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.

arXiv comment16 pages, to appear in Transactions of the AMS

References in corpus (2)

• -1-Phenomena for the pluri ${\chi }_y$-genus and elliptic genus
• The Hirzebruch ${\chi }_y$-genus and Poincare polynomial revisited

Cited by in corpus (4)

• Chern class inequalities on polarized manifolds and nef vector bundles
• Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold
• Characteristic numbers, Jiang subgroup and non-positive curvature
• Compactification of homology cells, Fujita's conjectures and the complex projective space