A compact complex manifold X is called hyperbolic if every holomorphic map from the complex plane into X is constant. A long-standing folklore conjecture attributed to Kobayashi predicts that all such manifolds admit a Kähler–Einstein metric with negative scalar curvature and in particular, embed into complex projective space. A strictly weaker diffeo-geometric version of the conjecture states that Hermitian manifolds with negative holomorphic sectional curvature should satisfy the same conclusion. In this talk, I will present some recent progress on this conjecture for pluriclosed manifolds, and the key technical ingredient in the proof:

an improved Schwarz lemma for Hermitian metrics. This is based on joint work with Kyle Broder.