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 Kähler 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 Kähler--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 $s$.