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Transactions of the American Mathematical Society

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Kähler hyperbolic manifolds and Chern number inequalities

Author: Ping Li
Journal: Trans. Amer. Math. Soc. 372 (2019), 6853-6868
MSC (2010): Primary 32Q45, 57R20, 58J20.
Published electronically: August 28, 2019
MathSciNet review: 4024540
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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 $L^2$-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 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.

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Additional Information

Ping Li
Affiliation: School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s Republic of China
MR Author ID: 902503

Keywords: Kähler hyperbolic manifold, Kähler nonelliptic manifold, Chern number inequality, Hirzebruch $\chi _y$-genus, Hirzebruch’s proportionality principle, negative sectional curvature, nonpositive sectional curvature, $L^2$-Hodge number, Kobayashi hyperbolicity.
Received by editor(s): May 26, 2018
Published electronically: August 28, 2019
Additional Notes: The author was partially supported by the National Natural Science Foundation of China (Grant No. 11722109).
Part of this article was completed when the author visited the Fields Institute in Toronto in May 2018. The author would like to thank the Institute for the hospitality.
Article copyright: © Copyright 2019 American Mathematical Society