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arXiv · 1709.05670

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

Vìctor González-Alonso·Lidia Stoppino·Sara Torelli
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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.

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Vìctor González-Alonso, Lidia Stoppino, Sara Torelli. 2017-09-17. On the rank of the flat unitary summand of the Hodge bundle. https://doi.org/10.1090/tran%2F7868

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