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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?

Espresso
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  • 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, Memoirs of the AMS 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
  • @David Speyer Ah. I guess this is fine. Either we just consider geometric points, or we just define "curve of genus $g$" to be a family $C\to S$ whose geometric fibers are smooth connected curves of genus $g$. It is slightly misleading to say a point on $M_g$ is a single curve ... that is, unless there is some non-standard construction in which a point really does correspond to a single curve. – Matt Oct 26 '11 at 21:19
  • A $\mathbb{C}$-point corresponds to an isomorphism class of curves over $\mathbb{C}$. Using "point" to mean $\mathbb{C}$-point is hardly nonstandard; it's what anyone who learned algebraic geometry from Griffiths-Harris or Shafareivich would do. Moreover, for any more general notion of point you want to use, I think there should be a correspondingly more general notion of curve which makes the statement right again :). – David E Speyer Oct 26 '11 at 21:33
  • @Matt: Although I used 'curves' in the title, I did modify it to 'Riemann surface' in the body. In complex geometry or Teichmuller theory, people do not usually use the language of representable functors. – Espresso Oct 27 '11 at 04:19
  • @DavidSpeyer: Thanks for the mention of Hodge bundle. That is exactly what I wanted. I hope it is also possible to prove a de Rham type theorem on that bundle. I was motivated by some dynamical considerations, and after searching google, found something called "Kontsevich-Zorich cocycle" that helps me. Thanks a lot once again. – Espresso Oct 27 '11 at 05:10

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