The algebraic hull of the Kontsevich–Zorich cocycle

Alex Eskin; Simion Filip; Alex Wright

doi:https://doi.org/10.4007/annals.2018.188.1.5

Abstract

We compute the algebraic hull of the Kontsevich–Zorich cocycle over any $ \mathrm {GL}^+_2(\mathbb {R}) $ invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

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@book {SGA3, TITLE = {Schémas en groupes ({SGA} 3). {T}ome {I}. {P}ropriétés générales des schémas en groupes}, SERIES = {Documents Mathématiques (Paris) [Mathematical Documents(Paris)]}, VOLUME = {7}, EDITOR = {Gille, Philippe and Polo, Patrick}, NOTE = {Séminaire de Géométrie Algébrique du Bois Marie 1962--64. [Algebraic Geometry Seminar of Bois Marie 1962--64]; a seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; revised and annotated edition of the 1970 French original}, PUBLISHER = {Société Mathématique de France, Paris}, YEAR = {2011}, PAGES = {xxviii+610}, ISBN = {978-2-85629-323-2}, MRCLASS = {14L15}, MRNUMBER = {2867621}, ZBLNUMBER = {1241.14003}, }
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@ARTICLE{Wcyl, author = {Wright, Alex}, title = {Cylinder deformations in orbit closures of translation surfaces}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {19}, year = {2015}, number = {1}, pages = {413--438}, issn = {1465-3060}, mrclass = {32G15 (37D40)}, mrnumber = {3318755}, mrreviewer = {Jayadev S. Athreya}, doi = {10.2140/gt.2015.19.413}, url = {https://doi.org/10.2140/gt.2015.19.413}, zblnumber = {1318.32021}, }
[Wsurvey] $Go to document$ A. Wright, "Translation surfaces and their orbit closures: an introduction for a broad audience," EMS Surv. Math. Sci., vol. 2, iss. 1, pp. 63-108, 2015.

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@ARTICLE{Wsurvey, author = {Wright, Alex}, title = {Translation surfaces and their orbit closures: an introduction for a broad audience}, journal = {EMS Surv. Math. Sci.}, fjournal = {EMS Surveys in Mathematical Sciences}, volume = {2}, year = {2015}, number = {1}, pages = {63--108}, issn = {2308-2151}, mrclass = {32G15 (30D05 30F30 30F60)}, mrnumber = {3354955}, mrreviewer = {Jayadev S. Athreya}, doi = {10.4171/EMSS/9}, url = {https://doi.org/10.4171/EMSS/9}, zblnumber = {1372.37090}, }
[Zimmer_book] $Go to document$ R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel, 1984, vol. 81.

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@BOOK{Zimmer_book, author = {Zimmer, Robert J.}, title = {Ergodic Theory and Semisimple Groups}, series = {Monogr. Math.}, volume = {81}, publisher = {Birkhäuser Verlag, Basel}, year = {1984}, pages = {x+209}, isbn = {3-7643-3184-4}, mrclass = {22E40 (22D40 28D15)}, mrnumber = {0776417}, mrreviewer = {S. G. Dani}, doi = {10.1007/978-1-4684-9488-4}, url = {https://doi.org/10.1007/978-1-4684-9488-4}, zblnumber = {0571.58015}, }
[Zorich_survey] A. Zorich, "Flat surfaces," in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, pp. 437-583.

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@INCOLLECTION{Zorich_survey, author = {Zorich, Anton}, title = {Flat surfaces}, booktitle = {Frontiers in Number Theory, Physics, and Geometry. {I}}, pages = {437--583}, publisher = {Springer, Berlin}, year = {2006}, mrclass = {37D40 (30F30 32G15 37D50 57M50)}, mrnumber = {2261104}, mrreviewer = {Thomas A. Schmidt}, zblnumber = {1129.32012}, }

The algebraic hull of the Kontsevich–Zorich cocycle

Abstract

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