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Nonvarying sums of Lyapunov exponents of abelian differentials in low genus. (English) Zbl 1266.14018

Let \(\Omega M_g\) denote the vector bundle of holomorphic one-forms over the moduli space \(M_g\) of genus \(g\) curves minus the zero section. The space \(\Omega M_g\) is stratified according to the zeros of one-forms. Each stratum contains many Teichmüller curves, for which an algorithm given in [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] calculates the sum of Lyapunov exponents (one Teichmüller curve at a time). Based on a limited number of computer experiments about a decade ago, Kontsevich and Zorich observed that the sum of Lyapunov exponents is nonvarying among all the Teichmüller curves in a stratum roughly if the genus plus the number of zeros is less than seven, while the sum varies if this sum is greater than seven.
The authors of the paper under review find an explanation of this observation, and give a detailed description of what happens for all strata of the moduli space of genera less than or equal to five. More precisely, they give a list of strata with nonvarying sum of Lyapunov exponents, and for some other strata they give upper bounds for these sums. Closely related results are also obtained by the authors in [“Quadratic differentials in low genus: exceptional and non-varying”, arXiv:1204.1707] and by F. Yu and K. Zuo in [“Weierstrass filtration on Teichmüller curves and Lyapunov exponents”, arXiv:1203.6053].

MSC:

14H10 Families, moduli of curves (algebraic)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
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References:

[1] E Arbarello, M Cornalba, The Picard groups of the moduli spaces of curves, Topology 26 (1987) 153 · Zbl 0625.14014 · doi:10.1016/0040-9383(87)90056-5
[2] E Arbarello, M Cornalba, P A Griffiths, J Harris, Geometry of algebraic curves. Vol. I, Grundl. Math. Wissen. 267, Springer (1985) · Zbl 0559.14017
[3] M Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007) 1887 · Zbl 1131.32007 · doi:10.2140/gt.2007.11.1887
[4] A Beauville, Complex algebraic surfaces, London Mathematical Society Student Texts 34, Cambridge Univ. Press (1996) · Zbl 0849.14014
[5] I Bouw, M Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. 172 (2010) 139 · Zbl 1203.37049 · doi:10.4007/annals.2010.172.139
[6] L Caporaso, Linear series on semistable curves, Int. Math. Res. Not., rnq 188, 49 pages (2010) · Zbl 1231.14007
[7] D Chen, Covers of elliptic curves and the moduli space of stable curves, J. reine angew. Math. 649 (2010) 167 · Zbl 1208.14024 · doi:10.1515/CRELLE.2010.092
[8] D Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math. 228 (2011) 1135 · Zbl 1227.14030 · doi:10.1016/j.aim.2011.06.002
[9] D Chen, M Möller, Quadratic differentials in low genus: exceptional and non-varying strata · Zbl 1395.14008
[10] M Cornalba, Moduli of curves and theta-characteristics, World Sci. Publ., Teaneck, NJ (1989) 560 · Zbl 0800.14011
[11] I Coskun, Degenerations of surface scrolls and the Gromov-Witten invariants of Grassmannians, J. Algebraic Geom. 15 (2006) 223 · Zbl 1105.14072 · doi:10.1090/S1056-3911-06-00426-7
[12] F Cukierman, Families of Weierstrass points, Duke Math. J. 58 (1989) 317 · Zbl 0687.14026 · doi:10.1215/S0012-7094-89-05815-8
[13] V Delecroix, P Hubert, S Leli‘evre, Diffusion in the periodic wind-tree model · Zbl 1351.37159
[14] D Eisenbud, J Harris, The Kodaira dimension of the moduli space of curves of genus \(\geq\nb 23\), Invent. Math. 90 (1987) 359 · Zbl 0631.14023 · doi:10.1007/BF01388710
[15] A Eskin, M Kontsevich, A Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow · Zbl 1305.32007 · doi:10.1007/s10240-013-0060-3
[16] A Eskin, H Masur, 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 · Zbl 1037.32013 · doi:10.1007/s10240-003-0015-1
[17] A Eskin, A Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001) 59 · Zbl 1019.32014 · doi:10.1007/s002220100142
[18] G Farkas, Koszul divisors on moduli spaces of curves, Amer. J. Math. 131 (2009) 819 · Zbl 1176.14006 · doi:10.1353/ajm.0.0053
[19] G Farkas, The birational type of the moduli space of even spin curves, Adv. Math. 223 (2010) 433 · Zbl 1183.14020 · doi:10.1016/j.aim.2009.08.011
[20] G Farkas, A Verra, The geometry of the moduli space of odd spin curves · Zbl 1325.14045
[21] H M Farkas, Unramified double coverings of hyperelliptic surfaces, J. Analyse Math. 30 (1976) 150 · Zbl 0348.32006 · doi:10.1007/BF02786710
[22] G Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, Elsevier B. V., Amsterdam (2006) 549 · Zbl 1130.37302 · doi:10.1016/S1874-575X(06)80033-0
[23] G Forni, C Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum
[24] J Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983) 221 · Zbl 0533.57003 · doi:10.1007/BF01389321
[25] J Harris, Families of smooth curves, Duke Math. J. 51 (1984) 409 · Zbl 0548.14009 · doi:10.1215/S0012-7094-84-05120-2
[26] J Harris, I Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990) 321 · Zbl 0705.14026 · doi:10.1007/BF01234422
[27] J Harris, I Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer (1998) · Zbl 0913.14005
[28] J Harris, D Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23 · Zbl 0506.14016 · doi:10.1007/BF01393371
[29] B Hassett, Stable log surfaces and limits of quartic plane curves, Manuscripta Math. 100 (1999) 469 · Zbl 0973.14014 · doi:10.1007/s002290050213
[30] K Kodaira, On compact analytic surfaces. II, III, Ann. of Math. 77 \((1963)\), 563-626; ibid. 78 (1963) 1 · Zbl 0171.19601
[31] M Kontsevich, Lyapunov exponents and Hodge theory, Adv. Ser. Math. Phys. 24, World Sci. Publishing (1997) 318 · Zbl 1058.37508
[32] M Kontsevich, A Zorich, Lyapunov exponents and Hodge theory · Zbl 1058.37508
[33] M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631 · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x
[34] A Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003) 105 · Zbl 1066.14030 · doi:10.1353/ajm.2003.0005
[35] H Masur, On a class of geodesics in Teichmüller space, Ann. of Math. 102 (1975) 205 · Zbl 0322.32010 · doi:10.2307/1971031
[36] M Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006) 327 · Zbl 1090.32004 · doi:10.1090/S0894-0347-05-00512-6
[37] M Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5 (2011) 1 · Zbl 1221.14033 · doi:10.3934/jmd.2011.5.1
[38] S Morita, Families of Jacobian manifolds and characteristic classes of surface bundles, II, Math. Proc. Cambridge Philos. Soc. 105 (1989) 79 · Zbl 0775.57001 · doi:10.1017/S0305004100001389
[39] D Mumford, Stability of projective varieties, Enseignement Math. 23 (1977) 39 · Zbl 0497.14004
[40] M Reid, Chapters on algebraic surfaces, IAS/Park City Math. Ser. 3, Amer. Math. Soc. (1997) 3 · Zbl 0910.14016
[41] F Yu, K Zuo, Weierstrass filtration on Teichmüller curves and Lyaponov exponents · Zbl 1273.32019 · doi:10.3934/jmd.2013.7.209
[42] A Zorich, Flat surfaces, Springer (2006) 439 · Zbl 1129.32012
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