Abstract
We prove that the zero locus of an admissible normal function over an algebraic parameter space $S$ is algebraic in the case where $S$ is a curve.
-
[CBH] N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Paris: Hermann, 1972.
@book{CBH, MRKEY = {0573068},
AUTHOR = {Bourbaki, N.},
TITLE = {\'{E}léments de mathématique. {F}asc. {\rm XXXVII}. {G}roupes et algèbres de {L}ie. {C}hapitre {\rm II}: {A}lgèbres de {L}ie libres. {C}hapitre {\rm III}: {G}roupes de {L}ie},
SERIES = {Actualit{é}s Scientifiques et Industrielles},
NUMBER = {1349},
PUBLISHER = {Hermann},
ADDRESS = {Paris},
YEAR = {1972},
PAGES = {320},
MRCLASS = {17-01 (22-01)},
MRNUMBER = {58 \#28083a},
ZBLNUMBER = {0244.22007},
MRREVIEWER = {H. Reitberger},
} -
[CKS]
E. Cattani, A. Kaplan, and W. Schmid, "Degeneration of Hodge structures," Ann. of Math., vol. 123, iss. 3, pp. 457-535, 1986.
@article{CKS, MRKEY = {840721},
AUTHOR = {Cattani, Eduardo and Kaplan, Aroldo and Schmid, Wilfried},
TITLE = {Degeneration of {H}odge structures},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {123},
YEAR = {1986},
NUMBER = {3},
PAGES = {457--535},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {32G20 (14C30 14D05 32J25)},
MRNUMBER = {88a:32029},
DOI = {10.2307/1971333},
ZBLNUMBER = {0617.14005},
MRREVIEWER = {Masaki Kashiwara},
} -
[Clemens]
H. Clemens, "The Néron model for families of intermediate Jacobians acquiring “algebraic” singularities," Inst. Hautes Études Sci. Publ. Math., vol. 58, pp. 5-18 (1984), 1983.
@article{Clemens, MRKEY = {720929},
AUTHOR = {Clemens, Herbert},
TITLE = {The {N}éron model for families of intermediate {J}acobians acquiring ``algebraic'' singularities},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {58},
YEAR = {1983},
PAGES = {5--18 (1984)},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {14K30 (14C30 32J25)},
MRNUMBER = {86d:14042},
URL = {http://www.numdam.org/item?id=PMIHES_1983__58__5_0},
ZBLNUMBER = {0529.14025},
MRREVIEWER = {Steven M. Zucker},
} -
[D] P. Deligne, Letter to E. Cattani and A. Kaplan, 1993.
@misc{D,
author = {Deligne, Pierre},
Title = {Letter to {E}. {C}attani and {A}. {K}aplan},
year = {1993},
} -
[Hodge2]
P. Deligne, "Théorie de Hodge, II," Inst. Hautes Études Sci. Publ. Math., vol. 40, pp. 5-57, 1971.
@article{Hodge2, MRKEY = {0498551},
AUTHOR = {Deligne, Pierre},
TITLE = {Théorie de {H}odge, {II}},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {40},
YEAR = {1971},
PAGES = {5--57},
ISSN = {0073-8301},
MRCLASS = {14C30 (14F15)},
MRNUMBER = {58 \#16653a},
URL = {http://www.numdam.org/item?id=PMIHES_1971__40__5_0},
ZBLNUMBER = {0219.14007},
MRREVIEWER = {J. H. M. Steenbrink},
} -
[K]
M. Kashiwara, "A study of variation of mixed Hodge structure," Publ. Res. Inst. Math. Sci., vol. 22, iss. 5, pp. 991-1024, 1986.
@article{K, MRKEY = {866665},
AUTHOR = {Kashiwara, Masaki},
TITLE = {A study of variation of mixed {H}odge structure},
JOURNAL = {Publ. Res. Inst. Math. Sci.},
FJOURNAL = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
VOLUME = {22},
YEAR = {1986},
NUMBER = {5},
PAGES = {991--1024},
ISSN = {0034-5318},
CODEN = {KRMPBV},
MRCLASS = {32G20 (32C20 32C35 32C42)},
MRNUMBER = {89i:32050},
DOI = {10.2977/prims/1195177264},
ZBLNUMBER = {0621.14007},
} -
[KNU] K. Kato, C. Nakayama, and S. Usui, "${ SL}(2)$-orbit theorem for degeneration of mixed Hodge structure," J. Algebraic Geom., vol. 17, iss. 3, pp. 401-479, 2008.
