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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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$L^{2}$-metrics, projective flatness and families of polarized abelian varieties
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by Wing-Keung To and Lin Weng PDF
Trans. Amer. Math. Soc. 356 (2004), 2685-2707 Request permission

Abstract:

We compute the curvature of the $L^{2}$-metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the $L^{2}$-metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.
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Additional Information
  • Wing-Keung To
  • Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
  • MR Author ID: 267228
  • Email: mattowk@nus.edu.sg
  • Lin Weng
  • Affiliation: Graduate School of Mathematics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan
  • Email: weng@math.kyushu-u.ac.jp
  • Received by editor(s): November 14, 2002
  • Published electronically: December 9, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2685-2707
  • MSC (2000): Primary 14K99, 32G08, 32G13, 32Q20
  • DOI: https://doi.org/10.1090/S0002-9947-03-03488-3
  • MathSciNet review: 2052193