functionals that are needed, see [31] for a collection ”
On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, methodology): “By Theorem B in [28], ω(ǫ,t) is a strong conical Kähler-Einstein metric.”
Tian's properness conjectures and Finsler geometry of the space of Kahler metrics (2015, background): “ (P3): regularity of minimizers The regularity condition (P3) for Fβ and Eβ follows after combining together the result of Berman [5, Theorem 1.1] and the regularity theorems [39, Corollary 6.9],[38],[35] (see [48] for more details). Theorem 5.31. Suppose (M,J,D,ω) satisﬁes (48) with µ= 1 and ϕ∈ E 1”