2013, 7(1): 135-152. doi: 10.3934/jmd.2013.7.135

Strata of abelian differentials and the Teichmüller dynamics

1. 

Department of Mathematics, Boston College, Chestnut Hill, MA 02467, United States

Received  January 2013 Published  May 2013

This paper focuses on the interplay between the intersection theory and the Teichmüller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a nonsimple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.
Citation: Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
References:
[1]

E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves,", Vol. I, 267 (1985).

[2]

D. Chen, Covers of elliptic curves and the moduli space of stable curves,, J. Reine Angew. Math., 649 (2010), 167. doi: 10.1515/CRELLE.2010.092.

[3]

D. Chen, Square-tiled surfaces and rigid curves on moduli spaces,, Adv. Math., 228 (2011), 1135. doi: 10.1016/j.aim.2011.06.002.

[4]

D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus,, Geom. Topol., 16 (2012), 2427.

[5]

D. Chen and M. Moeller, Quadratic differentials in low genus: Exceptional and non-varying strata,, \arXiv{1204.1707}, (2012).

[6]

D. Chen, M. Moeller and D. Zagier, Siegel-Veech constants and quasimodular forms,, in preparation., ().

[7]

F. Cukierman, Families of Weierstrass points,, Duke Math. J., 58 (1989), 317. doi: 10.1215/S0012-7094-89-05815-8.

[8]

S. Diaz, Porteous's formula for maps between coherent sheaves,, Michigan Math. J., 52 (2004), 507. doi: 10.1307/mmj/1100623410.

[9]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint, (2011).

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants,, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61. doi: 10.1007/s10240-003-0015-1.

[11]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2,\mathbb R)$ action on Moduli space,, \arXiv{1302.3320}, (2013).

[12]

G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves,, to appear in Comm. Math. Helv., ().

[13]

U. Hamenstädt, Signatures of surface bundles and Milnor Wood inequalities,, \arXiv{1206.0263}, (2012).

[14]

J. Harris and I. Morrison, "Moduli of Curves,", Graduate Texts in Mathematics, 187 (1998).

[15]

J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves,, With an appendix by William Fulton, 67 (1982), 23. doi: 10.1007/BF01393371.

[16]

David Jensen, Rational fibrations of $\overlineM_{5,1}$ and $\overlineM_{6,1}$,, J. Pure Appl. Algebra, 216 (2012), 633. doi: 10.1016/j.jpaa.2011.07.015.

[17]

David Jensen, Birational contractions of $\overlineM_{3,1}$ and $\overlineM_{4,1}$,, Trans. Amer. Math. Soc., 365 (2013), 2863. doi: 10.1090/S0002-9947-2012-05581-4.

[18]

A. Kokotov, D. Korotkin and P. Zograf, Isomonodromic tau function on the space of admissible covers,, Adv. Math., 227 (2011), 586. doi: 10.1016/j.aim.2011.02.005.

[19]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318.

[20]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x.

[21]

D. Korotkin and P. Zograf, Tau function and moduli of differentials,, Math. Res. Lett., 18 (2011), 447.

[22]

R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 48 (2004).

[23]

A. Logan, The Kodaira dimension of moduli spaces of curves with marked points,, Amer. J. Math., 125 (2003), 105. doi: 10.1353/ajm.2003.0005.

[24]

W. Rulla, "The Birational Geometry of Moduli Space $M(3)$ and Moduli Space $M(2,1)$,", Ph.D. Thesis, (2001).

[25]

B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$,, Michigan Math. J., 61 (2012), 359. doi: 10.1307/mmj/1339011531.

[26]

G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces,, Pure Appl. Math. Q., 7 (2011), 1297.

[27]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents,, to appear in J. Mod. Dyn., ().

[28]

A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

[29]

, D. Zvonkine,, personal communication., ().

show all references

References:
[1]

E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves,", Vol. I, 267 (1985).

[2]

D. Chen, Covers of elliptic curves and the moduli space of stable curves,, J. Reine Angew. Math., 649 (2010), 167. doi: 10.1515/CRELLE.2010.092.

[3]

D. Chen, Square-tiled surfaces and rigid curves on moduli spaces,, Adv. Math., 228 (2011), 1135. doi: 10.1016/j.aim.2011.06.002.

[4]

D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus,, Geom. Topol., 16 (2012), 2427.

[5]

D. Chen and M. Moeller, Quadratic differentials in low genus: Exceptional and non-varying strata,, \arXiv{1204.1707}, (2012).

