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

Abstract

Let $f:S\to B$ be a non-isotrivial fibred surface. We prove that the genus $g$, the rank $u_f$ of the unitary summand of the Hodge bundle $f_{\ast }{\omega }_f$ and the Clifford index $c_f$ satisfy the inequality $u_f\le g-c_f$. Moreover, we prove that if the general fibre is a plane curve of degree $\ge 5$ then the stronger bound $u_f\le g-c_f-1$ 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.

19 pages, revised version

References in corpus (5)

• Dihedral Monodromy and Xiao Fibrations
• On the Oort conjecture for Shimura varieties of unitary and orthogonal types
• On the Xiao conjecture for plane curves
• Xiao's conjecture for general fibred surfaces
• Massey Products and Fujita decompositions on fibrations of curves