The famous Lang conjecture states that a projective manifold is hyperbolic if and only if it is of general type and each of its subvarieties is of general type. Here, \(Y\) is of general type if the canonical bundle \(K_{\widetilde Y}\) of any smooth birational model \(\widetilde{Y}\) is big.
The first main result in this paper (Theorem A) says that if \((M,\omega)\) is a compact Kähler manifold with negative holomorphic sectional curvature (which is automatically projective and Brody hyperbolic) then any irreducible subvariety \(Y\) of \(M\) is of general type, in agreement with the Lang conjecture. The main new idea in the proof consists in constructing on a desingularization \(\widetilde Y\) of \(Y\) a family of singular Kähler-Einstein metrics \((\omega_b)_{b>0}\) generically with cone singularities along a given ample divisor \(B\) and whose cone angle tends to \(2\pi\) as \(b\to 0\) and then showing that the volume of \(\omega_b\) does not go to 0 when \(b\) tends to \(0\).
The author then generalizes Theorem A in two directions. First of all, he proves (Theorem 3.1) that only assuming that \((M,\omega)\) has quasinegative holomorphic sectional curvature then any irreducible subvariety intersecting the negative curvature locus is of general type.
The second generalization concerns the quasiprojective case. Indeed, the author proves (Theorem B) that if \(X\) is a projective manifold, \(D\) a reduced divisor with simple normal crossings and there exists a Kähler metric \(\omega\) on \(X\setminus D\) with holomorphic sectional curvature bounded above by a negative constant then \((X,D)\) is of log-general type, that is \(K_X+D\) is big. Moreover, if \(\omega\) is bounded near \(D\) then \(K_X\) itself is big.