On the rank of the flat unitary summand of the Hodge bundle

TL;DR

This paper establishes bounds on the rank of the unitary part of the Hodge bundle for fibred surfaces, relating it to the genus and Clifford index, with improvements for certain plane curve fibers.

Contribution

It proves new inequalities linking the rank of the unitary summand of the Hodge bundle to the genus and Clifford index, strengthening previous bounds for specific fiber types.

Findings

The inequality u_f ≤ g - c_f is established.

For plane curve fibers of degree ≥ 5, the bound improves to u_f ≤ g - c_f - 1.

The results extend and strengthen previous bounds in the literature.

Abstract

Let $f\colon S\to B$ be a non-isotrivial fibred surface. We prove that the genus $g$, the rank $u_f$ of the unitary summand of the Hodge bundle $f_*\omega_f$ and the Clifford index $c_f$ satisfy the inequality $u_f \leq g - c_f$. Moreover, we prove that if the general fibre is a plane curve of degree $\geq 5$ then the stronger bound $u_f \leq g - c_f-1$ holds. In particular, this provides a strengthening of the bounds of \cite{BGN} and of \cite{FNP}. The strongholds of our arguments are the deformation techniques developed by the first author in \cite{Rigid} and by the third author and Pirola in \cite{PT}, which display here naturally their power and depht.