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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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