# de Rham cohomology on moduli of curves

I do not know the constructions of Deligne-Mumford; so let us suppose that the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$, with $g>1$, is constructed using the moduli of abelian varieties.

Now given a point $x \in \mathcal{M}_g$, which actually corresponds to some Riemann surface $X$ of genus $g$, consider the vector space $H^1_{\mathrm{dR}}(X)$. This associates a real vector space to each point in $\mathcal{M}_g$. Is there a natural way to make this into a vector bundle?

• Do you have a reference for this construction? I'm not sure I understand it. A point on $\mathcal{M}_g$ is usually a family of curves. – Matt Oct 26 '11 at 16:23
• @Matt What Espresso is writing is correct. A point on $M_g$ is a single curve. – David E Speyer Oct 26 '11 at 19:25
• @Espresso Yes, there is such a bundle. I am hope someone will post a reference which treats this topic carefully and rigorously. Here is the intuition. Let $\pi: \mathcal{C} \to \mathcal{M}_g$ be the universal family. I will pretend that $\mathcal{C}$ and $\mathcal{M}$ are schemes, although in fact they are stacks. For any open set $U \subseteq \mathcal{M}$, let $H(U)$ be the hypercohomology $\mathbb{H}^1{\LARGE (}\mathcal{O}(\pi^{-1}(U)) \to \Omega^1_{\mathcal{C}/\mathcal{M}}(\pi^{-1}(U)) {\LARGE )}$. Then $U \mapsto H(U)$ is a sheaf, and what you want is the corresponding vector bundle. – David E Speyer Oct 26 '11 at 19:35
• PS A bit of googling turned up "The Hodge theory of Stable Curves", by Jerome Hoffman, <i>Memoirs of the AMS</i> Volume 308. I've never read it, but it looks relevant. – David E Speyer Oct 26 '11 at 19:36
• PPS Are you familiar with the construction of the Hodge bundle? That's similar but easier, because you use sheaf cohomology instead of hypercohomology. – David E Speyer Oct 26 '11 at 19:47