Catalogue et commandes en ligne (paiement sécurisé, VISA ou MASTERCARD uniquement)

Revues disponibles par abonnement

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Séries de livres

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Série Chaire Jean Morlet

SMF/AMS Texts and Monographs

La Série T

Fascicules « Journée Annuelle »

Autres livres

Donald E. Knuth - traductions françaises

Rééditions du Séminaire Nicolas Bourbaki

Rééditions des Œuvres de Jean Leray

Revue de l'Institut Elie Cartan

Editions électroniques

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

Plus d'information / Abonnement

Publications grand public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

Pour les auteurs

Soumission des manuscrits

Formats et documentation

Plus d'info

Liste de diffusion électronique (smf.emath.fr)

Information pour les libraires et diffuseurs (smf.emath.fr)

Publications de la SMF
fr en
Votre numéro IP : 54.224.99.70
Accès aux édit. élec. : SémCong

Annales scientifiques de l'ENS

Présentation de la publication

Parutions

Dernières parutions

Comité de rédaction / Secrétariat

Série 4 :
Série 3 :
Série 2 :
Série 1 :

Faire une recherche


Catalogue & commande

Annales scientifiques de l'ENS - Parutions - série 4, 47 (2014)

Parutions < série 4, 47

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 47, fascicule 2 (2014)

Dawei Chen, Martin Möller
Quadratic differentials in low genus: exceptional and non-varying strata
Annales scientifiques de l'ENS 47, fascicule 2 (2014), 309-369

Télécharger cet article : Fichier PDF

Résumé :
Différentielles quadratiques en petit genre : strates exceptionnelles et strates sans variance
Nous présentons une façon algébrique de distinguer les composantes exceptionnelles des strates de l'espace de modules des différentielles quadratiques en genres trois et quatre. La liste complète de ces strates est (9,-1), (6,3,-1) et (3,3,3,-1) en genre trois, (12), (9,3), (6,6), (6,3,3) et (3,3,3,3) en genre quatre, respectivement. La distinction est basée sur des propriétés géométriques du modèle canonique de ces courbes. Ce résultat fait partie de la détermination de la somme des exposants de Lyapunov des courbes de Teichmüller, dans la continuité de [73], [48] et [49]. Pour beaucoup de strates en petit genre les courbes de Teichmüller sont disjointes des diviseurs de type Brill-Noether. On en déduit que la somme des exposants de Lyapunov de toute courbe de Teichmüller dans ces strates est égale à la somme des exposants pour la mesure à support sur toute la strate.

Mots-clefs : Courbe de Teichmüller, somme des exposants de Lyapunov, modèle canonique, diviseur de Brill-Noether, strates exceptionnelles.

Abstract:
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. The upshot of our method is a detailed study regarding the geometry of canonical curves. This result is part of a more general investigation about the sum of Lyapunov exponents of Teichmüller curves, building on [73], [48] and [49]. Using disjointness of Teichmüller curves with divisors of Brill-Noether type on the moduli space of curves, we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmüller geodesic flow is the same for all Teichmüller curves in that stratum.

Keywords: Teichmüller curve, sum of Lyapunov exponents, canonical model, Brill-Noether divisor, exceptional strata.

