Quasi-projective manifolds with negative holomorphic sectional curvature

Let $(M,\omega)$ be a compact K\"ahler 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. Moreover, we can extend the theorem to the quasi-negative curvature case building on earlier results of Diverio-Trapani. Finally, we investigate the more general setting of a quasi-projective manifold $X^{\circ}$ endowed with a K\"ahler metric with negative holomorphic sectional curvature and we prove that such a manifold $X^{\circ}$ is necessarily of log general type.