Quasiprojective manifolds with negative holomorphic sectional curvature

Abstract

Let $(M,\mathit{\omega })$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of M is of general type, thus confirming in this particular case a celebrated conjecture of Lang. Moreover, we can extend the theorem to the quasinegative curvature case building on earlier results of Diverio–Trapani. Finally, we investigate the more general setting of a quasiprojective manifold $X^{\circ }$ endowed with a Kähler metric with negative holomorphic sectional curvature, and we prove that such a manifold $X^{\circ }$ is necessarily of log-general type.