Quasi-negative holomorphic sectional curvature and ampleness of the canonical class

This note is an extended version of a 50 min talk given at the INdAM Meeting “Complex and Symplectic Geometry”, held in Cortona from June 12th to June 18th, 2016. What follows was the abstract of our talk. Let X be a compact Kähler manifold with a Kähler metric whose holomorphic sectional curvature is strictly negative. Very recent results by Wu–Yau and Tosatti– Yang confirmed an old conjecture by S.-T. Yau which claimed that under this curvature assumption X should be projective and canonically polarized. We will explain how one can relax the assumption on the holomorphic sectional curvature to the weakest possible, i.e. non positive and strictly negative in at least one point, in order to have the same conclusions. We shall also try to motivate this generalization by arguments coming from birational geometry, such as the abundance conjecture. The results presented here were originally contained in the joint work with Diverio and Trapani (Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, 2016, ArXiv e-prints 1606.01381v3).