# Variations of Hodge structures of a Teichmueller curve

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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.

Autor: Martin Moeller

Fuente: https://archive.org/