Venue
Sala Seminari (Dip. Matematica).
Abstract
The moduli space $M_g$ of Riemann surfaces of genus g is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension (i.e. the highest k such that $H^{0,k}(M_g,E)$ does not vanish for some holomorphic vector bundle $E$ on $M_g$). The conjecturally optimal bound is g-2, which is verified for g=2,3,4,5. In this talk, I will show that such dimension is at most 2g-2. The key point is to show that the Dolbeault cohomological dimension of each stratum of the Hodge bundle is at most g (still non-optimal bound). In order to do that, I produce an exhaustion function, whose complex Hessian has controlled index: in the construction of such a function basic geometric properties of translation surfaces come into play.