Class. math. : 14H10; 14H51, 37D40


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Abikoff, W.
Augmented Teichmüller spaces
Bull. Amer. Math. Soc. 82 (1976) 333–334
Math Reviews MR0432919
2
Abikoff, W.
Degenerating families of Riemann surfaces
Ann. of Math. 105 (1977) 29–44
Math Reviews MR0442293
3
Accola, R.
Differentials and extremal length on Riemann surfaces
Proc. Nat. Acad. Sci. U.S.A. 46 (1960) 540–543
Math Reviews MR0118829
4
Ahlfors, L.
The complex analytic structure of the space of closed Riemann surfaces.
in Analytic functions
(1960) 45–66
Math Reviews MR0124486
5
Ahlfors, L.
Lectures on quasiconformal mappings
D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966
Math Reviews MR0200442
6
Athreya, J.
Quantitative recurrence and large deviations for Teichmuller geodesic flow
Geom. Dedicata 119 (2006) 121–140
Math Reviews MR2247652
7
Arbarello, E. and Cornalba, M.
The Picard groups of the moduli spaces of curves
Topology 26 (1987) 153–171
Math Reviews MR895568 (88e:14032)
8
Arbarello, E. and Cornalba, M.
Calculating cohomology groups of moduli spaces of curves via algebraic geometry
Publ. Math. I.H.É.S. 88 (1998) 97–127
Math Reviews MR1733327 (2001h:14030)
9
Arbarello, E. and Cornalba, M. and Griffiths, P. A. and Harris, J.
Geometry of algebraic curves. Vol. I
Springer, 1985
Math Reviews MR770932 (86h:14019)
10
Avila, A. and Viana, M.
Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture
Acta Math. 198 (2007) 1–56
Math Reviews MR2316268 (2008m:37010)
11
Baily, W. Jr.
Satake's compactification of Vn
Amer. J. Math. 80 (1958) 348–364
Math Reviews MR0099451
12
Baily, W. Jr.
On the moduli of Jacobian varieties
Ann. of Math. 71 (1960) 303–314
Math Reviews MR0110715
13
Baily, W. Jr.
On the theory of -functions, the moduli of abelian varieties, and the moduli of curves
Ann. of Math. 75 (1962) 342–381
Math Reviews MR0162799
14
Bainbridge, Matt
Euler characteristics of Teichmüller curves in genus two
Geom. Topol. 11 (2007) 1887–2073
Math Reviews MR2350471
15
Bainbridge, Matt
Billiards in L-shaped tables with barriers
Geom. Funct. Anal. 20 (2010) 299–356
Math Reviews MR2671280 (2011h:32011)
16
Bainbridge, Matt and Möller, M.
The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three
(2009)
17
Barth, W. P. and Hulek, K. and Peters, C. A. M. and Van de Ven, A.
Compact complex surfaces
Springer, 2004
Math Reviews MR2030225 (2004m:14070)
18
Barth, W. P.
Abelian surfaces with (1,2)-polarization
in Algebraic geometry, Sendai, 1985
Adv. Stud. Pure Math. 10 (1987) 41–84
Math Reviews MR946234 (89i:14038)
19
Bass, H.
Torsion free and projective modules
Trans. Amer. Math. Soc. 102 (1962) 319–327
Math Reviews MR0140542
20
Beauville, Arnaud
Prym varieties and the Schottky problem
Invent. Math. 41 (1977) 149–196
Math Reviews MR0572974 (58 \#27995)
21
Belabas, K.
A fast algorithm to compute cubic fields
Math. Comp. 66 (1997) 1213–1237
Math Reviews MR1415795
22
Bers, L.
Correction to ``Spaces of Riemann surfaces as bounded domains''
Bull. Amer. Math. Soc. 67 (1961) 465–466
Math Reviews MR0130972
23
Bers, L.
Holomorphic differentials as functions of moduli
Bull. Amer. Math. Soc. 67 (1961) 206–210
Math Reviews MR0122989
24
Bers, L.
Spaces of degenerating Riemann surfaces
in Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973)
Ann. of Math. Studies 79 (1974) 43–55
Math Reviews MR0361051
25
Bers, L.
Fiber spaces over Teichmüller spaces
Acta. Math. 130 (1973) 89–126
Math Reviews MR0430318
26
Bers, L.
On spaces of Riemann surfaces with nodes
Bull. Amer. Math. Soc. 80 (1974) 1219–1222
Math Reviews MR0361165
27
Bers, L.
Finite-dimensional Teichmüller spaces and generalizations
Bull. Amer. Math. Soc. (N.S.) 5 (1981) 131–172
Math Reviews MR621883
28
Birkenhake, C. and Lange, H.
Complex Abelian Varieties
Springer, 2004
Math Reviews MR2062673
29
Birkenhake, C. and Lange, H.
The dual polarization of an abelian variety
Arch. Math. (Basel) 73 (1999) 380–389
Math Reviews MR1712138 (2000j:14069)
30
