Abstract
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.
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@misc{ANW_H^4,
author = {Aulicino, D. and Nguyen, D.-M. and Wright, A.},
title = {Classification of higher rank orbit closures in {$\mathcal{H}^{\rm odd}$(4)}},
year = {2013},
arxiv = {1308.5879},
} -
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author = {Carlson, James A.},
title = {Extensions of mixed {H}odge structures},
booktitle = {Journées de {G}éometrie {A}lgébrique d'{A}ngers, {J}uillet 1979/{A}lgebraic {G}eometry, {A}ngers, 1979},
pages = {107--127},
publisher = {Sijthoff \& Noordhoff, Alphen aan den Rijn},
address = {Germantown, Md.},
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author = {Eskin, A. and Mirzakhani, M. and Mohammadi, A.},
title = {Isolation, equidistribution, and orbit closures for the {$\mathrm{SL}(2,\mathbb{R})$} action on moduli space},
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note = {{\em Ann. Sci. {É}c. Norm. Sup{é}r.} 4 {\bf 49},
to appear 1er fascicule (janvier-février), 2016},
arxiv= {1301.0783},
SORTYEAR={2016},
} -
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title = {Teichmüller curves generated by {W}eierstrass {P}rym eigenforms in genus 3 and genus 4},
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volume = {7},
year = {2014},
number = {2},
pages = {475--522},
issn = {1753-8416},
mrclass = {32G15},
mrnumber = {3217628},
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doi = {10.1112/jtopol/jtt036},
zblnumber = {06366504},
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volume = {115},
year = {1982},
number = {1},
pages = {169--200},
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author = {McMullen, Curtis T.},
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[McMullen_spin]
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year = {2009},
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author = {M{ö}ller, Martin},
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journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {165},
year = {2006},
number = {3},
pages = {633--649},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {14D07 (32G15)},
mrnumber = {2242629},
mrreviewer = {Thomas A. Schmidt},
doi = {10.1007/s00222-006-0510-3},
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author = {M{ö}ller, Martin},
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journal = {J. Amer. Math. Soc.},
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volume = {19},
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journal = {Duke Math. J.},
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volume = {144},
year = {2008},
number = {3},
pages = {447--487},
issn = {0012-7094},
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mrclass = {32G15 (14D07 32G20)},
mrnumber = {2444303},
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doi = {10.1215/00127094-2008-041},
zblnumber = {1148.32007},
} -
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year = {2014},
number = {4},
pages = {1316--1335},
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author = {Peters, Chris A. M. and Steenbrink, Joseph H. M.},
title = {Mixed {H}odge Structures},
series = {Ergeb. Math. Grenzgeb.},
volume = {52},
publisher = {Springer-Verlag},
year = {2008},
pages = {xiv+470},
isbn = {978-3-540-77015-2},
mrclass = {14C30 (14D07 32G20 32J25 32S35)},
mrnumber = {2393625},
mrreviewer = {Matt Kerr},
address = {New York},
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} -
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author = {Shafarevich, Igor R.},
title = {Basic {A}lgebraic {G}eometry. 2},
edition = {Second},
note = {Schemes and complex manifolds, translated from the 1988 {R}ussian edition by {M}iles {R}eid},
publisher = {Springer-Verlag},
year = {1994},
pages = {xiv+269},
isbn = {3-540-57554-5},
mrclass = {14-01},
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address = {New York},
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} -
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year = {2004},
pages = {249--260},
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mrnumber = {2085718},
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author = {Veech, William A.},
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journal = {Ann. of Math.},
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volume = {115},
year = {1982},
number = {1},
pages = {201--242},
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zblnumber = {0486.28014},
} -
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author = {Veech, William A.},
title = {Geometric realizations of hyperelliptic curves},
booktitle = {Algorithms, {F}ractals, and {D}ynamics},
venue = {{O}kayama/{K}yoto, 1992},
pages = {217--226},
publisher = {Plenum},
address = {New York},
year = {1995},
mrclass = {14H45},
mrnumber = {1402493},
mrreviewer = {G. Ewald},
zblnumber = {0859.30039},
} -
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author = {Wright, Alex},
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fjournal = {Geometry \& Topology},
volume = {18},
year = {2014},
number = {3},
pages = {1323--1341},
issn = {1465-3060},
mrclass = {32G15 (30F60 37D40)},
mrnumber = {3254934},
mrreviewer = {Jayadev S. Athreya},
doi = {10.2140/gt.2014.18.1323},
zblnumber = {1320.32019},
} -
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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},
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doi = {10.4171/EMSS/9},
zblnumber = {06455787},
} -
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author = {Zorich, Anton},
title = {Flat surfaces},
booktitle = {Frontiers in {N}umber {T}heory, {P}hysics, and {G}eometry. {I}},
pages = {439--585},
publisher = {Springer-Verlag},
year = {2006},
mrclass = {37D40 (30F30 32G15 37D50 57M50)},
mrnumber = {2261104},
mrreviewer = {Thomas A. Schmidt},
address = {New York},
zblnumber = {1129.32012},
}