On the cohomological dimension of the moduli space of Riemann surfaces

The moduli space of Riemann surfaces of genus g ≥ 2 is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is g−2. This expectation is verified in low genus and supported by Harer’s computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this paper we prove that such dimension is at most 2g−2. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most g. In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: in the construction of such a function some basic geometric properties of flat surfaces come into play.