Some Remarks on the Wu--Yau Theorem

Abstract

The Wu--Yau theorem asserts that a compact Kähler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and present several examples verifying the conjecture. We also produce an abundance of examples of Kähler--Einstein metrics on Kähler surfaces of general type that do not have negative holomorphic sectional curvature. This begins to address a question raised by Diverio.

arXiv comment10 pages; Updated to include some additional details and references