Variations of Hodge structures of a Teichmueller curve

Details Variations of Hodge structures of a Teichmueller curve Variations of Hodge structures of a Teichmueller curve

2004-01-22

arxiv.org/abs/math/0401290v2

Mathematics

Algebraic Geometry

Scientific paper

Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the tautological bundle $\Omega\M_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmueller curves.

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