Erratum for Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich’s conjecture$^\ast$

Abstract

We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich’s original conjecture holds for infinitesimally rigid families of canonically polarized varieties.

*This article was first printed in 172, 1719–1748.  This reprinting should be considered the version of record as it corrects a font error that resulted in notational problems throughout the paper.

  • [abramovich-olsson-vistoli] Go to document D. Abramovich, M. Olsson, and A. Vistoli, "Tame stacks in positive characteristic," Ann. Inst. Fourier $($Grenoble$)$, vol. 58, iss. 4, pp. 1057-1091, 2008.
    @article {abramovich-olsson-vistoli, MRKEY = {2427954},
      AUTHOR = {Abramovich, Dan and Olsson, Martin and Vistoli, Angelo},
      TITLE = {Tame stacks in positive characteristic},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {58},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {1057--1091},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {14A20 (14L15)},
      MRNUMBER = {2427954},
      MRREVIEWER = {Hsian-Hua Tseng},
      URL = {http://aif.cedram.org/item?id=AIF_2008__58_4_1057_0},
      ZBLNUMBER = {05303669},
      }
  • [MR1862797] Go to document D. Abramovich and A. Vistoli, "Compactifying the space of stable maps," J. Amer. Math. Soc., vol. 15, iss. 1, pp. 27-75, 2002.
    @article {MR1862797, MRKEY = {1862797},
      AUTHOR = {Abramovich, Dan and Vistoli, Angelo},
      TITLE = {Compactifying the space of stable maps},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {15},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {27--75},
      ISSN = {0894-0347},
      MRCLASS = {14H10 (14A20 14D20 14N35)},
      MRNUMBER = {1862797},
      MRREVIEWER = {Tyler J. Jarvis},
      DOI = {10.1090/S0894-0347-01-00380-0},
      ZBLNUMBER = {0991.14007},
      }
  • [Aesop] Aesop, Fables, 600 B.C.
    @misc{Aesop,
      author={Aesop},
      TITLE={Fables},
      YEAR={600 B.C.},
      }
  • [AIM04] M. A. Van Opstall, Open problems.
    @misc{AIM04,
      author={Van Opstall, M. A.},
      TITLE={Open problems},
      NOTE={AIM 5-day workshop "Compact moduli spaces and birational geometry", December 6-10, 2004},
      ARXIV={http://www.aimath.org/pastworkshops/birational.html},
      }
  • [Arakelov71] Go to document J. S. Arakelov, "Families of algebraic curves with fixed degeneracies," Izv. Akad. Nauk SSSR Ser. Mat., vol. 35, pp. 1269-1293, 1971.
    @article {Arakelov71, MRKEY = {0321933},
      AUTHOR = {Arakelov, S. Ju.},
      TITLE = {Families of algebraic curves with fixed degeneracies},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {35},
      YEAR = {1971},
      PAGES = {1269--1293},
      ISSN = {0373-2436},
      MRCLASS = {14H10 (14G05 14H25)},
      MRNUMBER = {0321933},
      MRREVIEWER = {J. W. S. Cassels},
      ZBLNUMBER={0248.14004},
      DOI={10.1007/s002229900046},
      }
  • [Bedulev-Viehweg00] Go to document E. Bedulev and E. Viehweg, "On the Shafarevich conjecture for surfaces of general type over function fields," Invent. Math., vol. 139, iss. 3, pp. 603-615, 2000.
    @article {Bedulev-Viehweg00, MRKEY = {1738062},
      AUTHOR = {Bedulev, Egor and Viehweg, Eckart},
      TITLE = {On the {S}hafarevich conjecture for surfaces of general type over function fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {139},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {603--615},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14J10 (14J29)},
      MRNUMBER = {2001f:14065},
      MRREVIEWER = {S{á}ndor J. Kov{á}cs},
      DOI = {10.1007/s002229900046},
      ZBLNUMBER = {1057.14044},
      }
  • [math.AG/0610203] Go to document C. Birker, P. Cascini, C. D. Hacon, and J. McKernan, "Existence of minimal models for varieties of log general type," J. Amer. Math. Soc., vol. 23, pp. 405-468, 2010.
