×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Paula Cohen and Jürgen Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. 56 (1990), no. 2, 93 – 110. · Zbl 0717.14014
[2] Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). · Zbl 0244.14004
[3] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5 – 57 (French). Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 – 77 (French).
[4] P. Deligne, Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 1 – 19 (French). · doi:10.1007/978-1-4899-6664-3_1 · doi.org
[5] G. Faltings, Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), no. 3, 337 – 347. · Zbl 0588.14025 · doi:10.1007/BF01388431 · doi.org
[6] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001
[7] Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191 – 213. · Zbl 0965.30019 · doi:10.1215/S0012-7094-00-10321-3 · doi.org
[8] János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361 – 398. · Zbl 0666.14003
[9] Lochak, P., On arithmetic curves in the moduli space of curves, Journal of the Math. Inst. of Jussieu 4, No. 3 (2005), 443-508. · Zbl 1094.14018
[10] Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. · Zbl 0779.14012
[11] Curtis T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857 – 885. · Zbl 1030.32012
[12] Curtis T. McMullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003), no. 2, 191 – 223. · Zbl 1131.37052 · doi:10.1007/BF02392964 · doi.org
[13] McMullen, C., Dynamics of \( {\mathrm SL}_2({\mathbb{R}})\) over moduli space in genus two, preprint (2003).
[14] Ben Moonen, Linearity properties of Shimura varieties. I, J. Algebraic Geom. 7 (1998), no. 3, 539 – 567. · Zbl 0956.14016
[15] Martin Möller, Teichmüller curves, Galois actions and \Hat \?\?-relations, Math. Nachr. 278 (2005), no. 9, 1061 – 1077. · Zbl 1081.14039 · doi:10.1002/mana.200310292 · doi.org
[16] David Mumford, Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 347 – 351.
[17] D. Mumford, A note of Shimura’s paper ”Discontinuous groups and abelian varieties”, Math. Ann. 181 (1969), 345 – 351. · Zbl 0169.23301 · doi:10.1007/BF01350672 · doi.org
[18] Chad Schoen, Varieties dominated by product varieties, Internat. J. Math. 7 (1996), no. 4, 541 – 571. · Zbl 0907.14002 · doi:10.1142/S0129167X9600030X · doi.org
[19] Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211 – 319. · Zbl 0278.14003 · doi:10.1007/BF01389674 · doi.org
[20] Paul Schmutz Schaller and Jürgen Wolfart, Semi-arithmetic Fuchsian groups and modular embeddings, J. London Math. Soc. (2) 61 (2000), no. 1, 13 – 24. · Zbl 0968.11022 · doi:10.1112/S0024610799008315 · doi.org
[21] Carlos T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713 – 770. · Zbl 0713.58012
[22] Goro Shimura, Moduli of abelian varieties and number theory, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 312 – 332.
[23] John Smillie and Barak Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004), 249 – 260. · Zbl 1052.37025 · doi:10.1007/BF02771535 · doi.org
[24] Kisao Takeuchi, On some discrete subgroups of \?\?\(_{2}\)(\?), J. Fac. Sci. Univ. Tokyo Sect. I 16 (1969), 97 – 100. · Zbl 0185.06302
[25] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553 – 583. , https://doi.org/10.1007/BF01388890 W. A. Veech, Erratum: ”Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards”, Invent. Math. 103 (1991), no. 2, 447. · Zbl 0709.32014 · doi:10.1007/BF01239521 · doi.org
[26] Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. · Zbl 0634.14022
[27] Eckart Viehweg and Kang Zuo, A characterization of certain Shimura curves in the moduli stack of abelian varieties, J. Differential Geom. 66 (2004), no. 2, 233 – 287. · Zbl 1078.11043
[28] Viehweg, E., Zuo, K., Numerical bounds for semi-stable families of curves or of higher-dimensional manifolds, preprint (2005). · Zbl 1200.14065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.