Lower bounds for Lyapunov exponents of flat bundles on curves

Eskin, Alex¹ ; Kontsevich, Maxim² ; Möller, Martin³ ; Zorich, Anton⁴

¹ Department of Mathematics, University of Chicago, Chicago, IL, United States
² Institut des Hautes Études Scientifiques, le Bois Marie, Bures-sur-Yvette, France
³ Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany
⁴ Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia, Institut de Mathématiques de Jussieu, Université Paris 6, Paris, France

Geometry & topology, Tome 22 (2018) no. 4, pp. 2299-2338.

Résumé

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top $k$ Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank-$k$ holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.

Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi–Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.

Détail

DOI : 10.2140/gt.2018.22.2299

Classification : 37D25
Keywords: Lyapunov exponents, hypergeometric differential equations, Hodge bundles, parabolic structure

Bibliographie

[1] F Beukers, G Heckman, Monodromy for the hypergeometric function nFn−1, Invent. Math. 95 (1989) 325 | DOI

[2] C Bonatti, A Eskin, A Wilkinson, Projective cocycles over SL2(R)–actions : measures invariant under the upper triangular group, preprint (2017)

[3] I I Bouw, M Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. 172 (2010) 139 | DOI

[4] C Brav, H Thomas, Thin monodromy in Sp(4), Compos. Math. 150 (2014) 333 | DOI

[5] J Carlson, S Müller-Stach, C Peters, Period mappings and period domains, 85, Cambridge Univ. Press (2003)

[6] J Daniel, B Deroin, Lyapunov exponents of the Brownian motion on a Kähler manifold, preprint (2017)

[7] P Deligne, Équations différentielles à points singuliers réguliers, 163, Springer (1970)

[8] B Deroin, R Dujardin, Complex projective structures : Lyapunov exponent, degree, and harmonic measure, Duke Math. J. 166 (2017) 2643 | DOI

[9] C F Doran, J W Morgan, Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi–Yau threefolds, from: "Mirror symmetry, V" (editors N Yui, S T Yau, J D Lewis), AMS/IP Stud. Adv. Math. 38, Amer. Math. Soc. (2006) 517

[10] A Eskin, M Kontsevich, A Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn. 5 (2011) 319 | DOI

[11] A Eskin, M Kontsevich, A Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014) 207 | DOI

[12] R Fedorov, Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles, Int. Math. Res. Not. (2017) | DOI

[13] S Filip, Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle, Duke Math. J. 166 (2017) 657 | DOI

[14] G Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. 155 (2002) 1 | DOI

[15] C Fougeron, Parabolic degrees and Lyapunov exponents for hypergeometric local systems, preprint (2017)

[16] P Griffiths, W Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253 | DOI

[17] A Kappes, M Möller, Lyapunov spectrum of ball quotients with applications to commensurability questions, Duke Math. J. 165 (2016) 1 | DOI

[18] Y Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982) 57 | DOI

[19] M Kontsevich, Homological algebra of mirror symmetry, from: "Proceedings of the International Congress of Mathematicians, 1" (editor S D Chatterji), Birkhäuser (1995) 120

[20] M Kontsevich, Lyapunov exponents and Hodge theory, from: "The mathematical beauty of physics" (editors J M Drouffe, J B Zuber), Adv. Ser. Math. Phys. 24, World Scientific (1997) 318

[21] M Kontsevich, A Zorich, Lyapunov exponents and Hodge theory, preprint (1997)

[22] M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631 | DOI

[23] R Krikorian, Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich–Zorich, from: "Séminaire Bourbaki, 2003/2004", Astérisque 299, Soc. Math. France (2005) 59

[24] V B Mehta, C S Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980) 205 | DOI

[25] M Möller, Teichmüller curves, mainly from the viewpoint of algebraic geometry, from: "Moduli spaces of Riemann surfaces" (editors B Farb, R Hain, E Looijenga), IAS/Park City Math. Ser. 20, Amer. Math. Soc. (2013) 267

[26] V I Oseledets, A multiplicative ergodic theorem : Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obshch. 19 (1968) 179

[27] C A M Peters, A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann. 268 (1984) 1 | DOI

[28] H L Royden, The Ahlfors–Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980) 547 | DOI

[29] D Ruelle, Chaotic evolution and strange attractors, Cambridge Univ. Press (1989) | DOI

[30] W Schmid, Variation of Hodge structure : the singularities of the period mapping, Invent. Math. 22 (1973) 211 | DOI

[31] C S Seshadri, Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc. 83 (1977) 124 | DOI

[32] C T Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988) 867 | DOI

[33] C T Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713 | DOI

[34] S Singh, T N Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163 (2014) 591 | DOI

[35] M Yoshida, Fuchsian differential equations, E11, Vieweg (1987) | DOI

[36] M Yoshida, Hypergeometric functions, my love, E32, Vieweg (1997) | DOI

[37] F Yu, Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations, Geom. Topol. 22 (2018) 2253 | DOI

[38] F Yu, K Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents, J. Mod. Dyn. 7 (2013) 209 | DOI

[39] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437 | DOI