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Complete families of indecomposable non-simple abelian varieties

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Abstract

Given a fixed product of non-isogenous abelian varieties at least one of which is general, we show how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to the given product and whose connected monodromy group is a product of symplectic groups or is a unitary group. As a consequence, we show how to realize any product of symplectic groups of total rank g as the connected monodromy group of a complete family of \(g'\)-dimensional abelian varieties for any \(g'\ge g\). These methods also yield a construction of a new Kodaira fibration with fiber genus 4.

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Acknowledgements

The author would like to thank Giulia Saccà and Nick Salter whose conversations inspired the writing down of this paper. The author gratefully acknowledges support of the National Science Foundation through award DMS-1803082. Additionally, this material is based upon work supported by the National Science Foundation under Grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2019 semester.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Laure Flapan

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  1. Laure Flapan
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Correspondence to Laure Flapan.

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Communicated by Jean-Yves Welschinger.

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Flapan, L. Complete families of indecomposable non-simple abelian varieties. Math. Ann. 382, 255–283 (2022). https://doi.org/10.1007/s00208-021-02253-z

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