Weierstrass filtration on Teichmüller curves and Lyapunov exponents

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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

Abstract
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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.
[2]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.
[3]	E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009.
[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.
[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.
[6]	D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.
[7]	D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().
[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.
[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}., ().
[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.
[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.
[12]	R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.
[13]	D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.
[14]	M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().
[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.
[16]	E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().
[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.
[18]	C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.
[19]	C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215. doi: 10.1353/ajm.2007.0002.
[20]	M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.
[21]	M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().
[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}., ().
[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.
[24]	G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466. doi: 10.1007/BF01450841.
[25]	F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().

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.
[2]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.
[3]	E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009.
[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.
[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.
[6]	D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479.
[7]	D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().
[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.
[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}., ().
[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.
[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.
[12]	R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.
[13]	D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.
[14]	M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().
[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.
[16]	E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().
[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.
[18]	C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.
[19]	C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215. doi: 10.1353/ajm.2007.0002.
[20]	M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.
[21]	M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().
[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}., ().
[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.
[24]	G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466. doi: 10.1007/BF01450841.
[25]	F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().

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