Kähler hyperbolic manifolds and Chern number inequalities

Abstract

We show in this article that Kähler hyperbolic manifolds satisfy a family of sharp Chern number inequalities and the equality cases can be attained by the compact quotients of the unit balls in the complex Euclidean spaces. 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 χy-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 general type Kähler exact manifold is ample. Some of its consequences and remarks are discussed as well.2010 Mathematics Subject Classification. 32Q45, 57R20, 58J20.

Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold

“…Following the idea of Li in [24], we then have Theorem 3.17. Let (X, ω) be a closed Kähler manifold with sectional curvature bounded from above by a negative constant, i.e., sec ≤ −K, for some K > 0.…”

“…All the equality cases in (1.1) hold if and only if χ p (X) = (−1) n−p , 0 ≤ p ≤ n. (see [24,Theorem 2.1]).…”

Characteristic numbers, Jiang subgroup and non-positive curvature

“…We only briefly recall Hirzebruch's χ y -genus for compact complex manifolds, which is enough for our purpose in this article. We refer the reader to [Li17] and [Li19,§3] for its definition on general almost-complex manifolds and the summary on its various properties.…”

“…For the reader's convenience more related details are included in Section 6.1. The reader may also consult [Li19,§4].…”