Luca Rizzi, An Approach to the Fujita Decomposition via Massey Products and Castelnuovo-de Franchis Theorem

Let f be a semistable fibration between a smooth complex variety of dimension n and a smooth complex curve. By a famous result of Fujita the direct image of the relative dualizing sheaf is the direct sum of a unitary flat vector bundle and an ample vector bundle. In this talk we focus our attention on the former: unitary flat bundles are in one-to-one correspondence with local systems and monodromy representations and I will show how these local systems are related to the de Rham closed one forms and top forms on the fibers of f. I will also give an introduction on the construction of Massey products and show its close relation with both a famous result by Castelnuovo and de Franchis and these local systems. Thanks to the presented techniques it is possible to give some results on the finiteness of the monodromy associated to the unitary flat bundle; equivalently on its semi-ampleness.