On bisectional nonpositively curved compact Kähler–Einstein surfaces

Daniel Guan

Vol. 288 (2017), No. 2, 343–353

DOI: 10.2140/pjm.2017.288.343

Abstract

We prove a conjecture on the pinching of the bisectional curvature of nonpositively curved Kähler–Einstein surfaces. We also prove that any compact Kähler–Einstein surface $M$ is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if $M$ has nonpositive Einstein constant and, at each point, the average holomorphic sectional curvature is closer to the minimum than to the maximum.

Keywords

Kähler–Einstein metrics, compact complex surfaces, bisectional curvature, pinching of the curvatures

Mathematical Subject Classification 2010

Primary: 32M15, 32Q20, 53C21, 53C55

Milestones

Received: 8 December 2014

Revised: 1 December 2016

Accepted: 14 December 2016

Published: 28 April 2017