Quasiprojective manifolds with negative holomorphic sectional curvature

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