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

Abstract

Let f : S → B f\colon S\to B be a nonisotrivial fibered surface. We prove that the genus g g , the rank u f u_f of the unitary summand of the Hodge bundle f ∗ ω f f_*\omega _f , and the Clifford index c f c_f satisfy the inequality u f ≤ g − c f u_f \leq g - c_f . Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5 \geq 5 , then the stronger bound u f ≤ g − c f − 1 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.

Families of curves with Higgs field of arbitrarily large kernel

“…Our motivation to study this question stems from the classification of fibered (irregular) surfaces. 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.…”

“…Lemma 2.3 and Theorem 2.4] or[15, Theorem 2.9]). Let C be a projective curve of genus g, ξ ∈ H 1 (C, T C ) a first-order infinitesimal deformation and ∪ξ : H 0 (C, ω C ) → H 1 (C, O C ) the map induced by cup-product.…”

Massey Products and Fujita decompositions on fibrations of curves

“…In this section we relate the local system U underlying the unitary flat bundle U in the second Fujita decomposition of a fibration f : S → B with the subsheaf Ω 1 S,d ⊂ Ω 1 S of the closed holomorphic 1-forms on S. We prove that it lifts to the direct image of the sheaf Ω 1 S,d so that it is described by closed holomorphic forms defined on tubular neighborhoods a fibres of f . In this section we restate a generalized version of the Castelnuovo de Franchis theorem for fibred surfaces (see [18]) and we use it to relate the geometry of Massey-trivial subspaces W ⊂ Γ(A, U) of flat local sections of the unitary summand U to the existence of a fibration from the surface into a smooth compact curve Σ of genus greater than 2, after a base change.…”

“…Let us first assume that B is a complex disk. In this case the theorem has been proven in [18], following the classical argument given in [3, Proposition X.6] for a compact S. This assumption is essentially used to conclude that the foliation defined by pointwise proportional holomorphic forms is integrable, since all holomorphic forms on compact surfaces are closed. By releasing it, one can assume that holomorphic forms be closed (that is no longer automatic) and check that this is sufficient to repeat the original argument.…”

“…In this section we relate the geometry of Massey-trivial subspaces W ⊂ Γ(A, U) of flat local sections of the unitary factor U of a fibration f : S → B of genus g(F ) ≥ 2 to the existence, up to base change, of morphisms from the surface into a smooth compact curve Σ of genus greater than 2. The construction is given by a Castelnuovo de Franchis theorem for fibred surfaces proved in [GST17], which we adapt to our setting.…”