Holomorphic one forms on moduli of curves

It is a classical result from the sixties that the moduli space of smooth projective curves Mg does not have non-trivial closed holomorphic 1-forms, for genus g > 2. In the talk I present a stronger result recently proven in collaboration with F. Favale and G.P. Pirola that the moduli space Mg does not have holomorphic 1-forms at all, even not closed, at least for genus g ≥ 5. Time permitting, I will then discuss further applications of the developed techniques.