Bombieri, E. and Masser, D. and Zannier, U.
Intersecting a curve with algebraic subgroups of multiplicative groups
Int. Math. Res. Not. 1999 (1999) 1119–1140
Math Reviews MR1728021
31
Bondy, J. and Murty, U.
Graph theory with applications
American Elsevier Publishing Co., Inc., New York, 1976
Math Reviews MR0411988
32
Borel, A. and Ji, L.
Compactifications of symmetric and locally symmetric spaces
Birkhäuser, 2006
Math Reviews MR2189882
33
Bott, R. and Tu, L.
Differential Forms in Algebraic Topology
Springer, 1982
Math Reviews MR658304
34
Bourbaki, N.
Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés
Hermann, 1969
Math Reviews MR0276436
35
Borevich, A. and Shafarevich, I.
Number Theory
Academic Press, 1966
Math Reviews MR0195803
36
Bouw, I.
The p-rank of ramified covers of curves
Compositio Math. 126 (2001) 295–322
Math Reviews MR1834740 (2002e:14045)
37
Bouw, I. and Möller, M.
Teichmüller curves, triangle groups, and Lyapunov exponents
Ann. of Math. 172 (2010) 139–185
Math Reviews MR2680418
38
Bouw, I. and Möller, M.
Differential equations associated with nonarithmetic Fuchsian groups
J. Lond. Math. Soc. 81 (2010) 65–90
Math Reviews MR2580454
39
Calta, K.
Veech surfaces and complete periodicity in genus two
J. Amer. Math. Soc. 17 (2004) 871–908
Math Reviews MR2083470
40
Canuto, G.
Weierstrass points on trigonal curves of genus five
Bull. Soc. Math. France 113 (1985) 157–182
Math Reviews MR820317 (87c:14024)
41
Calta, K. and Wortman, K.
On unipotent flows in H(1,1)
42
Caporaso, L.
Linear series on semistable curves
43
Chai, C.-L.
Compactification of Siegel Moduli Schemes
Cambridge Univ. Press, 1985
Math Reviews MR853543
44
Charney, R. and Lee, R.
Cohomology of the Satake compactification
Topology 22 (1983) 389–423
Math Reviews MR715247
45
Charney, R. and Lee, R.
Moduli space of stable curves from a homotopy viewpoint
J. Differential Geom. 20 (1984) 185–235
Math Reviews MR772131
46
Chen, W. and Ruan, Y.
Orbifold Gromov-Witten theory
in Orbifolds in mathematics and physics (Madison, WI, 2001)
Contemp. Math. 310 (2002) 25–85
Math Reviews MR1950941
47
Chen, D.
Covers of elliptic curves and the moduli space of stable curves
J. reine angew. Math. 649 (2010) 167–205
48
Chen, D.
Square-tiled surfaces and rigid curves on moduli spaces
Adv. Math. 228 (2011) 1135–1162
49
Chen, D. and Moeller, M.
Non-varying sums of Lyapunov exponents of Abelian differentials in low genus
Geom. Topol. 16 (2012) 2427–2479
50
Morrison, I.
Mori theory of moduli spaces of stable curves
(2009)
51
Cohen, H.
Sums involving the values at negative integers of L-functions of quadratic characters
Math. Ann. 217 (1975) 271–285
Math Reviews MR0382192
52
Cohen, H.
Variations sur un thème de Siegel et Hecke
Acta Arith. 30 (1976/77) 63–93
Math Reviews MR0422215
53
Cohen, H.
Advanced topics in computational number theory
Springer, 2000
Math Reviews MR1728313
54
Cohen, P. and Wolfart, J.
Modular embeddings for some nonarithmetic Fuchsian groups
Acta Arith. 56 (1990) 93–110
Math Reviews MR1075639 (92d:11039)
55
Cornalba, M.
Moduli of curves and theta-characteristics
in Lectures on Riemann surfaces (Trieste, 1987)
(1989) 560–589
Math Reviews MR1082361 (91m:14037)
56
Cornalba, M. and Harris, J.
Divisor classes associated to families of stable varieties, with applications to the moduli space of curves
Ann. Sci. ENS 58 (1988) 455–475
57
Cukierman, F.
Families of Weierstrass points
Duke Math. J. 58 (1989) 317–346
Math Reviews MR1016424 (90g:14013)
58
Hubert, P. and Leliévre, S. and Troubetzkoy, S.
The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion
(2009)
59
Debarre, O.
Inégalités numériques pour les surfaces de type général
Bull. Soc. Math. France 110 (1982) 319–346
Math Reviews MR688038 (84f:14026)
60
Carlson, J. and Müller-Stach, S. and Peters, C. A. M.
Period mappings and period domains
Cambridge Univ. Press, 2003
Math Reviews MR2012297 (2005a:32014)
61
Delecroix, V. and Hubert, P. and Leliévre, S.
Diffusion for the periodic wind-tree model
