×
Filip, Simion

Global properties of some weight 3 variations of Hodge structure. (English) Zbl 1523.14016

Hujdurović, Ademir (ed.) et al., European congress of mathematics. Proceedings of the 8th congress, 8ECM, Portorož, Slovenia, June 20–26, 2021. Berlin: European Mathematical Society (EMS). 553-568 (2023).
Summary: We survey results on the global geometry of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). Included are uniformization results of domains in flag manifolds, a strong Torelli theorem, as well as the formula for the sum of Lyapunov exponents conjectured by Eskin, Kontsevich, Möller, and Zorich. Additionally, we establish the Anosov property of the monodromy representation, using gradient estimates of certain functions derived from the Hodge structure.
For the entire collection see [Zbl 1519.00033].

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
14L24 Geometric invariant theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry

Keywords:

variations of Hodge structure; Calabi-Yau; Anosov representation; GIT; Lyapunov exponents
Cite Review PDF
Full Text: DOI

References:

[1] F. Beukers and G. Heckman, Monodromy for the hypergeometric function n F n 1 . Invent. Math. 95 (1989), no. 2, 325-354 Zbl 0663.30044 MR 974906 · Zbl 0663.30044
[2] C. Brav and H. Thomas, Thin monodromy in Sp.4/. Compos. Math. 150 (2014), no. 3, 333-343 Zbl 1311.14010 MR 3187621 · Zbl 1311.14010
[3] Global properties of some weight 3 variations of Hodge structure 567
[4] M. Burger, A. Iozzi, F. Labourie, and A. Wienhard, Maximal representations of surface groups: symplectic Anosov structures. Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 543-590 Zbl 1157.53025 MR 2201327 · Zbl 1157.53025
[5] P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B 359 (1991), no. 1, 21-74 Zbl 1098.32506 MR 1115626 · Zbl 1098.32506
[6] B. Collier, N. Tholozan, and J. Toulisse, The geometry of maximal representations of surface groups into SO 0 .2; n/. Duke Math. J. 168 (2019), no. 15, 2873-2949 Zbl 07145323 MR 4017517 · Zbl 1516.32004
[7] A. Eskin, M. Kontsevich, M. Möller, and A. Zorich, Lower bounds for Lyapunov expo-nents of flat bundles on curves. Geom. Topol. 22 (2018), no. 4, 2299-2338 Zbl 1386.37036 MR 3784522 · Zbl 1386.37036
[8] R. Fedorov, Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles. Int. Math. Res. Not. IMRN 2018 (2018), no. 18, 5583-5608 Zbl 1408.32017 MR 3862114 · Zbl 1408.32017
[9] S. Filip, Uniformization of some weight 3 variations of Hodge structure, Anosov repre-sentations, and Lyapunov exponents. 2021, arXiv:2110.07533
[10] S. Filip and C. Fougeron, A cyclotomic family of thin hypergeometric monodromy groups in Sp 4 .R/. 2021, arXiv:2106.09181
[11] F. Guéritaud, O. Guichard, F. Kassel, and A. Wienhard, Anosov representations and proper actions. Geom. Topol. 21 (2017), no. 1, 485-584 Zbl 1373.37095 MR 3608719 · Zbl 1373.37095
[12] O. Guichard and A. Wienhard, Anosov representations: domains of discontinuity and applications. Invent. Math. 190 (2012), no. 2, 357-438 Zbl 1270.20049 MR 2981818 · Zbl 1270.20049
[13] M. Kapovich and B. Leeb, Relativizing characterizations of Anosov subgroups, I. 2018, arXiv:1807.00160
[14] M. Kapovich, B. Leeb, and J. Porti, Dynamics on flag manifolds: domains of proper dis-continuity and cocompactness. Geom. Topol. 22 (2018), no. 1, 157-234 Zbl 1381.53090 MR 3720343 · Zbl 1381.53090
[15] A. W. Knapp, Structure theory of semisimple Lie groups. In Representation Theory and Automorphic Forms (Edinburgh, 1996), pp. 1-27, Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, RI, 1997 Zbl 0902.17003 MR 1476489
[16] F. Labourie, Anosov flows, surface groups and curves in projective space. Invent. Math. 165 (2006), no. 1, 51-114 Zbl 1103.32007 MR 2221137 · Zbl 1103.32007
[17] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory. 3rd edn., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994 Zbl 0797.14004 MR 1304906 · Zbl 0797.14004
[18] P. Sarnak, Notes on thin matrix groups. In Thin Groups and Superstrong Approximation, pp. 343-362, Math. Sci. Res. Inst. Publ. 61, Cambridge Univ. Press, Cambridge, 2014 Zbl 1365.11039 MR 3220897
[19] C. T. Simpson, Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3 (1990), no. 3, 713-770 Zbl 0713.58012 MR 1040197 · Zbl 0713.58012
[20] S. Filip 568
[21] S. Usui, Generic Torelli theorem for quintic-mirror family. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 8, 143-146 Zbl 1164.14003 MR 2457802 · Zbl 1164.14003
[22] M. Yoshida, Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory. Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987 Zbl 0618.35001 MR 986252
[23] F. Zhu, Relatively dominated representations. Ann. Inst. Fourier (Grenoble) 71 (2021), no. 5, 2169-2235 Zbl 07492562 MR 4398259 · Zbl 1512.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.