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Gravitational instantons from rational elliptic surfaces

Author: Hans-Joachim Hein
Journal: J. Amer. Math. Soc. 25 (2012), 355-393
MSC (2010): Primary 53C25, 14J27
Published electronically: November 18, 2011
MathSciNet review: 2869021
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Abstract: Let $ X$ denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let $ D$ be any fiber of the resulting elliptic fibration on $ X$. Using ansatz metrics inspired by work of Gross-Wilson and a PDE method due to Tian-Yau, we prove that $ X \setminus D$ admits complete Ricci-flat Kähler metrics in most de Rham cohomology classes. If $ D$ is smooth, the metrics converge to split flat cylinders $ \mathbb{R}^+ \times S^1 \times D$ at an exponential rate. In this case, we also obtain a partial uniqueness result and a local description of the Einstein moduli space, which contains cylindrical metrics whose cross section does not split off a circle. If $ D$ is singular but of finite monodromy, they converge at least polynomially to flat $ T^2$-submersions over flat $ 2$-dimensional cones that need not be quotients of $ \mathbb{R}^2$. If $ D$ is singular of infinite monodromy, their volume growth rates are $ 4/3$ and $ 2$ for the Kodaira types $ {\rm I}_b$ and $ {{\rm I}_b}^*$, their injectivity radii decay like $ r^{-1/3}$ and $ (\log r)^{-1/2}$, and their curvature tensors decay like $ r^{-2}$ and $ r^{-2}(\log r)^{-1}$. In particular, the $ {\rm I}_b$ examples show that a curvature estimate due to Cheeger and Tian cannot be improved in general.

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Hans-Joachim Hein
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom

Received by editor(s): April 24, 2010
Received by editor(s) in revised form: August 25, 2010, September 30, 2011, October 19, 2011, and October 23, 2011
Published electronically: November 18, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.