## Duke Mathematical Journal

### Linear manifolds in the moduli space of one-forms

Martin Möller

#### Abstract

We study closures of ${\rm GL}^+_2(\mathbb{R})$-orbits in the total space $\Omega M_g$ of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus, or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to $\Omega M_g$. For nongeneric strata, similar results can be shown by a case-by-case inspection. We also propose to study a notion of linear manifold that comprises Teichmüller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow [DM]. Moreover, we give an explanation for the difference between Hilbert modular surfaces and Hilbert modular threefolds with respect to this notion of linearity

#### Article information

Source
Duke Math. J., Volume 144, Number 3 (2008), 447-487.

Dates
First available in Project Euclid: 15 August 2008

https://projecteuclid.org/euclid.dmj/1218811401

Digital Object Identifier
doi:10.1215/00127094-2008-041

Mathematical Reviews number (MathSciNet)
MR2444303

Zentralblatt MATH identifier
1148.32007

#### Citation

Möller, Martin. Linear manifolds in the moduli space of one-forms. Duke Math. J. 144 (2008), no. 3, 447--487. doi:10.1215/00127094-2008-041. https://projecteuclid.org/euclid.dmj/1218811401

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