# Complete families of indecomposable non-simple abelian varieties

## TL;DR

This paper constructs complete families of indecomposable abelian varieties with prescribed monodromy groups, demonstrating realizability of certain symplectic and unitary groups as monodromy groups and producing a new Kodaira fibration.

## Contribution

It introduces a method to construct complete families of indecomposable abelian varieties with specified monodromy groups, including symplectic and unitary groups, and constructs a new Kodaira fibration.

## Key findings

- Realized any product of symplectic groups as monodromy of abelian families.
- Constructed a new Kodaira fibration with fiber genus 4.
- Provided explicit methods for building indecomposable abelian families with desired properties.

## 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 $4$.