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

González-Alonso, Víctor; Stoppino, Lidia; Torelli, Sara

doi:10.1090/tran/7868

On the rank of the flat unitary summand of the Hodge bundle
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by Víctor González-Alonso, Lidia Stoppino and Sara Torelli PDF

Trans. Amer. Math. Soc. 372 (2019), 8663-8677 Request permission

Abstract:

Let $f\colon S\to B$ be a nonisotrivial fibered 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 fiber 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 bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth.

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