Some Remarks on the Wu--Yau Theorem

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.