## Geometry & Topology

### Euler characteristics of Teichmüller curves in genus two

Matt Bainbridge

#### Abstract

We calculate the Euler characteristics of all of the Teichmüller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves are naturally embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmüller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies.

The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmüller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmüller curves.

We apply these results to calculate the Siegel–Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich–Zorich cocycle for any ergodic, $SL2(ℝ)$–invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).

#### Article information

Source
Geom. Topol., Volume 11, Number 4 (2007), 1887-2073.

Dates
Accepted: 12 July 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799977

Digital Object Identifier
doi:10.2140/gt.2007.11.1887

Mathematical Reviews number (MathSciNet)
MR2350471

Zentralblatt MATH identifier
1131.32007

#### Citation

Bainbridge, Matt. Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11 (2007), no. 4, 1887--2073. doi:10.2140/gt.2007.11.1887. https://projecteuclid.org/euclid.gt/1513799977

#### References

• W Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. $(2)$ 105 (1977) 29–44
• L V Ahlfors, The complex analytic structure of the space of closed Riemann surfaces., from: “Analytic functions”, Princeton Univ. Press, Princton, N.J. (1960) 45–66
• J S Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119 (2006) 121–140
• W L Baily, Jr, Satake's compactification of $V\sb{n}$, Amer. J. Math. 80 (1958) 348–364
• W L Baily, Jr, On the moduli of Jacobian varieties, Ann. of Math. $(2)$ 71 (1960) 303–314
• W L Baily, Jr, On the theory of $\theta$-functions, the moduli of abelian varieties, and the moduli of curves, Ann. of Math. $(2)$ 75 (1962) 342–381
• M Bainbridge, Billiards in L-shaped tables with barriers
• L Bers, Correction to “Spaces of Riemann surfaces as bounded domains”, Bull. Amer. Math. Soc. 67 (1961) 465–466
• L Bers, Holomorphic differentials as functions of moduli, Bull. Amer. Math. Soc. 67 (1961) 206–210
• L Bers, Fiber spaces over Teichmüller spaces, Acta. Math. 130 (1973) 89–126
• L Bers, On spaces of Riemann surfaces with nodes, Bull. Amer. Math. Soc. 80 (1974) 1219–1222
• L Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. $($N.S.$)$ 5 (1981) 131–172
• C Birkenhake, H Lange, Complex abelian varieties, volume 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], second edition, Springer, Berlin (2004)
• A I Borevich, I R Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York (1966)
• N Bourbaki, Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés, Actualités Scientifiques et Industrielles, No. 1343, Hermann, Paris (1969)
• I Bouw, M M öller, Teichmüller curves, triangle groups, and Lyapunov exponents
• K Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004) 871–908
• W Chen, Y Ruan, Orbifold Gromov-Witten theory, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, Contemp. Math. 310, Amer. Math. Soc., Providence, RI (2002) 25–85
• H Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975) 271–285
• P Deligne, D Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) 75–109
• A Eskin, H Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001) 443–478
• A Eskin, H Masur, M Schmoll, Billiards in rectangles with barriers, Duke Math. J. 118 (2003) 427–463
• A Eskin, H Masur, A Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. (2003) 61–179
• A Eskin, A Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001) 59–103
• A Eskin, A Okounkov, R Pandharipande, The theta characteristic of a branched covering
• H Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331–368
• P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, New York (1978)
• R C Gunning, Lectures on complex analytic varieties: The local parametrization theorem, Mathematical Notes, Princeton University Press, Princeton, N.J. (1970)
• R C Gunning, H Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N.J. (1965)
• E Gutkin, C Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000) 191–213
• J Harris, I Morrison, Moduli of curves, Graduate Texts in Mathematics 187, Springer, New York (1998)
• R Hartshorne, Residues and duality, Lecture Notes in Mathematics 20, Springer, Berlin (1966) Lecture notes of a seminar on the work of A Grothendieck, given at Harvard 1963/64. With an appendix by P Deligne
• R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York (1977)
• F E P Hirzebruch, Hilbert modular surfaces, Enseignement Math. $(2)$ 19 (1973) 183–281
• P Hubert, S Lelièvre, Prime arithmetic Teichmüller discs in $\mathcal H(2)$, Israel J. Math. 151 (2006) 281–321
• K Ireland, M Rosen, A classical introduction to modern number theory, second edition, Graduate Texts in Mathematics 84, Springer, New York (1990)
• E Kani, The number of curves of genus two with elliptic differentials, J. Reine Angew. Math. 485 (1997) 93–121