[6]

D. Chen, M. Moeller and D. Zagier, Siegel-Veech constants and quasimodular forms,, in preparation., ().

[7]

F. Cukierman, Families of Weierstrass points,, Duke Math. J., 58 (1989), 317. doi: 10.1215/S0012-7094-89-05815-8.

[8]

S. Diaz, Porteous's formula for maps between coherent sheaves,, Michigan Math. J., 52 (2004), 507. doi: 10.1307/mmj/1100623410.

[9]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint, (2011).

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants,, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61. doi: 10.1007/s10240-003-0015-1.

[11]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2,\mathbb R)$ action on Moduli space,, \arXiv{1302.3320}, (2013).

[12]

G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves,, to appear in Comm. Math. Helv., ().

[13]

U. Hamenstädt, Signatures of surface bundles and Milnor Wood inequalities,, \arXiv{1206.0263}, (2012).

[14]

J. Harris and I. Morrison, "Moduli of Curves,", Graduate Texts in Mathematics, 187 (1998).

[15]

J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves,, With an appendix by William Fulton, 67 (1982), 23. doi: 10.1007/BF01393371.

[16]

David Jensen, Rational fibrations of $\overlineM_{5,1}$ and $\overlineM_{6,1}$,, J. Pure Appl. Algebra, 216 (2012), 633. doi: 10.1016/j.jpaa.2011.07.015.

[17]

David Jensen, Birational contractions of $\overlineM_{3,1}$ and $\overlineM_{4,1}$,, Trans. Amer. Math. Soc., 365 (2013), 2863. doi: 10.1090/S0002-9947-2012-05581-4.

[18]

A. Kokotov, D. Korotkin and P. Zograf, Isomonodromic tau function on the space of admissible covers,, Adv. Math., 227 (2011), 586. doi: 10.1016/j.aim.2011.02.005.

[19]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318.

[20]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x.

[21]

D. Korotkin and P. Zograf, Tau function and moduli of differentials,, Math. Res. Lett., 18 (2011), 447.

[22]

R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 48 (2004).

[23]

A. Logan, The Kodaira dimension of moduli spaces of curves with marked points,, Amer. J. Math., 125 (2003), 105. doi: 10.1353/ajm.2003.0005.

[24]

W. Rulla, "The Birational Geometry of Moduli Space $M(3)$ and Moduli Space $M(2,1)$,", Ph.D. Thesis, (2001).

[25]

B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$,, Michigan Math. J., 61 (2012), 359. doi: 10.1307/mmj/1339011531.

[26]

G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces,, Pure Appl. Math. Q., 7 (2011), 1297.

[27]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents,, to appear in J. Mod. Dyn., ().

[28]

A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

[29]

, D. Zvonkine,, personal communication., ().

[1]

Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433

[2]

Ferrán Valdez. Veech groups, irrational billiards and stable abelian differentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1055-1063. doi: 10.3934/dcds.2012.32.1055

[3]

Jonathan Chaika, Yitwah Cheung, Howard Masur. Winning games for bounded geodesics in moduli spaces of quadratic differentials. Journal of Modern Dynamics, 2013, 7 (3) : 395-427. doi: 10.3934/jmd.2013.7.395

[4]

Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71

[5]

Anton Zorich. Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials. Journal of Modern Dynamics, 2008, 2 (1) : 139-185. doi: 10.3934/jmd.2008.2.139

[6]

Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619

[7]

Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125

[8]

Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 53-59.

[9]

Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405

[10]

Christopher Kumar Anand. Unitons and their moduli. Electronic Research Announcements, 1996, 2: 7-16.

[11]

V. Balaji, P. Barik, D. S. Nagaraj. On degenerations of moduli of Hitchin pairs. Electronic Research Announcements, 2013, 20: 103-108. doi: 10.3934/era.2013.20.105

[12]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511

[13]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443

[14]

Julien Grivaux, Pascal Hubert. Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum. Journal of Modern Dynamics, 2014, 8 (1) : 61-73. doi: 10.3934/jmd.2014.8.61

[15]

Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933

[16]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139

[17]

S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361

[18]

Lisa C. Jeffrey and Frances C. Kirwan. Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electronic Research Announcements, 1995, 1: 57-71.

[19]

Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155

[20]

Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma. Moduli of 3-dimensional diffeomorphisms with saddle-foci. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5021-5037. doi: 10.3934/dcds.2018220

2017 Impact Factor: 0.425

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]