On the cohomological dimension of the moduli space of Riemann surfaces

TL;DR

This paper establishes an upper bound for the Dolbeault cohomological dimension of the moduli space of Riemann surfaces, advancing understanding of its geometric and topological properties, especially in relation to translation surfaces.

Contribution

It proves that the cohomological dimension is at most 2g-2 for the moduli space and g for each stratum of translation surfaces, providing new bounds and methods.

Findings

Bound of 2g-2 for the moduli space's cohomological dimension.

Bound of g for each stratum of translation surfaces.

Construction of exhaustion functions with controlled complex Hessian.

Abstract

The moduli space of Riemann surfaces of genus $g\geq 2$ is (up to a finite \'etale 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: the construction of such a function relies on some basic geometric properties of translation surfaces.