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?



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


