On the universal covering of projective manifolds of general type (Q1916071)

The author gives a partial affirmative answer for Kobayashi's conjecture: Let \(M\) be a compact Kähler manifold. Suppose that \(M\) is measure hyperbolic. Then \(M\) is of general type [\textit{S. Kobayashi}, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker (New York, 1970; Zbl 0207.37902)]. Recently M. Gromov introduced the notion of Kähler hyperbolicity and proved that Kähler hyperbolic manifolds are projective of general type [\textit{M. Gromov}, Kähler hyperbolicity and \(L^2\)-Hodge theory, J. Differ. Geom. 33, No. 1, 263-292 (1991; Zbl 0719.53042)]. Although Gromov's theorem is a partial affirmative answer of Kobayashi's conjecture it seems to be hard to check that a given complex manifold is Kähler hyperbolic. Theorem 1. Let \(X\) be a projective manifold of general type and let \(\pi : D \to X\) be the universal covering of \(X\). Then any compact unramified quotient of \(D\) is of general type.