Duke Mathematical Journal

Linear manifolds in the moduli space of one-forms

Martin Möller

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We study closures of GL2+(R)-orbits in the total space ΩMg 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 ΩMg. 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

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

First available in Project Euclid: 15 August 2008

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Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 14D07: Variation of Hodge structures [See also 32G20] 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]


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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