(2011)
62
Deligne, P. and Mumford, D.
The irreducibility of the space of curves of given genus
Publ. Math. I.H.É.S. 36 (1969) 75–109
Math Reviews MR0262240
63
Deligne, P.
Un théorème de finitude pour la monodromie
in Discrete groups in geometry and analysis (New Haven, Conn., 1984)
Progr. Math. 67 (1987) 1–19
Math Reviews MR900821 (88h:14013)
64
Deligne, P.
Équations différentielles à points singuliers réguliers
Springer, 1970
Math Reviews MR0417174 (54 \#5232)
65
Deligne, P.
Théorie de Hodge. II
Publ. Math. I.H.É.S. 40 (1971) 5–57
Math Reviews MR0498551 (58 \#16653a)
66
Deligne, P.
Théorie de Hodge. III
Publ. Math. I.H.É.S. 44 (1974) 5–77
Math Reviews MR0498552 (58 \#16653b)
67
Douady, A. and Hubbard, J. H.
A proof of Thurston's topological characterization of rational functions
Acta Math. 171 (1993) 263–297
Math Reviews MR1251582
68
Eisenbud, D. and Harris, J.
The Kodaira dimension of the moduli space of curves of genus 23
Invent. Math. 90 (1987) 359–387
Math Reviews MR0910206
69
Eskin, A. and Marklof, J. and Morris, D.
Unipotent flows on the space of branched covers of Veech surfaces
Ergodic Theory Dynam. Systems 26 (2006) 129–162
Math Reviews MR2201941
70
Eskin, A. and Masur, H.
Asymptotic formulas on flat surfaces
Ergodic Theory Dynam. Systems 21 (2001) 443–478
Math Reviews MR1827113
71
Eskin, A. and Masur, H. and Schmoll, M.
Billiards in rectangles with barriers
Duke Math. J. 118 (2003) 427–463
Math Reviews MR1983037
72
Eskin, A. and Masur, H. and Zorich, A.
Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants
Publ. Math. I.H.É.S. 97 (2003) 61–179
Math Reviews MR2010740
73
Eskin, A. and Kontsevich, M. and Zorich, A.
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow
to appear in Publ. Math. IHÉS
74
Eskin, A. and Kontsevich, M. and Zorich, A.
Lyapunov spectrum of square-tiled cyclic covers
(2010)
75
Eskin, A. and McMullen, C.
Mixing, counting, and equidistribution in Lie groups
Duke Math. J. 71 (1993) 181–209
Math Reviews MR1230290
76
Eskin, A. and Okounkov, A.
Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
Invent. Math. 145 (2001) 59–103
Math Reviews MR1839286
77
Eskin, A. and Okounkov, A. and Pandharipande, R.
The theta characteristic of a branched covering
78
Faltings, G.
Arakelov's theorem for abelian varieties
Invent. Math. 73 (1983) 337–347
Math Reviews MR718934 (85m:14061)
79
Farkas, G. and Verra, A.
The geometry of the moduli space of odd spin curves
80
Farkas, G. and Popa, M.
Effective divisors on M_g, curves on K3 surfaces, and the slope conjecture
J. Algebraic Geom. 14 (2005) 241–267
Math Reviews MR2123229 (2006a:14043)
81
Farkas, G.
Birational aspects of the geometry of M_g
in Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces
Surv. Differ. Geom. 14 (2009) 57–110
Math Reviews MR2655323 (2011g:14065)
82
Farkas, H. M.
Unramified double coverings of hyperelliptic surfaces
J. Analyse Math. 30 (1976) 150–155
Math Reviews MR0437741 (55 \#10664)
83
Farkas, G.
Koszul divisors on moduli spaces of curves
Amer. J. Math. 131 (2009) 819–867
Math Reviews MR2530855 (2010f:14030)
84
Farkas, G.
The birational type of the moduli space of even spin curves
Adv. Math. 223 (2010) 433–443
Math Reviews MR2565536
85
Farb, B. and Masur, H.
Superrigidity and mapping class groups
Topology 37 (1998) 1169–1176
Math Reviews MR1632912 (99f:57017)
86
Forni, G.
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
Ann. of Math. 155 (2002) 1–103
Math Reviews MR1888794
87
Forni, G.
On the Lyapunov exponents of the Kontsevich-Zorich cocycle
in Handbook of dynamical systems. Vol. 1B
(2006) 549–580
Math Reviews MR2186248 (2010c:37004)
88
Forni, G. and Matheus, C.
An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum
(2008)
89
Freitag, E.
Hilbert modular forms
Springer, 1990
Math Reviews MR1050763
90
van der Geer, G.
Hilbert Modular Surfaces
Springer, 1988
Math Reviews MR930101
91
Gerritzen, L.
Equations defining the periods of totally degenerate curves
Israel J. Math. 77 (1992) 187–210
