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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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    @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.2011-10-27

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