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Teichmüller curves, mainly from the viewpoint of algebraic geometry. (English) Zbl 1279.14031
Farb, Benson (ed.) et al., Moduli spaces of Riemann surfaces. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (IAS) (ISBN 978-0-8218-9887-1/hbk). IAS/Park City Mathematics Series 20, 267-318 (2013).
Denote by $$M_g$$ the moduli space of smooth projective algebraic curves of genus $$g$$. Equivalently, $$M_g$$ may be viewed as the moduli space of compact Riemann surfaces of genus $$g$$ and as such it can be constructed as the quotient of the Teichmüller space $$T_g$$ modulo the action of the mapping class group $$\text{Mod}_g$$. Then a Teichmüller curve in $$M_g$$ is the projection of a geodesic disc in $$T_g$$ under the quotient map. There are various algebraic quantities that can be associated with any curve in $$M_g$$, and from this viewpoint one might ask the question of what distinguishes Teichmüller curves in $$M_g$$ from arbitrary curves.
The purpose of the paper under review is to show how some of the algebraic invariants are related to dynamical invariants of Teichmüller curves. The material presented here grew out of the author’s lectures delivered at the IAS/Park City Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann Surfaces, in July 2011.
After a brief introduction to guiding questions and to the main results discussed in this article, Section 2 provides the necessary background material on flat surfaces à la W. P. Thurston and W. Veech, affine groups, of flat surfaces and translation structures on flat (Riemann) surfaces. Section 3 introduces further background material on the algebraic geometry of curves and (special) divisors in the moduli space $$M_g$$ with particular emphasis on equations and inequalities for slopes of these objects. Section 4 is still of preparatory nature in that it explains the concept of a variation of Hodge structure, together with the related notions of period domains, period mappings and period coordinates. Furthermore, Hilbert modular varieties and the locus of real multiplication are briefly touched upon. Section 5 finally introduces Teichmüller curves in $$M_g$$ and describes the author’s own results on the special properties of the variation of Hodge structures over such a Teichmüller curve [the author, J. Am. Math. Soc. 19, No. 2, 327–344 (2006; Zbl 1090.32004)] in the context of Section 4. The state of the art concerning the classification problem for Teichmüller curves, together with some related, still open problems is briefly sketched at the end of this section. The concluding Section 6 turns to the theory of Lyapunov exponents for the geodesic Teichmüller flow (à la Eskin-Kontsevich-Zorich), thereby establishing a bridge between the dynamical defintion of Lyapunov exponents for Teichmüller curves, on the one hand, and an algebraic approach via variations of Hodge structures on the other. The results on sums of Lyapunov exponents for Teichmüller curves presented here are fairly recent, and partly due to the author and D. Chen as published in [Non-varying sums of Lyapunov exponents for abelian differentials in low genus (2011), arxiv:1104.3932]. This final section ends with an outlook towards a characterization of Shimura curves in terms of Lyapunov exponents. Also, some open problems concerning Lyapunov exponents of Teichmüller curves are discussed along these lines, and in the course of these lectures it is pointed out that slope estimates for divisors in $$M_g$$ can be obtained by using the special properties of Teichmüller curves as well.
For the entire collection see [Zbl 1272.30002].

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14H15 Families, moduli of curves (analytic) 14D07 Variation of Hodge structures (algebro-geometric aspects) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 32G20 Period matrices, variation of Hodge structure; degenerations 30F20 Classification theory of Riemann surfaces