Abstract We prove a general Schwarz lemma for holomorphic maps between Hermitian manifolds. As a consequence, we show that a compact Kähler manifold with a pluriclosed metric of negative holomorphic sectional curvature is canonically polarized. In particular, such manifolds are projective and admit a Kähler–Einstein metric with negative scalar curvature.
Broder et al. (Thu,) studied this question.