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I'm trying to understand the paper

Arbarello, Enrico, Cornalba, Maurizio, Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 97–127 (1999).

At the very top of page 103 (of the journal; this is the 7th page of the paper) they assert without proof that $H^k(\overline{\mathcal{M}}_{g,p})$ is pure of weight $k$. I'm not at all an expert in Hodge theory, so I'm probably missing something obvious here, but why is this clear?

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  • $\begingroup$ See en.wikipedia.org/wiki/Hodge_structure and especially the second example in "Examples" on that page. $\endgroup$
    – Ariyan Javanpeykar
    Commented Apr 19, 2014 at 21:13
  • $\begingroup$ @Ari : Yes, but $\overline{\mathcal{M}}_{g,p}$ is not smooth; it has singularities along its boundary. $\endgroup$
    – Gina
    Commented Apr 19, 2014 at 21:25
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    $\begingroup$ Gina, yes, the purity follows from Deligne, Theorie de Hodge III, Thm 8.2.4 (iv) + plus the fact that orbifolds are rational homology manifolds. I'm sure it's Peters-Steenbrink also if that is preferable. $\endgroup$
    – Donu Arapura
    Commented Apr 19, 2014 at 22:16
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    $\begingroup$ @Gina Yes the whole theory of weights works also for stacks. But a remark is that you can bypass stacks completely in this case, since (as you mention) the spaces $\overline M_{g,n}$ are quotients of a smooth projective variety by a finite group. In general one has for rational cohomology $H^\bullet(X/G) = H^\bullet(X)^G$ ($G$-invariants); when $X$ is an algebraic variety, $H^\bullet(X)^G$ is a sub-Hodge structure of $H^\bullet(X)$ and in particular it is pure if $H^\bullet(X)$ is pure. $\endgroup$
    – Dan Petersen
    Commented Apr 20, 2014 at 9:14
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    $\begingroup$ By the way, the fact that $\overline M_{g,n}$ is a quotient of a smooth projective variety by a finite group action was first proven (over the complex numbers) by Looijenga (the paper on "Prym level structures"). $\endgroup$
    – Dan Petersen
    Commented Apr 20, 2014 at 9:20

1 Answer 1

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Let me elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbb{Q})$ admits a pure Hodge structure of weight $k$.

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