Math Reviews MR1194792
92
Grauert, H.
Über Modifikationen und exzeptionelle analytische Mengen
Math. Ann. 146 (1962) 331–368
Math Reviews MR0137127
93
Griffiths, P. A. and Harris, J.
Principles of Algebraic Geometry
Wiley-Interscience, 1978 Pure and Applied Mathematics
Math Reviews MR507725
94
Griffiths, P. A.
Periods of integrals on algebraic manifolds. II. Local study of the period mapping
Amer. J. Math. 90 (1968) 805–865
Math Reviews MR0233825 (38 \#2146)
95
Goren, E.
Lectures on Hilbert modular varieties and modular forms
Amer. Math. Soc., 2002
Math Reviews MR1863355
96
Gunning, R.
Lectures on Complex Analytic Varieties: The Local parametrization theorem
Princeton Univ. Press, 1970
Math Reviews MR0273060
97
Gunning, R. and Rossi, H.
Analytic Functions of Several Complex Variables
Prentice-Hall Inc., 1965
Math Reviews MR0180696
98
Gutkin, E. and Judge, C.
Affine mappings of translation surfaces: geometry and arithmetic
Duke Math. J. 103 (2000) 191–213
Math Reviews MR1760625
99
Habegger, P.
Intersecting subvarieties of <b>G</b>nm with algebraic subgroups
Math. Ann. 342 (2008) 449–466
Math Reviews MR2425150
100
Hain, R.
Locally symmetric families of curves and Jacobians
in Moduli of curves and abelian varieties
Aspects Math., E33 (1999) 91–108
Math Reviews MR1722540 (2000i:14040)
101
Harer, J. and Zagier, D.
The Euler characteristic of the moduli space of curves
Invent. Math. 85 (1986) 457–485
Math Reviews MR848681
102
Harris, J. and Morrison, I.
Moduli of Curves
Springer, 1998
Math Reviews MR1631825
103
Harris, J. and Mumford, D.
On the Kodaira dimension of the moduli space of curves
Invent. Math. 67 (1982) 23–88
Math Reviews MR664324 (83i:14018)
104
Harris, J. and Morrison, I.
Slopes of effective divisors on the moduli space of stable curves
Invent. Math. 99 (1990) 321–355
Math Reviews MR1031904 (91d:14009)
105
Hartshorne, R.
Residues and Duality
Springer, 1966
Math Reviews MR0222093
106
Hartshorne, R.
Algebraic Geometry
Springer, 1977 Graduate Texts in Mathematics, No. 52
Math Reviews MR0463157
107
Hassett, B.
Stable log surfaces and limits of quartic plane curves
Manuscripta Math. 100 (1999) 469–487
Math Reviews MR1734796 (2000j:14045)
108
Hindry, M.and Silverman, J. H.
Diophantine geometry
Springer, 2000 An introduction
Math Reviews MR1745599
109
Herrlich, F. and Schmithüsen, G.
An extraordinary origami curve
Math. Nachr. 281 (2008) 219–237
Math Reviews MR2387362 (2008k:14059)
110
Hirzebruch, F.
Hilbert modular surfaces
Enseignement Math. 19 (1973) 183–281
Math Reviews MR0393045
111
Hubbard, J. H.
Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1
Matrix Editions, Ithaca, NY, 2006
Math Reviews MR2245223 (2008k:30055)
112
Hubbard, J. H. and Masur, H.
Quadratic differentials and foliations
Acta Math. 142 (1979) 221–274
Math Reviews MR523212 (80h:30047)
113
Hubert, P. and Lanneau, E.
Veech groups without parabolic elements
Duke Math. J. 133 (2006) 335–346
Math Reviews MR2225696
114
Hubert, P. and Lelièvre, S.
Prime arithmetic Teichmüller discs in H(2)
Israel J. Math. 151 (2006) 281–321
Math Reviews MR2214127
115
Ireland, K. and Rosen, M.
A Classical Introduction to Modern Number Theory
Springer, 1990
Math Reviews MR1070716
116
de Jong, J. and Zhang, S.-W.
Generic abelian varieties with real multiplication are not Jacobians
in Diophantine geometry
CRM Series 4 (2007) 165–172
Math Reviews MR2349653
117
Kani, E.
Hurwitz spaces of genus 2 covers of an elliptic curve
Collect. Math. 54 (2003) 1–51
Math Reviews MR1962943
118
Kenyon, R. and Smillie, J.
Billiards on rational-angled triangles
Comment. Math. Helv. 75 (2000) 65–108
Math Reviews MR1760496
119
Kerckhoff, S. and Masur, H. and Smillie, J.
Ergodicity of billiard flows and quadratic differentials
Ann. of Math. 124 (1986) 293–311
Math Reviews MR855297
120
Knudsen, F. and Mumford, D.
The projectivity of the moduli space of stable curves. I. Preliminaries on ``det'' and ``Div''
Math. Scand. 39 (1976) 19–55
Math Reviews MR0437541
121
Kobayashi, S. and Nomizu, K.
