K\"ahler hyperbolic manifolds and Chern number inequalities

Abstract

We show in this article that K\"{a}hler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively-curved compact K\"{a}hler manifolds, thus providing evidence to the rigidity conjecture of S.-T. Yau. The main ingredients in our proof are Gromov's results on the $L^2$-Hodge numbers, the $-1$-phenomenon of the $\chi_y$-genus and Hirzebruch's proportionality principle. Similar methods can be applied to obtain parallel results on K\"{a}hler non-elliptic manifolds. In addition to these, we term a condition called ``K\"{a}hler exactness", which includes K\"{a}hler hyperbolic and non-elliptic manifolds and has been used by B.-L. Chen and X. Yang in their work, and show that the canonical bundle of a K\"{a}hler exact manifold of general type is ample. Some of its consequences and remarks are discussed as well.