Duke Mathematical Journal

On the cohomological dimension of the moduli space of Riemann surfaces

Gabriele Mondello

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The moduli space of Riemann surfaces of genus g2 is (up to a finite étale cover) a complex manifold, so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is g2. This expectation is verified in low genus and is supported by Harer’s computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this article, we prove that such a dimension is at most 2g2. 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.

Article information

Duke Math. J. Volume 166, Number 8 (2017), 1463-1515.

Received: 12 May 2014
Revised: 31 March 2016
First available in Project Euclid: 18 March 2017

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Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 32F10: $q$-convexity, $q$-concavity 30F30: Differentials on Riemann surfaces

Riemann surfaces moduli space translation surfaces cohomological dimension


Mondello, Gabriele. On the cohomological dimension of the moduli space of Riemann surfaces. Duke Math. J. 166 (2017), no. 8, 1463--1515. doi:10.1215/00127094-0000004X. https://projecteuclid.org/euclid.dmj/1489802635

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