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Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors. (English) Zbl 1344.53053

Let \((X, D)\) be a log smooth klt pair, i.e., \(X\) is a compact Kähler manifold and \(D\) is an \(\mathbb R\)-divisor with simple normal crossing support. A natural question to ask in this setting is whether one can find a Kähler-Einstein metric on \(X\setminus \operatorname{Supp}(D)\) having conic singularities along \(D\). Such a metric will be reffered to as a conic Kähler-Einstein metric.
This very interesting paper provides a complete (positive) answer to this question (Theorem A) and also derives finer regularity estimates for this type of metric by proving a conic analogue of the Evans-Krylov theorem for complex Monge-Ampère equations (Theorem B).

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q20 Kähler-Einstein manifolds
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