Massey Products and Fujita decompositions on fibrations of curves

TL;DR

This paper investigates the relationship between Massey products, monodromy groups, and Fujita decompositions in fibrations of curves, establishing conditions under which the monodromy group is finite or infinite, with applications to hyperelliptic fibrations and normal functions.

Contribution

It introduces a new link between Massey products and monodromy actions in fibrations, providing criteria for finiteness of monodromy groups and applications to hyperelliptic cases.

Findings

Vanishing Massey products imply finite monodromy groups.

Monodromy group of hyperelliptic fibrations is finite.

Normal function is non-torsion when monodromy is infinite.

Abstract

Let $f:S\to B$ be a fibration of curves and let $f_\ast\omega_{S/B}=\cU\oplus \cA$ be the second Fujita decomposition of $f.$ In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $\cU$. The main result states that their vanishing on a general fibre of $f$ implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole $\cU$ of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).