# Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

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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See en.wikipedia.org/wiki/Hodge_structure and especially the second example in "Examples" on that page. – Ariyan Javanpeykar Apr 19 '14 at 21:13
@Ari : Yes, but $\overline{\mathcal{M}}_{g,p}$ is not smooth; it has singularities along its boundary. – Gina Apr 19 '14 at 21:25
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. – Donu Arapura Apr 19 '14 at 22:16
@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. – Dan Petersen Apr 20 '14 at 9:14
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"). – Dan Petersen Apr 20 '14 at 9:20

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.
Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbb{Q})$ admits a pure Hodge structure of weight $k$.
Great, thanks!$\$ – Gina Apr 19 '14 at 23:40