    @article{math.AG/0610203,
      author={Birker, C. and Cascini, P. and Hacon, C. D. and McKernan, J.},
      TITLE={Existence of minimal models for varieties of log general type},
      JOURNAL={J. Amer. Math. Soc.},
      VOLUME={23},
      YEAR={2010},
      PAGES={405--468},
      MRNUMBER={2601039},
      DOI={10.1090/S0894-0347-09-00649-3},
      }
  • [MR2306040] Go to document C. Cadman, "Using stacks to impose tangency conditions on curves," Amer. J. Math., vol. 129, iss. 2, pp. 405-427, 2007.
    @article {MR2306040, MRKEY = {2306040},
      AUTHOR = {Cadman, Charles},
      TITLE = {Using stacks to impose tangency conditions on curves},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {129},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {405--427},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {14D20 (14A20 14N35)},
      MRNUMBER = {2306040},
      MRREVIEWER = {Michael A. Rose},
      DOI = {10.1353/ajm.2007.0007},
      ZBLNUMBER = {1127.14002},
      }
  • [Caporaso02] Go to document L. Caporaso, "On certain uniformity properties of curves over function fields," Compositio Math., vol. 130, iss. 1, pp. 1-19, 2002.
    @article {Caporaso02, MRKEY = {1883689},
      AUTHOR = {Caporaso, Lucia},
      TITLE = {On certain uniformity properties of curves over function fields},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {130},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--19},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14H05 (14H10)},
      MRNUMBER = {1883689},
      MRREVIEWER = {Carlo Gasbarri},
      DOI = {10.1023/A:1013703430436},
      ZBLNUMBER = {1067.14022},
      }
  • [MR861969] Arithmetic Geometry, Cornell, G. and Silverman, J. H., Eds., New York: Springer-Verlag, 1986.
    @book {MR861969, MRKEY = {861969},
      TITLE = {Arithmetic Geometry},
      NOTE = {papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30--August 10, 1984},
      EDITOR={Cornell, G. and Silverman, J. H.},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {xvi+353},
      ISBN = {0-387-96311-1},
      MRCLASS = {14Gxx (11D41 11Gxx 14G25 14K15)},
      MRNUMBER = {861969.},
      MRREVIEWER = {H. Gillet},
      ZBLNUMBER={0596.00007},
      }
  • [MR1423020] Go to document A. J. de Jong, "Smoothness, semi-stability and alterations," Inst. Hautes Études Sci. Publ. Math., iss. 83, pp. 51-93, 1996.
    @article {MR1423020, MRKEY = {1423020},
      AUTHOR = {de Jong, A. J.},
      TITLE = {Smoothness, semi-stability and alterations},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {83},
      YEAR = {1996},
      PAGES = {51--93},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {14E15 (14B05 14H10)},
      MRNUMBER = {1423020},
      MRREVIEWER = {Marko Roczen},
      URL = {http://www.numdam.org/item?id=PMIHES_1996__83__51_0},
      ZBLNUMBER = {0916.14005},
      }
  • [Faltings83] Go to document G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern," Invent. Math., vol. 73, iss. 3, pp. 349-366, 1983.
    @article {Faltings83, MRKEY = {718935},
      AUTHOR = {Faltings, G.},
      TITLE = {Endlichkeitssätze für abelsche {V}arietäten über {Z}ahlkörpern},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {73},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {349--366},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11D41 (11G30 14G25)},
      MRNUMBER = {1423020},
      MRREVIEWER = {James Milne},
      DOI = {10.1007/BF01388432},
      ZBLNUMBER={0588.14026},
      }
  • [Faltings84] Go to document G. Faltings, "Erratum: “Finiteness theorems for abelian varieties over number fields”," Invent. Math., vol. 75, iss. 2, p. 381, 1984.
    @article {Faltings84, MRKEY = {732554},
      AUTHOR = {Faltings, G.},
      TITLE = {Erratum: ``{F}initeness theorems for abelian varieties over number fields''},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {75},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {381},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11D41 (11G30 14G25)},
      MRNUMBER = {732554},
      MRREVIEWER = {James Milne},
      DOI = {10.1007/BF01388572},
      ZBLNUMBER={0588.14026},
      }
  • [EGAIV4] A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV," Inst. Hautes Études Sci. Publ. Math., vol. 32, p. 361, 1967.
    @article {EGAIV4, MRKEY = {0238860},
      AUTHOR = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas {IV}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {32},
      YEAR = {1967},
      PAGES = {361 pp.},
      ISSN = {0073-8301},
      MRCLASS = {14.55},
      MRNUMBER = {0238860},
      MRREVIEWER = {J. P. Murre},
      }
  • [Heier04b] Go to document G. Heier, "Uniformly effective Shafarevich conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus," J. Math. Pures Appl., vol. 83, iss. 7, pp. 845-867, 2004.
