1 June 2017 On the cohomological dimension of the moduli space of Riemann surfaces
Gabriele Mondello
Duke Math. J. 166(8): 1463-1515 (1 June 2017). DOI: 10.1215/00127094-0000004X


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.


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Gabriele Mondello. "On the cohomological dimension of the moduli space of Riemann surfaces." Duke Math. J. 166 (8) 1463 - 1515, 1 June 2017. https://doi.org/10.1215/00127094-0000004X


Received: 12 May 2014; Revised: 31 March 2016; Published: 1 June 2017
First available in Project Euclid: 18 March 2017

zbMATH: 06754737
MathSciNet: MR3659940
Digital Object Identifier: 10.1215/00127094-0000004X

Primary: 32G15
Secondary: 30F30 , 32F10

Keywords: cohomological dimension , moduli space , Riemann surfaces , translation surfaces

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 8 • 1 June 2017
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