Foundations of Differential Geometry. Vol I
Interscience Publishers, New York-London, 1963
Math Reviews MR0152974
122
Kodaira, K.
On compact analytic surfaces. II, III
Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963) 1–40
Math Reviews MR0184257 (32 \#1730)
123
Kontsevich, M.
Lyapunov exponents and Hodge theory
in The mathematical beauty of physics (Saclay, 1996)
Adv. Ser. Math. Phys. 24 (1997) 318–332
Math Reviews MR1490861
124
Kontsevich., M. and Zorich, A.
Lyapunov exponents and Hodge theory
(1997)
125
Kontsevich, M. and Zorich, A.
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Invent. Math. 153 (2003) 631–678
Math Reviews MR2000471
126
Kra, I.
Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups
J. Amer. Math. Soc. 3 (1990) 499–578
Math Reviews MR1049503
127
Laufer, H.
Normal Two-Dimensional Singularities
Princeton Univ. Press, 1971 Annals of Mathematics Studies, No. 71
Math Reviews MR0320365
128
Laurent, M.
Équations diophantiennes exponentielles
Invent. Math. 78 (1984) 299–327
Math Reviews MR767195
129
Lanneau, E.
Connected components of the strata of the moduli spaces of quadratic differentials
Ann. Sci. Éc. Norm. Supér. 41 (2008) 1–56
Math Reviews MR2423309 (2009e:30094)
130
Lanneau, E.
Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities
Comment. Math. Helv. 79 (2004) 471–501
Math Reviews MR2081723 (2005g:32015)
131
Lelièvre, S.
Siegel-Veech constants in H(2)
Geom. Topol. 10 (2006) 1157-1172
132
Leliévre, S. and Royer, E.
Orbitwise countings in H(2) and quasimodular forms
International Mathematics Research Notices 2006 (2006) Article ID 42151, 30 pages
133
Logan, A.
The Kodaira dimension of moduli spaces of curves with marked points
Amer. J. Math. 125 (2003) 105–138
Math Reviews MR1953519 (2003j:14035)
134
Lanneau, E. and Manh, D.-N.
Teichmüller curves generated by WeierstraPrym eigenforms in genus three
(2011)
135
Masur, H.
On a class of geodesics in Teichmüller space
Ann. of Math. 102 (1975) 205–221
Math Reviews MR0385173
136
Masur, H.
Extension of the Weil-Petersson metric to the boundary of Teichmuller space
Duke Math. J. 43 (1976) 623–635
Math Reviews MR0417456
137
Masur, H.
The growth rate of trajectories of a quadratic differential
Ergodic Theory Dynam. Systems 10 (1990) 151–176
Math Reviews MR1053805
138
McMullen, C.
Billiards and Teichmüller curves on Hilbert modular surfaces
J. Amer. Math. Soc. 16 (2003) 857–885
Math Reviews MR1992827
139
McMullen, C.
Dynamics of SL_2(R) over moduli space in genus two
Ann. of Math. 165 (2007) 397–456
140
McMullen, C.
Teichmüller curves in genus two: the decagon and beyond
J. reine angew. Math. 582 (2005) 173–199
Math Reviews MR2139715
141
McMullen, C.
Foliations of Hilbert modular surfaces
Amer. J. Math. 129 (2007) 183–215
Math Reviews MR2288740
142
McMullen, C.
Teichmüller geodesics of infinite complexity
Acta Math. 191 (2003) 191–223
Math Reviews MR2051398
143
McMullen, C.
Prym varieties and Teichmüller curves
Duke Math. J. 133 (2006) 569–590
Math Reviews MR2228463
144
McMullen, C.
Teichmüller curves in genus two: Discriminant and spin
Math. Ann. 333 (2005) 87–130
145
McMullen, C.
Teichmüller curves in genus two: torsion divisors and ratios of sines
Invent. Math. 165 (2006) 651–672
Math Reviews MR2242630
146
Marden, A.
Geometric complex coordinates for Teichmüller space
in Mathematical aspects of string theory (San Diego, Calif., 1986)
Adv. Ser. Math. Phys. 1 (1987) 341–354
Math Reviews MR915828
147
Masur, H.
Interval exchange transformations and measured foliations
Ann. of Math. 115 (1982) 169–200
Math Reviews MR644018
148
Masur, H. and Tabachnikov, S.
Rational billiards and flat structures
in Handbook of dynamical systems, Vol.1A
(2002) 1015–1089
Math Reviews MR1928530
149
Miranda, R.
The basic theory of elliptic surfaces
ETS Editrice, 1989
Math Reviews MR1078016 (92e:14032)
150
Miyake, T.
Modular Forms
Springer, 1989
Math Reviews MR1021004
151
Möller, M.
Shimura and Teichmüller curves
J. Mod. Dyn. 5 (2011) 1–32