• E Kani, Hurwitz spaces of genus 2 covers of an elliptic curve, Collect. Math. 54 (2003) 1–51
• R Kenyon, J Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000) 65–108
• S Kerckhoff, H Masur, J Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. $(2)$ 124 (1986) 293–311
• F F Knudsen, D Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976) 19–55
• M Kontsevich, Lyapunov exponents and Hodge theory, from: “The mathematical beauty of physics (Saclay, 1996)”, Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ (1997) 318–332
• M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631–678
• I Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990) 499–578
• H B Laufer, Normal two-dimensional singularities, Annals of Mathematics Studies 71, Princeton University Press, Princeton, N.J. (1971)
• S Lelièvre, Siegel–Veech constants in $\mathcal H(2)$, Geom. Topol. 10 (2006) 1157–1172
• S Lelièvre, E Royer, Orbitwise countings in $\mathcal H(2)$ and quasimodular forms, Int. Math. Res. Not. (2006) Art. ID 42151, 30
• A Marden, Geometric complex coordinates for Teichmüller space, from: “Mathematical aspects of string theory (San Diego, CA, 1986)”, Adv. Ser. Math. Phys. 1, World Sci. Publishing, Singapore (1987) 341–354
• H Masur, Interval exchange transformations and measured foliations, Ann. of Math. $(2)$ 115 (1982) 169–200
• H Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems 10 (1990) 151–176
• H Masur, S Tabachnikov, Rational billiards and flat structures, from: “Handbook of dynamical systems, Vol. 1A”, North-Holland, Amsterdam (2002) 1015–1089
• C T McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003) 857–885
• C T McMullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003) 191–223
• C T McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005) 87–130
• C T McMullen, Teichmüller curves in genus two: the decagon and beyond, J. Reine Angew. Math. 582 (2005) 173–199
• C T McMullen, Teichmüller curves in genus two: torsion divisors and ratios of sines, Invent. Math. 165 (2006) 651–672
• C T McMullen, Dynamics of ${\rm SL}\sb 2(\Bbb R)$ over moduli space in genus two, Ann. of Math. $(2)$ 165 (2007) 397–456
• C T McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math. 129 (2007) 183–215
• T Miyake, Modular forms, Springer, Berlin (1989) Translated from the Japanese by Yoshitaka Maeda
• M M öller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006) 633–649
• D Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961) 5–22
• D Mumford, The structure of the moduli spaces of curves and Abelian varieties, from: “Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1”, Gauthier-Villars, Paris (1971) 457–465
• D Mumford, Hirzebruch's proportionality theorem in the noncompact case, Invent. Math. 42 (1977) 239–272
• D Mumford, Towards an enumerative geometry of the moduli space of curves, from: “Arithmetic and geometry, Vol. II”, Progr. Math. 36, Birkhäuser, Boston (1983) 271–328
• D Mumford, The red book of varieties and schemes, expanded edition, Lecture Notes in Mathematics 1358, Springer, Berlin (1999) Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello
• Y Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52 (1973) 197–259
• M Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994) 236–257
• I Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 359–363
• I Satake, On the compactification of the Siegel space, J. Indian Math. Soc. $($N.S.$)$ 20 (1956) 259–281
• M Schmoll, Spaces of elliptic differentials, from: “Algebraic and topological dynamics”, Contemp. Math. 385, Amer. Math. Soc., Providence, RI (2005) 303–320
• J-P Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956) 1–42
• J-P Serre, Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris (1959)
• C L Siegel, The volume of the fundamental domain for some infinite groups, Trans. Amer. Math. Soc. 39 (1936) 209–218
• C L Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969) 87–102
• G van der Geer, Hilbert modular surfaces, volume 16 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer, Berlin (1988)
• W A Veech, The Teichmüller geodesic flow, Ann. of Math. $(2)$ 124 (1986) 441–530
• W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553–583
• W A Veech, Moduli spaces of quadratic differentials, J. Analyse Math. 55 (1990) 117–171
• W A Veech, The billiard in a regular polygon, Geom. Funct. Anal. 2 (1992) 341–379
• W A Veech, Siegel measures, Ann. of Math. $(2)$ 148 (1998) 895–944
• H Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont. (1972)
• S A Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. Math. 107 (1985) 1485–1507 (1986)
• O Zariski, P Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Princeton, NJ-Toronto-London-New York (1960)
• A Zorich, How do the leaves of a closed $1$–form wind around a surface?, from: “Pseudoperiodic topology”, Amer. Math. Soc. Transl. Ser. 2 197, Amer. Math. Soc., Providence, RI (1999) 135–178
• A Zorich, Flat surfaces, from: “Frontiers in number theory, physics, and geometry. I”, Springer, Berlin (2006) 437–583