    @article {Heier04b, MRKEY = {2074680},
      AUTHOR = {Heier, Gordon},
      TITLE = {Uniformly effective {S}hafarevich conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus},
      JOURNAL = {J. Math. Pures Appl.},
      FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      VOLUME = {83},
      YEAR = {2004},
      NUMBER = {7},
      PAGES = {845--867},
      ISSN = {0021-7824},
      CODEN = {JMPAAM},
      MRCLASS = {14H10 (14C05 14G05)},
      MRNUMBER = {2074680},
      MRREVIEWER = {Flaminio Flamini},
      DOI = {10.1016/j.matpur.2004.02.001},
      ZBLNUMBER = {1066.14028},
      }
  • [MR0491680] L. Illusie, Complexe Cotangent et Déformations. I, New York: Springer-Verlag, 1971, vol. 239.
    @book {MR0491680, MRKEY = {0491680},
      AUTHOR = {Illusie, Luc},
      TITLE = {Complexe Cotangent et Déformations. {\rm I}},
      SERIES = {Lecture Notes in Math.},
      VOLUME={239},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {xv+355},
      MRCLASS = {14B10 (14D15 14F20 55D20)},
      MRNUMBER = {0491680},
      MRREVIEWER = {Larry Breen},
      ZBLNUMBER = {0224.13014},
      }
  • [KKMS73] G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings. I, New York: Springer-Verlag, 1973, vol. 339.
    @book {KKMS73, MRKEY = {0335518},
      AUTHOR = {Kempf, G. and Knudsen, Finn Faye and Mumford, D. and Saint-Donat, B.},
      TITLE = {Toroidal Embeddings. {\rm I}},
      SERIES = {Lecture Notes in Math.},
      VOLUME={339},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      PAGES = {viii+209},
      MRCLASS = {14E15 (14D20 14E05 14M20 20G15)},
      MRNUMBER = {0335518},
      MRREVIEWER = {G. Harder},
      ZBLNUMBER = {0271.14017},
      }
  • [Kollar90] Go to document J. Kollár, "Projectivity of complete moduli," J. Differential Geom., vol. 32, iss. 1, pp. 235-268, 1990.
    @article {Kollar90, MRKEY = {1064874},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Projectivity of complete moduli},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {32},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {235--268},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {14D22 (14H10 14J10)},
      MRNUMBER = {1064874},
      MRREVIEWER = {Autorreferat},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214445046},
      ZBLNUMBER = {0684.14002},
      }
  • [MR1658959] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
    @book {MR1658959, MRKEY = {1658959},
      AUTHOR = {Koll{á}r, J{á}nos and Mori, Shigefumi},
      TITLE = {Birational Geometry of Algebraic Varieties},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {134},
      NOTE = {with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {viii+254},
      ISBN = {0-521-63277-3},
      MRCLASS = {14E30},
      MRNUMBER = {1658959},
      MRREVIEWER = {Mark Gross},
      ZBLNUMBER = {0926.14003},
      DOI={10.1017/CBO9780511662560},
      }
  • [Kovacs03c] S. J. Kovács, "Families of varieties of general type: the Shafarevich conjecture and related problems," in Higher Dimensional Varieties and Rational Points, New York: Springer-Verlag, 2003, vol. 12, pp. 133-167.
    @incollection {Kovacs03c, MRKEY = {2011746},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Families of varieties of general type: the {S}hafarevich conjecture and related problems},
      BOOKTITLE = {Higher Dimensional Varieties and Rational Points},
      VENUE={{B}udapest, 2001},
      SERIES = {Bolyai Soc. Math. Stud.},
      VOLUME = {12},
      PAGES = {133--167},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      MRCLASS = {14J10 (14H10)},
      MRNUMBER = {2011746},
      MRREVIEWER = {Valmecir A. Bayer},
      ZBLNUMBER = {1058.14057},
      }
  • [Kovacs06a] S. J. Kovács, "Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich’s conjecture," in Algebraic Geometry—Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 685-709.
