Fujita decompositions and infinitesimal invariants on fibred surfaces

The thesis is devoted to the study of infinitesimal invariants in relation to the second Fujita decompositions of fibered surfaces. The study is developed in the contest of variation of the Hodge structure and the aim is to capture geometric information of the unitary flat summand in the decomposition above. The results are a Lifting property that provides a description of such bundle linked to the geometry of the surface, a monodromy criterion of finiteness and description of the action in terms of morphisms of curves and a numerical bound on the rank involving geometric invariants.