Massey Products and Fujita decompositions on fibrations of curves

Abstract

Let f : S → B be a fibred surface and f * ω S/B = U ⊕A be the second Fujita decomposition of f. We study a Massey product related with variation of the Hodge structure over flat sections of U. We prove that the vanishing of the Massey product implies that the monodromy of U is finite and described by morphisms over a fixed curve. The main tools are a lifting lemma of flat sections of U to closed holomorphic forms of S and two classical results due (essentially) to de Franchis. As applications we find a new proof of a theorem of Luo and Zuo for hyperelliptic fibrations. We also analyze, as for the surfaces constructed by Catanese and Dettweiler, the case when U has not finite monodromy.Abstract. Let f : S → B be a fibration of curves and let f * ω S/B = U ⊕ A 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 U.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 U 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).

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

“…The proof of the first inequality (1.4) follows the lines of the original argument of [BGAN15], but in this more general setting new results are needed. The key points of our arguments are the deformation thecnhniques developed by the first author in [GA16] and by the third author and Pirola in [PT17], together with an ad-hoc Castelnuovo-de Franchis theorem for tubular surfaces.…”

“…The main difference between the present case and [BGAN15] stems from the fact that sections of U do not correspond to global differential 1-forms on S, while sections of the trivial part do. Instead, after the recent work of Pirola and Torelli [PT17], local flat sections of U can be identified with closed holomorphic 1-forms on "tubes" f −1 (∆) ⊂ S around smooth fibres (see Lemma 2.4). This local liftability of sections of U is enough to deal with the movable case.…”

“…Lemma 2.4 (Lifting Lemma in [PT17]). Let f : S → B be a semistable fibration, U the local system associated to the flat unitary summand, and Ω 1 S,d the sheaf of closed holomorphic 1-forms.…”

Families of curves with Higgs field of arbitrarily large kernel

“…(2) The argument follows the line of [15, section 3.1, case 1] (see also [21,Lemma 3.2]). Assume that rank U 2, otherwise there is nothing to prove.…”

“…Indeed, in the recent work [15] an upper bound for the rank of is obtained, depending on geometric invariants of the fibers like their genus and the general Clifford index, generalizing a previous result of [1] on the relative irregularity. A closer look at the proof of that result shows that in some cases the inequality can be proved using Massey products of sections of [14, 21] combined with Castelnuovo–de Franchis fibration type theorems. Similar constructions are used in [18] to study hyperelliptic fibrations.…”