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Science China Mathematics

, Volume 62, Issue 11, pp 2211–2228 | Cite as

Metric distortion in the geometric Schottky problem

  • Lizhen JiEmail author
Articles
  • 19 Downloads

Abstract

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus \(J(\mathcal{M}_g)\) in the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties as an algebraic subvariety. By viewing \(\mathcal{A}_g\) as a noncompact metric space coming from its structure as a locally symmetric space and \(J(\mathcal{M}_g)\) as a metric subspace, we compare the subspace metric d and the induced length metric ℓ on \(J(\mathcal{M}_g)\). Consequently, we clarify the nature of the metric distortion of the subspace \(J(\mathcal{M}_g)\) and hence settle a problem posed by Farb (2006) on the metric distortion of \(J(\mathcal{M}_g)\) inside \(\mathcal{A}_g\) in a certain sense (see Theorem 1.5 and Corollary 1.6).

Keywords

metric distortion Schottky problem noncompact metric space 

MSC(2010)

14H42 14H10 14K10 

Notes

Acknowledgements

This work was supported by the Simons Foundation (Grant No. 353785). The author thanks an anonymous referee for his careful reading of this paper and his constructive suggestions.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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