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.

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).

We prove that Dolbeault formality, weakly Dolbeault formality and geometrically Dolbeault formality are not stable under small deformations of the complex structure.

In this paper, we study totally geodesic subvarieties [Formula: see text] of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for [Formula: see text]. We prove that if [Formula: see text] is generically contained in the Torelli locus, then [Formula: see text].

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