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Asymptotics of complete Kähler-Einstein metrics: Negativity of the holomorphic sectional curvature

We consider complete Kahler-Einstein metrics on the complements of smooth divisors in projective manifolds. The estimates proven earlier by the author [5] imply that in directions parallel to the divisor at infinity the metric tensor converges to the Kahler-Einstein metric on the divisor. Here we show that the holomorphic sectional curvature is bounded from above by a negative constant near infinity.