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Transactions of the American Mathematical Society

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$L^{2}$-metrics, projective flatness and families of polarized abelian varieties


Authors: Wing-Keung To and Lin Weng
Translated by:
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
Published electronically: December 9, 2003
MathSciNet review: 2052193
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Abstract | References | Similar Articles | Additional Information

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
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

DOI: https://doi.org/10.1090/S0002-9947-03-03488-3
Received by editor(s): November 14, 2002
Published electronically: December 9, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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