## Duke Mathematical Journal

### On the cohomological dimension of the moduli space of Riemann surfaces

Gabriele Mondello

#### Abstract

The moduli space of Riemann surfaces of genus $g\geq2$ 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 $g-2$. 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 $2g-2$. 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.

#### Article information

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

Dates
Revised: 31 March 2016
First available in Project Euclid: 18 March 2017

https://projecteuclid.org/euclid.dmj/1489802635

Digital Object Identifier
doi:10.1215/00127094-0000004X

Mathematical Reviews number (MathSciNet)
MR3659940

Zentralblatt MATH identifier
06754737

#### Citation

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

#### References

• [1] A. Andreotti and H. Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259.
• [2] E. Arbarello, Weierstrass points and moduli of curves, Compos. Math. 29 (1974), 325–342.
• [3] E. Arbarello and G. Mondello, “Two remarks on the Weierstrass flag” in Compact Moduli Spaces and Vector Bundles, Contemp. Math. 564, Amer. Math. Soc., Providence, 2012, 137–144.
• [4] B. H. Bowditch and D. B. A. Epstein, Natural triangulations associated to a surface, Topology 27 (1988), 91–117.
• [5] D. Chen, Strata of abelian differentials and the Teichmüller dynamics, J. Mod. Dyn. 7 (2013), 135–152.
• [6] J.-P. Demailly, Complex analytic and differential geometry, preprint, 2012, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf.
• [7] S. Diaz, A bound on the dimensions of complete subvarieties of $\mathcal{M}_{g}$, Duke Math. J. 51 (1984), 405–408.
• [8] C. Faber, “A conjectural description of the tautological ring of the moduli space of curves” in Moduli of Curves and Abelian Varieties, Aspects Math. E33, Vieweg, Braunschweig, 1999, 109–129.
• [9] C. Fontanari, “Moduli of curves via algebraic geometry” in Liaison and Related Topics (Turin, 2001), Rend. Semin. Mat. Univ. Politec. Torino 59, 2001, 137–139.
• [10] C. Fontanari and E. Looijenga, A perfect stratification of $\mathcal{M}_{g}$ for $g\leq5$, Geom. Dedicata 136 (2008), 133–143.
• [11] C. Fontanari and S. Pascolutti, An affine open covering of $\mathcal{M}_{g}$ for $g\leq5$, Geom. Dedicata 158 (2012), 61–68.
• [12] T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), 1–37.
• [13] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. Inst. Hautes Études Sci. 5 (1960), 64.
• [14] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois-Marie 1962 (SGA 2), Doc. Math. (Paris) 4, Soc. Math. France, Paris, 2005.
• [15] S. Grushevsky and I. Krichever, “The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces” in Geometry of Riemann Surfaces and their Moduli Spaces, Surv. Differ. Geom. 14, Int. Press, Somerville, Mass., 2009, 111–129.
• [16] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157–176.
• [17] R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math. 156, Springer, Berlin, 1970.
• [18] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221–274.
• [19] E.-N. Ionel, Topological recursive relations in $H^{2g}(\mathcal{M}_{g,n})$, Invent. Math. 148 (2002), 627–658.
• [20] S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), 293–311.
• [21] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1–23.
• [22] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631–678.
• [23] D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett. 18 (2011), 447–458.
• [24] E. Looijenga, On the tautological ring of $\mathcal{M}_{g}$, Invent. Math. 121 (1995), 411–419.
• [25] E. Looijenga and G. Mondello, The fine structure of the moduli space of abelian differentials in genus 3, Geom. Dedicata 169 (2014), 109–128.
• [26] H. Masur, “Ergodic theory of translation surfaces” in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 527–547.
• [27] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2) 134 (1991), 455–543.
• [28] M. Möller, Linear manifolds in the moduli space of one-forms, Duke Math. J. 144 (2008), 447–487.
• [29] G. Mondello, A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces, J. Eur. Math. Soc. (JEMS) 10 (2008), 231–241.
• [30] G. Mondello, Riemann surfaces with boundary and natural triangulations of the Teichmüller space, J. Eur. Math. Soc. (JEMS) 13 (2011), 635–684.
• [31] G. Mondello, Stratifications of the moduli space of curves and related questions, Rend. Mat. Appl. (7) 35 (2014), 131–158.
• [32] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299–339.
• [33] A. Polishchuk, “Moduli spaces of curves with effective $r$-spin structures” in Gromov-Witten Theory of Spin Curves and Orbifolds, Contemp. Math. 403, Amer. Math. Soc., Providence, 2006, 1–20.
• [34] R. Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257–286.
• [35] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278.
• [36] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955–1956), 1–42.
• [37] W. A. Veech, Flat surfaces, Amer. J. Math. 115 (1993), 589–689.
• [38] C. Voisin, A counterexample to the Hodge conjecture extended to Kähler varieties, Int. Math. Res. Not. IMRN 2002, no. 20, 1057–1075.