# Families of curves with Higgs field of arbitrarily large kernel

## TL;DR

This paper constructs families of curves with a Higgs field whose kernel can have arbitrarily large dimension, exploring the relationship between the flat bundle and Higgs field in variations of Hodge structures.

## Contribution

It demonstrates the existence of non-isotrivial deformations of curves where the kernel of the Higgs field can be arbitrarily large, showing the potential strictness of the inclusion of the flat bundle.

## Key findings

- Constructed families with kernel rank k for any 0 ≤ k ≤ g-1
- Showed the flat bundle U can have rank at most (g+1)/2
- Provided examples of non-isotrivial deformations with specified properties

## Abstract

In this note we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion of U in K can be in the geometric case. More precisely, for any smooth projective curve C of genus g (at least 2) and any k (between 0 and g-1), we construct non-isotrivial deformations of C over a quasi-projecive base such that K has rank k and U has rank at most (g+1)/2.