@article{KNU, MRKEY = {2395135},
AUTHOR = {Kato, Kazuya and Nakayama, Chikara and Usui, Sampei},
TITLE = {{${\rm SL}(2)$}-orbit theorem for degeneration of mixed {H}odge structure},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {17},
YEAR = {2008},
NUMBER = {3},
PAGES = {401--479},
ISSN = {1056-3911},
MRCLASS = {14Dxx (14Cxx)},
MRNUMBER = {2395135},
ZBLNUMBER = {1144.14005},
} -
[KP] A. Kaplan and G. Pearlstein, "Singularities of variations of mixed Hodge structure," Asian J. Math., vol. 7, iss. 3, pp. 307-336, 2003.
@article{KP, MRKEY = {2129326},
AUTHOR = {Kaplan, Aroldo and Pearlstein, Gregory},
TITLE = {Singularities of variations of mixed {H}odge structure},
JOURNAL = {Asian J. Math.},
FJOURNAL = {The Asian Journal of Mathematics},
VOLUME = {7},
YEAR = {2003},
NUMBER = {3},
PAGES = {307--336},
ISSN = {1093-6106},
MRCLASS = {14D07 (32J25)},
MRNUMBER = {2007b:14021},
ZBLNUMBER = {1076.14018},
} -
[P3]
G. Pearlstein, "${ SL}_2$-orbits and degenerations of mixed Hodge structure," J. Differential Geom., vol. 74, iss. 1, pp. 1-67, 2006.
@article{P3, MRKEY = {2260287},
AUTHOR = {Pearlstein, Gregory},
TITLE = {{${\rm SL}\sb 2$}-orbits and degenerations of mixed {H}odge structure},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {74},
YEAR = {2006},
NUMBER = {1},
PAGES = {1--67},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {32G20 (14D07)},
MRNUMBER = {2008f:32016},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1175266181},
ZBLNUMBER = {1107.14010},
} -
[P1]
G. J. Pearlstein, "Variations of mixed Hodge structure, Higgs fields, and quantum cohomology," Manuscripta Math., vol. 102, iss. 3, pp. 269-310, 2000.
@article{P1, MRKEY = {1777521},
AUTHOR = {Pearlstein, Gregory J.},
TITLE = {Variations of mixed {H}odge structure, {H}iggs fields, and quantum cohomology},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {102},
YEAR = {2000},
NUMBER = {3},
PAGES = {269--310},
ISSN = {0025-2611},
CODEN = {MSMHB2},
MRCLASS = {32G20 (14D07 14J32 14N35 32Q25)},
MRNUMBER = {2001m:32033},
DOI = {10.1007/PL00005852},
ZBLNUMBER = {0973.32008},
MRREVIEWER = {Andreas Gathmann},
} -
[MS] M. Saito, "Admissible normal functions," J. Algebraic Geom., vol. 5, iss. 2, pp. 235-276, 1996.
@article{MS, MRKEY = {1374710},
AUTHOR = {Saito, Morihiko},
TITLE = {Admissible normal functions},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {5},
YEAR = {1996},
NUMBER = {2},
PAGES = {235--276},
ISSN = {1056-3911},
MRCLASS = {14K30 (14C30 14D07 32G20)},
MRNUMBER = {97c:14044},
ZBLNUMBER = {0918.14018},
MRREVIEWER = {Claire Voisin},
} -
[SZ]
J. Steenbrink and S. Zucker, "Variation of mixed Hodge structure, I," Invent. Math., vol. 80, iss. 3, pp. 489-542, 1985.
@article{SZ, MRKEY = {791673},
AUTHOR = {Steenbrink, Joseph and Zucker, Steven},
TITLE = {Variation of mixed {H}odge structure, {I}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {80},
YEAR = {1985},
NUMBER = {3},
PAGES = {489--542},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32G20 (14C30 32J25)},
MRNUMBER = {87h:32050a},
DOI = {10.1007/BF01388729},
ZBLNUMBER = {0626.14007},
MRREVIEWER = {Sampei Usui},
} -
[U]
S. Usui, "Variation of mixed Hodge structure arising from family of logarithmic deformations. II: Classifying space," Duke Math. J., vol. 51, iss. 4, pp. 851-875, 1984.
@article{U, MRKEY = {771384},
AUTHOR = {Usui, Sampei},
TITLE = {Variation of mixed {H}odge structure arising from family of logarithmic deformations. {II}: {C}lassifying space},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {51},
YEAR = {1984},
NUMBER = {4},
PAGES = {851--875},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14D05 (14C30 32G20 32J25)},
MRNUMBER = {86h:14005},
DOI = {10.1215/S0012-7094-84-05137-8},
ZBLNUMBER = {0558.14005},
MRREVIEWER = {Werner Kleinert},
}