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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, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985.  Google Scholar

[2]

D. Chen, Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., 649 (2010), 167-205. doi: 10.1515/CRELLE.2010.092.  Google Scholar

[3]

D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162. 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), 2427-2479. 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), 317-346. 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), 507-514. 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, arXiv:1112.5872, (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), 61-179. 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, Springer-Verlag, New York, 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, Invent. Math., 67 (1982), 23-88. 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), 633-642. 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), 2863-2879. 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), 586-600. doi: 10.1016/j.aim.2011.02.005.  Google Scholar

[19]

M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, (1997), 318-332.  Google Scholar

[20]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. 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), 447-458.  Google Scholar

[22]

R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004.  Google Scholar

[23]

A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math., 125 (2003), 105-138. 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, The University of Texas at Austin, 2001.  Google Scholar

[25]

B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$, Michigan Math. J., 61 (2012), 359-383. 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), 1297-1307.  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 "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583. 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, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985.  Google Scholar

[2]

D. Chen, Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., 649 (2010), 167-205. doi: 10.1515/CRELLE.2010.092.  Google Scholar

[3]

D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162. 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), 2427-2479. 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), 317-346. 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), 507-514. 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, arXiv:1112.5872, (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), 61-179. 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, Springer-Verlag, New York, 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, Invent. Math., 67 (1982), 23-88. 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), 633-642. 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), 2863-2879. 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), 586-600. doi: 10.1016/j.aim.2011.02.005.  Google Scholar

[19]

M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, (1997), 318-332.  Google Scholar

[20]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. 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), 447-458.  Google Scholar

[22]

R. Lazarsfeld, "Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004.  Google Scholar

[23]

A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math., 125 (2003), 105-138. 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, The University of Texas at Austin, 2001.  Google Scholar

[25]

B. Thomas, Excess porteous, coherent porteous, and the hyperelliptic locus in $\overline{\mathcal M}_3$, Michigan Math. J., 61 (2012), 359-383. 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), 1297-1307.  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 "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

[29]

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

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