Curvature of hyperbolic complex manifolds

Abstract

The article addresses the construction and geography of negatively curved metrics on hyperbolic complex manifolds. We introduce a mechanism for constructing complete K\"ahler metrics with negative bisectional curvature. This applies to some product complex manifolds, thereby resolving a longstanding problem attributed to N. Mok. We then construct projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes $c_1^2/c_2$ realize any $s \in \mathbf{Q} \cap \left( \frac{2}{7}, \frac{2}{3} \right)$. For slopes $s\in \mathbf{Q}\cap \left( \frac{2}{7},\frac{1}{3} \right)$, the corresponding surfaces admit a Hermitian metric with $\text{HSC}<0$, but their K\"ahler--Einstein metric cannot have $\text{HSC}<0$. We finally construct, for every $s \in \left( \frac{1}{2}, 3 \right)$, a sequence of projective Kobayashi hyperbolic surfaces that do not admit a Hermitian metric of nonpositive holomorphic sectional curvature, whose Chern slopes $c_1^2/c_2$ converge to $s$.