Duke Math. J. 171 (2), 417-442, (1 February 2022) DOI: 10.1215/00127094-2021-0041
KEYWORDS: complex hyperbolic manifold, holomorphic sectional curvature, Lang conjecture, singular subvarieties, 32Q05, 32Q20, 32Q45
Let 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 endowed with a Kähler metric with negative holomorphic sectional curvature, and we prove that such a manifold is necessarily of log-general type.