Let (M, !) be a compact Kahler 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 degrees endowed with a Kahler metric with negative holomorphic sectional curvature, and we prove that such a manifold X degrees is necessarily of log-general type.
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