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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Almost quarter-pinched Kähler metrics and Chern numbers


Authors: Martin Deraux and Harish Seshadri
Journal: Proc. Amer. Math. Soc. 139 (2011), 2571-2576
MSC (2010): Primary 53C21; Secondary 53C20
Published electronically: December 9, 2010
MathSciNet review: 2784826
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Abstract: Given $ n \in {\mathbb{Z}}^+$ and $ \varepsilon >0$, we prove that there exists $ \delta = \delta (\varepsilon,n) >0$ such that the following holds: If $ (M^n,g)$ is a compact Kähler $ n$-manifold whose sectional curvatures $ K$ satisfy

$\displaystyle -1 - \delta \le K \le - \frac {1}{4}$

and $ c_I(M)$, $ c_J(M)$ are any two Chern numbers of $ M$, then

$\displaystyle \Bigl \vert \frac {c_I(M)}{c_J(M)} - \frac {c_I^0}{c_J^0} \Bigr \vert < \varepsilon,$

where $ c_I^0$, $ c_J^0$ are the corresponding characteristic numbers of a complex hyperbolic space form.

It follows that the Mostow-Siu surfaces and the threefolds of Deraux do not admit Kähler metrics with pinching close to $ \frac {1}{4}$.


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Additional Information

Martin Deraux
Affiliation: Institut Fourier, Université de Grenoble I, 38402 Saint-Martin-d’Hères Cedex, France
Email: deraux@ujf-grenoble.fr

Harish Seshadri
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Email: harish@math.iisc.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10676-7
PII: S 0002-9939(2010)10676-7
Received by editor(s): December 18, 2009
Received by editor(s) in revised form: June 28, 2010
Published electronically: December 9, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society