Strata of abelian differentials and the Teichmüller dynamics

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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

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

Received January 2013 Published May 2013

Abstract
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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.
[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.
[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.
[4]	D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.
[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), 317-346. doi: 10.1215/S0012-7094-89-05815-8.
[8]	S. Diaz, Porteous's formula for maps between coherent sheaves, Michigan Math. J., 52 (2004), 507-514. 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, arXiv:1112.5872, (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), 61-179. 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, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett., 18 (2011), 447-458.
[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.
[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.
[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.
[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.
[26]	G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces, Pure Appl. Math. Q., 7 (2011), 1297-1307.
[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 "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13.
[29]	, D. Zvonkine,, personal communication., ().

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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.
[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.
[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.
[4]	D. Chen and M. Moeller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.
[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), 317-346. doi: 10.1215/S0012-7094-89-05815-8.
[8]	S. Diaz, Porteous's formula for maps between coherent sheaves, Michigan Math. J., 52 (2004), 507-514. 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, arXiv:1112.5872, (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), 61-179. 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, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett., 18 (2011), 447-458.
[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.
[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.
[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.
[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.
[26]	G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces, Pure Appl. Math. Q., 7 (2011), 1297-1307.
[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 "Frontiers in Number Theory, Physics and Geometry. 1," Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13.
[29]	, D. Zvonkine,, personal communication., ().

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