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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?

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  • $\begingroup$ 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. $\endgroup$ – Matt Oct 26 '11 at 16:23
  • $\begingroup$ @Matt What Espresso is writing is correct. A point on $M_g$ is a single curve. $\endgroup$ – David E Speyer Oct 26 '11 at 19:25
  • $\begingroup$ @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. $\endgroup$ – David E Speyer Oct 26 '11 at 19:35
  • $\begingroup$ 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. $\endgroup$ – David E Speyer Oct 26 '11 at 19:36
  • $\begingroup$ PPS Are you familiar with the construction of the Hodge bundle? That's similar but easier, because you use sheaf cohomology instead of hypercohomology. $\endgroup$ – David E Speyer Oct 26 '11 at 19:47

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