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Kähler-Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. (English) Zbl 1151.32009

Let \(M\) denote a quasi projective manifold which can be compactified by adding a divisor \(D\) with simple normal crossings.
In this paper the author considers a general positivity assumption under which \(M\) admits a Kähler Einstein metric of negative Ricci curvature. After that the positivity assumption is relaxed and the existence of such a metric with negative Ricci curvature is proved. Moreover the new conditions allow the construction of complete Kähler metric having bounded sectional curvature. In any case the proof of existence amounts to solve a degenerate Monge Ampère equation. It is also proved that this metric can be unique in some cases.

MSC:

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