K\"ahler hyperbolic manifolds and Chern number inequalities

TL;DR

This paper proves optimal Chern number inequalities for Kähler hyperbolic manifolds, providing evidence for Yau's rigidity conjecture, and explores implications for the canonical bundle in Kähler exact manifolds.

Contribution

It establishes new Chern number inequalities for Kähler hyperbolic manifolds and introduces the concept of Kähler exactness with implications for the canonical bundle.

Findings

Kähler hyperbolic manifolds satisfy optimal Chern number inequalities.

Equality cases are achieved by certain compact ball quotients.

The canonical bundle of a Kähler exact manifold of general type is ample.

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.