American Institute of Mathematical Sciences

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

Strata of abelian differentials and the Teichmüller dynamics

Dawei Chen 1,

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.
Keywords: Abelian differentials, moduli space of curves..
Mathematics Subject Classification: Primary: 14H10; Secondary: 14C1.
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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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).   Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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).   Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[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]

Sang-hyun Kim, Thomas Koberda. Integrability of moduli and regularity of denjoy counterexamples. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 6061-6088. doi: 10.3934/dcds.2020259
[13]

Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284
[14]

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
[15]

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
[16]

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
[17]

Sandra Carillo, Mauro Lo Schiavo, Cornelia Schiebold. Abelian versus non-Abelian Bäcklund charts: Some remarks. Evolution Equations & Control Theory, 2019, 8 (1) : 43-55. doi: 10.3934/eect.2019003
[18]

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
[19]

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
[20]

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

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences