Massey products and Fujita decomposition

On the Hodge bundle $f_*\omega_{S/B}$ attached to a fibration $f:S\to B$ over a projective curve $B$ there are two {\em Fujita decompositions}: the first one $f_*\omega_{S/B}=\mathcal{O}^{ \oplus q_f}\oplus E$ splits a trivial bundle $\mathcal{O}^{\oplus q_f}$ of rank equal to the relative irregularity $q_f$, while the second one splits $f_*\omega_{S/B}=U\oplus A$ into a unitary flat bundle $U$ and an ample bundle $A$. The monodromy of $U$ can be infinite in general, as Catanese and Dettweiler show with explicit examples. In this talk we first give a description of the structure of $U$ in terms of "tubular" closed holomorphic forms and then we discuss how a vanishing condition on "Massey products" (also known as adjoint images) forces the monodromy of $U$ to be finite. This is a joint work with Gian Pietro Pirola.