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Kähler-Einstein metrics with edge singularities. (English) Zbl 1337.32037
Let $$(M,\omega)$$ be a compact Kähler manifold and let $$D\subset M$$ be a smooth divisor. This paper is devoted to the study of the existence and regularity of Kähler-Einstein metrics on $$M$$ with edge singularities with angle $$2\pi \beta$$ along $$D$$. Precisely, let us suppose that $$\mu [\omega]+(1-\beta)[D]=c_1(M)$$ with $$0<\beta\leq 1$$, $$\mu \in \mathbb R$$ and, if $$\mu>0$$, that the twisted $$K$$-energy is proper. Then the main result of this article is the proof of Kähler-Einstein edge metrics with cone angle $$2\pi \beta$$ along $$D$$, and with Ricci curvature $$\mu$$. In particular, this metric is unique if $$\mu<0$$, is unique in its Kähler class if $$\mu=0$$ and, if $$\mu>0$$, then it is unique up to automorphisms preserving $$D$$. Moreover, it is proved that this metric is polyhomogeneous, and that it has a complete asymptotically expansion with smooth coefficients. As a consequence, the authors obtain a new proof of the result of Aubin-Yau-Tian on the existence of Kähler-Einstein metrics on smooth compact Kähler manifolds.

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