  The Wu--Yau theorem asserts that a compact K\"ahler 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 K\"ahler--Einstein metrics on K\"ahler surfaces of
general type that do not have negative holomorphic sectional curvature. This
begins to address a question raised by Diverio.
