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 that 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 for the rigidity conjecture of S.-T. Yau. The main ingredients in our proof are Gromov’s results on the L 2 L^2 -Hodge numbers, the − 1 -1 -phenomenon of the χ y \chi _y -genus and Hirzebruch’s proportionality principle. Similar methods can be applied to obtain parallel results on Kähler nonelliptic manifolds. In addition to these, we term a condition called “Kähler exactness”, which includes Kähler hyperbolic and nonelliptic manifolds and has been used by B.-L. Chen and X. Yang in their work, and we show that the canonical bundle of a Kähler exact manifold of the general type is ample. Some of its consequences and remarks are discussed as well.