Asymptotics of Complete Kähler-Einstein Metrics -- Negativity of the Holomorphic Sectional Curvature

We consider complete Kähler-Einstein metrics on the complements of smooth divisors in projective manifolds. The estimates proven earlier by the author \cite{framas} imply that in directions parallel to the divisor at infinity the metric tensor converges to the Kähler-Einstein metric on the divisor. Here we show that the holomorphic sectional curvature is bounded from above by a negative constant near infinity.

2000 Mathematics Subject Classification: 32Q05, 32Q20, 53C55

Keywords and Phrases: Complete Kähler-Einstein metrics, negative sectional curvature

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