Complete families of indecomposable non-simple abelian varieties

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 ′ ≥ g. These methods also yield a construction of a new Kodaira fibration with fiber genus 4.

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

“…Since the construction of the first examples of such fibrations by Atiyah and Kodaira [5,44], these complex surfaces have been widely studied, either from the point of view of complex geometry or from the point of view of the theory of surface bundles. See for instance [8,15,17,18,20,31,32,36,46,60] for a few works on this topic. In this text we study several geometric and group theoretical problems related to these complex surfaces.…”