Consider a fibration \(f\, :S \to B \,\) from a smooth complex projective surface to a smooth curve. Let \(g\geq 2\) be the genus of a general fibre and let \(q_f := q(S)-g(B)\) be the relative irregularity. In [M. Barja et al., J. Reine Angew. Math. 739, 297–308 (2018; Zbl 1437.14020)] it is proved that \(g\) , \(q_f\) and the Clifford index \(c_f \) of a {non-isotrivial} fibration satisfy the inequality \( q_f \leq g-c_f \) . This yields in particular a proof of Xiao’s conjecture for fibrations whose general fibres have maximal Clifford index.
The present paper is an interesting addition to [loc. cit.]. The starting point is to look at Fujita’s decompositions of the Hodge bundle \( f_*\omega_f= \mathcal{ O}_B^{\oplus q_f} \oplus \mathcal{ F} = \mathcal{ U} \oplus \mathcal{ A}\). Here \(\mathcal{ U} \) is a flat and unitary bundle of rank \(u_f\), its restriction over the open set of non critical points is the subbundle of \( f_*\omega_f\) spanned by flat sections. Now there is an inclusion \(\mathcal{ O}_B^{\oplus q_f} \subset \mathcal{ U}\), hence \(q_f \leq u_f\). The main result of the article is the proof of the stronger bound \( u_f \leq g-c_f \), again under the requirement of non isotriviality.
The authors indicate that they follow the strategy used in (loc. cit.), which is based on deformation methods. The proof, by contradiction, relies on the existence of certain supporting divisors for the family, in the sense of V. González-Alonso [Ann. Mat. Pura Appl. (4) 195, No. 1, 111–132 (2016; Zbl 1346.14029)]. A crucial new ingredient is needed here in order to deal with the case when this divisor is rigid. This is so because sections of \(\mathcal{ U}\) do not come from global 1-forms on \(S\), the property which holds instead is the fact that local flat sections of \(\mathcal{ U}\) amount to { closed} holomorphic 1-forms on “tubes” \(f^{-1}\left(\Delta\right) \subset S\) around smooth fibres, see [G. P. Pirola and S. Torelli [“Massey products and Fujita decompositions on fibrations of curves”, Preprint, arXiv:1710.02828]. It is proved here that such forms satisfy a tubular version of the Castelnuovo de Franchis theorem. This is the novel result which must be used to conclude the argument for the rigid case.
The paper witness, once again, to the efficacy of the adjoint techniques of Pirola and his school.