Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: K\"ahler hyperbolic manifolds and Chern number inequalities   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.

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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: chem-ph