Math Reviews MR2787595
152
Möller, M.
Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve
Invent. Math. 165 (2006) 633–649
Math Reviews MR2242629
153
Möller, M.
Affine groups of flat surfaces
in Handbook of Teichmüller theory. Vol. II
IRMA Lect. Math. Theor. Phys. 13 (2009) 369–387
Math Reviews MR2497782 (2011a:32023)
154
Möller, M.
Variations of Hodge structures of a Teichmüller curve
J. Amer. Math. Soc. 19 (2006) 327–344
Math Reviews MR2188128
155
Möller, M.
Finiteness results for Teichmüller curves
Ann. Inst. Fourier (Grenoble) 58 (2008) 63–83
Math Reviews MR2401216
156
Möller, M. and Viehweg, E.
Kobayashi geodesics in A_g
J. Differential Geom. 86 (2010) 355–579
Math Reviews MR2772554
157
Möller, M.
Teichmüller curves, mainly from the viewpoint of algebraic geometry
AMS, 2013
158
Möller, M. and Zagier, D.
Theta derivatives and Teichmüller curves
(2011)
159
Moonen, B. and Oort, Frans
The Torelli locus and special subvarieties.
(2010)
160
Mumford, D.
The topology of normal singularities of an algebraic surface and a criterion for simplicity
Publ. Math. I.H.É.S. 9 (1961) 5–22
Math Reviews MR0153682
161
Mumford, D.
The structure of the moduli spaces of curves and Abelian varieties
in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1
(1971) 457–465
Math Reviews MR0441983
162
Mumford, D.
Prym varieties. I
in Contributions to analysis (a collection of papers dedicated to Lipman Bers)
(1974) 325–350
Math Reviews MR0379510 (52 \#415)
163
Mumford, D.
Hirzebruch's proportionality theorem in the noncompact case
Invent. Math. 42 (1977) 239–272
Math Reviews MR471627
164
Mumford, D.
Towards an enumerative geometry of the moduli space of curves
in Arithmetic and geometry, Vol. II
Progr. Math. 36 (1983) 271–328
Math Reviews MR717614
165
Mumford, D.
The Red Book of Varieties and Schemes
Springer, 1999
Math Reviews MR1748380
166
Oort, Frans and Steenbrink, Joseph
The local Torelli problem for algebraic curves
in Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979
(1980) 157–204
Math Reviews MR605341 (82i:14014)
167
Namikawa, Y.
On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform
Nagoya Math. J. 52 (1973) 197–259
Math Reviews MR0337981
168
Philippon, P.
Sur des hauteurs alternatives. III
J. Math. Pures Appl. 74 (1995) 345–365
Math Reviews MR1341770
169
Ratner, M.
Invariant measures and orbit closures for unipotent actions on homogeneous spaces
Geom. Funct. Anal. 4 (1994) 236–257
Math Reviews MR1262705
170
Roman, S.
Advanced linear algebra
Springer, 1992
Math Reviews MR1169104
171
Royden, H. L.
The Ahlfors-Schwarz lemma in several complex variables
Comment. Math. Helv. 55 (1980) 547–558
Math Reviews MR604712 (82i:32049)
172
Satake, I.
On a generalization of the notion of manifold
Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 359–363
Math Reviews MR0079769
173
Satake, I.
On the compactification of the Siegel space
J. Indian Math. Soc. (N.S.) 20 (1956) 259–281
Math Reviews MR0084842
174
Schmithüsen, G.
An algorithm for finding the Veech group of an origami
Experiment. Math. 13 (2004) 459–472
Math Reviews MR2118271 (2006b:30080)
175
Schmoll, M.
Spaces of elliptic differentials
in Algebraic and topological dynamics
Contemp. Math. 385 (2005) 303–320
Math Reviews MR2180242
176
Schmid, W.
Variation of Hodge structure: the singularities of the period mapping
Invent. Math. 22 (1973) 211–319
Math Reviews MR0382272 (52 \#3157)
177
Serre, J. P.
Géométrie algébrique et géométrie analytique
Ann. Inst. Fourier, Grenoble 6 (1955–1956) 1–42
Math Reviews MR0082175
178
Serre, J. P.
Groupes algébriques et corps de classes
Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris, 1959
Math Reviews MR0103191
179
Siegel, C.
Über die analytische Theorie der quadratischen Formen
Ann. of Math. 36 (1935) 527–606
Math Reviews MR1503238
180
Siegel, C.
The volume of the fundamental domain for some infinite groups
Trans. Amer. Math. Soc. 39 (1936) 209–218
Math Reviews MR1501843
181
Siegel, C.
