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April  2013, 7(2): 209-237. doi: 10.3934/jmd.2013.7.209

Weierstrass filtration on Teichmüller curves and Lyapunov exponents

Fei Yu 1, and  Kang Zuo 2,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
2. 

Fachbereich 08-Physik Mathematik und Informatik, Universität Mainz, 55099 Mainz, Germany

Received  September 2012 Published  September 2013

We define the Weierstrass filtration for Teichmüller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmüller curves in these strata.
Keywords: Teichmüller curves, Harder-Narasimhan filtration, Lyapunov exponents.
Mathematics Subject Classification: Primary: 32G15; Secondary: 14H1.
Citation: Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209
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]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.  Google Scholar

[3]

E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009.  Google Scholar

[4]

I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents, Ann of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.  Google Scholar

[5]

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

[6]

D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.  Google Scholar

[7]

D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().   Google Scholar

[8]

A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.  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,, \arXiv{1112.5872}., ().   Google Scholar

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems and the Siegel-Veech constats, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1.  Google Scholar

[11]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285.  Google Scholar

[12]

R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[13]

D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.  Google Scholar

[14]

M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().   Google Scholar

[15]

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

[16]

E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().   Google Scholar

[17]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885. doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

[18]

C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar

[19]

C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215. doi: 10.1353/ajm.2007.0002.  Google Scholar

[20]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar

[21]

M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().   Google Scholar

[22]

M. Möller, Teichmüller curves, mainly from the view point of algebraic geometry,, to appear as PCMI Lecture Notes. Available from: \url{http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/moeller/summaries/PCMI.pdf}., ().   Google Scholar

[23]

E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties, J. Diff. Geometry, 66 (2004), 233-287.  Google Scholar

[24]

G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466. doi: 10.1007/BF01450841.  Google Scholar

[25]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().   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]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.  Google Scholar

[3]

E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009.  Google Scholar

[4]

I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents, Ann of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.  Google Scholar

[5]

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

[6]

D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.  Google Scholar

[7]

D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().   Google Scholar

[8]

A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.  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,, \arXiv{1112.5872}., ().   Google Scholar

[10]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems and the Siegel-Veech constats, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1.  Google Scholar

[11]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285.  Google Scholar

[12]

R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[13]

D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.  Google Scholar

[14]

M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().   Google Scholar

[15]

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

[16]

E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().   Google Scholar

[17]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885. doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

[18]

C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar

[19]

C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215. doi: 10.1353/ajm.2007.0002.  Google Scholar

[20]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar

[21]

M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().   Google Scholar

[22]

M. Möller, Teichmüller curves, mainly from the view point of algebraic geometry,, to appear as PCMI Lecture Notes. Available from: \url{http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/moeller/summaries/PCMI.pdf}., ().   Google Scholar

[23]

E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties, J. Diff. Geometry, 66 (2004), 233-287.  Google Scholar

[24]

G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466. doi: 10.1007/BF01450841.  Google Scholar

[25]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().   Google Scholar

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