Variations of Hodge structures of a Teichmüller curve

Möller, Martin

doi:10.1090/S0894-0347-05-00512-6

Variations of Hodge structures of a Teichmüller curve
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by Martin Möller

J. Amer. Math. Soc. 19 (2006), 327-344

DOI: https://doi.org/10.1090/S0894-0347-05-00512-6

Published electronically: December 12, 2005

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Abstract:

Teichmüller curves are geodesic discs in Teichmüller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmüller curves map to the locus of real multiplication in the moduli space of abelian varieties. Observe that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\textrm {SL}_2({\mathbb {R}})$-action on the tautological bundle $\Omega M_g$. We also show that Teichmüller curves are defined over number fields and we provide a completely algebraic description of Teichmüller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmüller curves.

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