Berechnung von Zetafunktionen an ganzzahligen Stellen
Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969) 87–102
Math Reviews MR0252349
182
Stankova-Frenkel, Zvezdelina E.
Moduli of trigonal curves
J. Algebraic Geom. 9 (2000) 607–662
Math Reviews MR1775309 (2001h:14031)
183
Smillie, J. and Weiss, B.
Minimal sets for flows on moduli space
Israel J. Math. 142 (2004) 249–260
Math Reviews MR2085718 (2005g:37067)
184
Thurston, W. P.
On the geometry and dynamics of diffeomorphisms of surfaces
Bull. Amer. Math. Soc. (N.S.) 19 (1988) 417–431
Math Reviews MR956596 (89k:57023)
185
Trapani, S.
On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces
Math. Ann. 293 (1992) 681–705
186
Veech, W. A.
Gauss measures for transformations on the space of interval exchange maps
Ann. of Math. 115 (1982) 201–242
Math Reviews MR644019 (83g:28036b)
187
Veech, W. A.
Geometric realizations of hyperelliptic curves
in Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992)
(1995) 217–226
Math Reviews MR1402493 (98f:14022)
188
Veech, W. A.
The Teichmüller geodesic flow
Ann. of Math. 124 (1986) 441–530
Math Reviews MR866707
189
Veech, W. A.
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
Invent. Math. 97 (1989) 553–583
Math Reviews MR1005006
190
Veech, W. A.
Moduli spaces of quadratic differentials
J. Analyse Math. 55 (1990) 117–171
Math Reviews MR1094714
191
Veech, W. A.
Erratum: ``Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards''
Invent. Math. 103 (1991) 447
Math Reviews MR1085115
192
Veech, W. A.
The billiard in a regular polygon
Geom. Funct. Anal. 2 (1992) 341–379
Math Reviews MR1177316
193
Veech, W. A.
Siegel measures
Ann. of Math. 148 (1998) 895–944
Math Reviews MR1670061
194
Vorobets, Ya. B.
Plane structures and billiards in rational polygons: the Veech alternative
Uspekhi Mat. Nauk 51 (1996) 3–42
Math Reviews MR1436653 (97j:58092)
195
Viana, M.
IMPA lecture notes on Lyapunov exponents
(2010)
196
Xiao, G.
Fibered algebraic surfaces with low slope
Math. Ann. 276 (1987) 449–466
Math Reviews MR875340 (88a:14046)
197
Xiao, G.
Surfaces fibrées en courbes de genre deux
Springer, 1985
Math Reviews MR872271 (88a:14042)
198
Viehweg, E. and Zuo, Kang
A characterization of certain Shimura curves in the moduli stack of abelian varieties
J. Differential Geom. 66 (2004) 233–287
Math Reviews MR2106125 (2006a:14015)
199
Ward, C.
Calculation of Fuchsian groups associated to billiards in a rational triangle
Ergodic Theory Dynam. Systems 18 (1998) 1019–1042
Math Reviews MR1645350
200
Weiss, C.
Twisted Teichmüller curves
(2012)
201
Whitney, H.
Complex Analytic Varieties
Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972
Math Reviews MR0387634
202
Wolpert, S.
On obtaining a positive line bundle from the Weil-Petersson class
Amer. J. Math. 107 (1985) 1485–1507 (1986)
Math Reviews MR815769
203
Wolpert, S.
Riemann surfaces, moduli and hyperbolic geometry
in Lectures on Riemann surfaces (Trieste, 1987)
(1989) 48–98
Math Reviews MR1082351
204
Yu, Fei and Zuo, Kang
Weierstrass filtration on Teichmüller curves and Lyapunov exponents
J. Mod. Dyn. 7 (2013) 209–237
Math Reviews MR3106711
205
Zariski, O. and Samuel, P.
Commutative Algebra. Vol. II
D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960
Math Reviews MR0120249
206
Zorich, A.
Asymptotic flag of an orientable measured foliation on a surface
in Geometric study of foliations (Tokyo, 1993)
(1994) 479–498
Math Reviews MR1363744
207
Zorich, A.
How do the leaves of a closed 1-form wind around a surface?
in Pseudoperiodic Topology
Amer. Math. Soc. Transl. Ser. 2 197 (1999) 135–178
Math Reviews MR1733872
208
Zorich, A.
Flat surfaces
in Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems
(2006) 439–586
209
Zorich, A.
Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials.
Journ. Modern Dyn. 2 (208) 139–185
210
211
Zucker, S.
Hodge theory with degenerating coefficients. L_2 cohomology in the Poincaré metric
Ann. of Math. 109 (1979) 415–476
Math Reviews MR534758 (81a:14002)