Kähler hyperbolic manifolds and Chern number inequalities

Abstract

We show in this article that Kähler 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 Kähler 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 Kähler non-elliptic manifolds. In addition to these, we term a condition called ``Kähler exactness", which includes Kähler 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 Kähler exact manifold of general type is ample. Some of its consequences and remarks are discussed as well.

16 pages, to appear in Transactions of the AMS