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The rational cohomology of \(\overline{\mathcal{M}}_4\). (English) Zbl 1126.14030

Denote by \(M_g\) the moduli space of smooth projective complex curves of genus \(g\), and let \(\overline M_g\) its Deligne-Mumford compactification, i.e., the moduli space of stable complex curves of genus \(g\). The topological study of these classifying spaces has turned out to be a rather delicate matter, involving individual case-by-case inspections (with respect to the genus \(g)\) and varying advanced techniques. In this context, one of the most important problems in the moduli theory of algebraic curves is to determine the structure of the (rational) cohomology ring of these moduli spaces, together with its natural Hodge decomposition. In the paper under review, the authors present two general approaches to the study of the cohomology of moduli spaces of curves, which are then applied to compute the rational cohomology of the particular moduli space \(\overline M_4\) of stable complex curves of genus 4, thereby adding a further explicit result to the overall picture.
Using the fact that \(\overline M_4\) is a proper and smooth Deligne-Mumford stack, it is argued that the rational cohomology of \(\overline M_4\) is completely determined by its Hodge-Euler characteristic \[ e(\overline M_4):=\sum_{j\in\mathbb{Z}}(-1)^j[H_c^j(M_4, \mathbb{Q})] \] of the cohomology with compact support in the Grothendieck group of rational mixed Hodge structures. On the other hand, the Hodge-Euler characteristic \(e(\overline M_4)\) is tackled by showing that the moduli space \(\overline M_4\) admits a natural stratification by finite quotients of products of moduli spaces \(M_{g,n}\) of smooth genus \(g\) curves with \(n\) marked points, where \(g\leq 4\) and \(n\leq 8-2g\), and by determining the \(S_n\)-equivariant Hodge-Euler characteristics of those spaces \(M_{g,n}\) separately.
The latter task, the crucial point of the entire approach, is accomplished in two different ways. The first method is based on the equivariant count of the number of points of \(M_{g,n}\) defined over finite fields, whereas the second method uses stratifications of moduli spaces by quotients of complements of discriminants in a complex vector space by the action of a reductive group. The computation of the rational cohomology of such strata is carried out by following an adapted version of a procedure due to A. Gorinov and V. Vassiliev [cf.: V. Vassiliev, Proc. Steklov Inst. Math. 225, 121–140 (1999; Zbl 0981.55008)], which is recalled at the end of the paper.
The main result finally states that the Hodge-Euler characteristic is given by the formula \[ e(\overline M_4)= \mathbb{L}^9+4\mathbb{L}^8+13\mathbb{L}^7+32 \mathbb{L}^6+50\mathbb{L}^5+50 \mathbb{L}^4+32\mathbb{L}^3+13 \mathbb{L}^2+4 \mathbb{L}+1 \] where \(\mathbb{L}\) denotes the class of the Tate-Hodge structure of weight 2 in the Grothendieck group of rational Hodge structures. This explicit formula enhances the picture shaped by previous related results of E. Arbarello and M. Cornalba [Publ. Math., Inst. Hautes Étud. Sci. 88, 97–127 (1998; Zbl 0991.14012)] and others.

MSC:

14H10 Families, moduli of curves (algebraic)
11G20 Curves over finite and local fields
55R80 Discriminantal varieties and configuration spaces in algebraic topology
14H25 Arithmetic ground fields for curves
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F45 Topological properties in algebraic geometry

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References:

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