Asymptotics of complete Kähler-Einstein metrics -- negativity of the holomorphic sectional curvature

Abstract

We consider complete Kähler-Einstein metrics on the complements of smooth divisors in projective manifolds. The estimates proven earlier by the author citeframas 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.