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Variations of Hodge structures of a Teichmüller curve. (English) Zbl 1090.32004
The bundle $$\Omega T^{*}_{g}$$ over the Teichmüller space, whose points parameterize pairs $$(X, \omega)$$ of a Riemann surface $$X$$ of genus $$g$$ with a Teichmüller marking and a nonzero hololmorphic 1-form $$\omega$$ has a natural $$SL_{2}({\mathbb R})$$-action. The orbit of $$(X, \omega) \in \Omega T^{*}_{g}$$ projected to $$T_{g}$$ is a holomorphic embedding of the upper half-plane into Teichmüller space $$\widetilde{j} : {\mathbb H} \rightarrow T_{g},$$ which is totally geodesic for the Teichmüller metric. Only rarely do these geodesics project to algebraic curves $$C = {\mathbb H}/ \text{Stab} (\widetilde{j})$$ in the moduli space $$M_{g}.$$ These curves are called Teichmüller curves and $$(X, \omega)$$ a Veech surface.
The author investigates the variation of Hodge structures (VHS) of the family of Jacobians over a Teichmüller curves $$C$$ or more precisely over an unramifed cover of $$C$$ where the universal family exists.
Theorem 2.7. The image of a Teichmüller curves $$C \rightarrow A_{g}$$ is contained in the locus of abelian varieties that split up to isogeny into $$A_{1} \times A_{2},$$ where $$A_{1}$$ has dimension $$r$$ and real multiplication by the trace field $$K = {\mathbb Q}(\text{tr(Stab}(\widetilde{j}))),$$ where $$r = [K:{\mathbb Q}].$$ The generating differential $$\omega \in H^{0}(X, \Omega^{1}_{X})$$ is an eigenform for the multiplication by $$K,$$ i.e., $$K \cdot \omega \subset {\mathbb C} \omega.$$ Teichmüller curves are in some sense similar to Shimura curves, since their VHS always contains a sub-VHS that is maximal Higgs. In fact, the author characterizes Teichmüller curves algebraically using this notion.
Theorem 5.3. Suppose that the Higgs bundle of a family of curves $$f : {X} \rightarrow C = {\mathbb H}/ \Gamma$$ has a rank two Higgs subbundle with maximal Higgs field. Then $$C \rightarrow M_{g}$$ is a finite covering of a Teichmüller curve.
Theorem 4.2. Suppose that $$C \rightarrow M_{g}$$ is a Teichmüller curve with $$r = g.$$ Then the Shimura variety parameterizing abelian varieties with real multiplication by $$K$$ is the smallest Shimura subvariety of $$A_{g}$$ that the Teichmüller curve $$C$$ maps to.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14D07 Variation of Hodge structures (algebro-geometric aspects)
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