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?

notsmooth; it has singularities along its boundary. – Gina Apr 19 '14 at 21:25