𝐿²-metrics, projective flatness and families of polarized abelian varieties

To, Wing-Keung; Weng, Lin

doi:10.1090/S0002-9947-03-03488-3

$L^{2}$-metrics, projective flatness and families of polarized abelian varieties
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by Wing-Keung To and Lin Weng

Trans. Amer. Math. Soc. 356 (2004), 2685-2707

DOI: https://doi.org/10.1090/S0002-9947-03-03488-3

Published electronically: December 9, 2003

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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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