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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Gravitational instantons from rational elliptic surfaces
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by Hans-Joachim Hein
J. Amer. Math. Soc. 25 (2012), 355-393
Published electronically: November 18, 2011


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 $\textrm {I}_b$ and ${\textrm {I}_b}^*$, 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 $\textrm {I}_b$ examples show that a curvature estimate due to Cheeger and Tian cannot be improved in general.
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Bibliographic Information
  • Hans-Joachim Hein
  • Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
  • MR Author ID: 938594
  • ORCID: 0000-0002-3719-9549
  • Email:
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 355-393
  • MSC (2010): Primary 53C25, 14J27
  • DOI:
  • MathSciNet review: 2869021