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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A momentum construction for circle-invariant Kähler metrics
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by Andrew D. Hwang and Michael A. Singer PDF
Trans. Amer. Math. Soc. 354 (2002), 2285-2325 Request permission

Abstract:

Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea—the “Calabi ansatz”—is to begin with an Hermitian line bundle $p:(L,h)\to (M,g_M)$ over a Kähler manifold, and to search for Kähler forms $\omega =p^*\omega _M+dd^c f(t)$ in some disk subbundle, where $t$ is the logarithm of the norm function and $f$ is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric $g$ arising from the Calabi ansatz. This suggests geometric hypotheses (which we call “$\sigma$-constancy”) to impose upon the base metric $g_M$ and Hermitian structure $h$ in order that the scalar curvature of $g$ be specified by solving an ODE. We show that $\sigma$-constancy is “necessary and sufficient for the Calabi ansatz to work” in the following sense. Under the assumption of $\sigma$-constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to $g_M$ on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of $\sigma$-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on $\mathbf {C}^2$.
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Additional Information
  • Andrew D. Hwang
  • Affiliation: Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, Massachusetts 01610-2395
  • Email: ahwang@mathcs.holycross.edu
  • Michael A. Singer
  • Affiliation: Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, EH9 3JZ, UK
  • MR Author ID: 215930
  • Email: michael@maths.ed.ac.uk
  • Received by editor(s): May 16, 2001
  • Received by editor(s) in revised form: October 18, 2001
  • Published electronically: February 13, 2002
  • Additional Notes: MAS is an EPSRC advanced fellow; ADH was supported in part by JSPS Fellowship #P-94016 and an NSERC Canada Research Grant.
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2285-2325
  • MSC (1991): Primary 53C55; Secondary 32L05, 53C21, 53C25
  • DOI: https://doi.org/10.1090/S0002-9947-02-02965-3
  • MathSciNet review: 1885653