Almost quarter-pinched Kähler metrics and Chern numbers
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- by Martin Deraux and Harish Seshadri PDF
- Proc. Amer. Math. Soc. 139 (2011), 2571-2576 Request permission
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 \[ -1 - \delta \le K \le - \frac {1}{4}\] and $c_I(M)$, $c_J(M)$ are any two Chern numbers of $M$, then \[ \Bigl \vert \frac {c_I(M)}{c_J(M)} - \frac {c_I^0}{c_J^0} \Bigr \vert < \varepsilon ,\] 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
- MR Author ID: 740008
- Email: deraux@ujf-grenoble.fr
- Harish Seshadri
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
- MR Author ID: 712201
- Email: harish@math.iisc.ernet.in
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2571-2576
- MSC (2010): Primary 53C21; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-2010-10676-7
- MathSciNet review: 2784826