    @incollection {Kovacs06a, MRKEY = {2483952},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Subvarieties of moduli stacks of canonically polarized varieties: generalizations of {S}hafarevich's conjecture},
      BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 2},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {80},
      PAGES = {685--709},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {14D23 (14J10)},
      MRNUMBER = {2483952},
      MRREVIEWER = {Stefan Kebekus},
      ZBLNUMBER = {1183.14002},
      }
  • [Kovacs06b] S. J. Kovács, "Young person’s guide to moduli of higher dimensional varieties," in Algebraic Geometry—Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 711-743.
    @incollection {Kovacs06b, MRKEY = {2483953},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Young person's guide to moduli of higher dimensional varieties},
      BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 2},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {80},
      PAGES = {711--743},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {14J10 (14-02 14D20 14E30)},
      MRNUMBER = {2483953},
      MRREVIEWER = {James McKernan},
      ZBLNUMBER = {1182.14034},
      }
  • [kresch-geometry-of-stacks] A. Kresch, "On the geometry of Deligne-Mumford stacks," in Algebraic Geometry—Seattle 2005. Part 1, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 259-271.
    @incollection {kresch-geometry-of-stacks, MRKEY = {2483938},
      AUTHOR = {Kresch, Andrew},
      TITLE = {On the geometry of {D}eligne-{M}umford stacks},
      BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 1},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {80},
      PAGES = {259--271},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {14A20 (14D20 14F22 14L24)},
      MRNUMBER = {2483938},
      ZBLNUMBER = {1169.14001},
      }
  • [MR886677] S. Lang, Number Theory. III. Diophantine Geometry, New York: Springer-Verlag, 1991, vol. 60.
    @book {MR886677, MRKEY = {1112552},
      AUTHOR = {Lang, Serge},
      TITLE = {Number Theory. {\rm III}. Diophantine Geometry},
      SERIES = {Encyclopaedia Math. Sci.},
      VOLUME = {60},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {xiv+296},
      ISBN = {3-540-53004-5},
      MRCLASS = {11Gxx (14Gxx)},
      MRNUMBER = {1112552},
      MRREVIEWER = {Joseph H. Silverman},
      ZBLNUMBER = {0744.14012},
      }
  • [Lau-MB00] G. Laumon and L. Moret-Bailly, Champs Algébriques, New York: Springer-Verlag, 2000, vol. 39.
    @book {Lau-MB00, MRKEY = {1771927},
      AUTHOR = {Laumon, G{é}rard and Moret-Bailly, Laurent},
      TITLE = {Champs Algébriques},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {39},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2000},
      PAGES = {xii+208},
      ISBN = {3-540-65761-4},
      MRCLASS = {14A20 (14D20)},
      MRNUMBER = {1771927},
      MRREVIEWER = {Dan Edidin},
      ZBLNUMBER = {0945.14005},
      }
  • [MR2388554] Go to document M. Lieblich, "Twisted sheaves and the period-index problem," Compos. Math., vol. 144, iss. 1, pp. 1-31, 2008.
    @article {MR2388554, MRKEY = {2388554},
      AUTHOR = {Lieblich, Max},
      TITLE = {Twisted sheaves and the period-index problem},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {144},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {1--31},
      ISSN = {0010-437X},
      MRCLASS = {14F22 (14A20 14D20 16K50)},
      MRNUMBER = {2388554},
      MRREVIEWER = {Stefan Schr{ö}er},
      DOI = {10.1112/S0010437X07003144},
      ZBLNUMBER = {1133.14018},
      }
  • [matsuki-olsson] K. Matsuki and M. Olsson, "Kawamata-Viehweg vanishing as Kodaira vanishing for stacks," Math. Res. Lett., vol. 12, iss. 2-3, pp. 207-217, 2005.
    @article {matsuki-olsson, MRKEY = {2150877},
      AUTHOR = {Matsuki, Kenji and Olsson, Martin},
      TITLE = {Kawamata-{V}iehweg vanishing as {K}odaira vanishing for stacks},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {12},
      YEAR = {2005},
      NUMBER = {2-3},
      PAGES = {207--217},
      ISSN = {1073-2780},
      MRCLASS = {14F17 (14A20 14F10)},
      MRNUMBER = {2150877},
      MRREVIEWER = {Ivan S. Kausz},
      ZBLNUMBER = {1080.14023},
      }
  • [MR1011461] H. Matsumura, Commutative Ring Theory, Second ed., Cambridge: Cambridge Univ. Press, 1989, vol. 8.
    @book {MR1011461, MRKEY = {1011461},
      AUTHOR = {Matsumura, Hideyuki},
      TITLE = {Commutative Ring Theory},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME= {8},
      EDITION = {Second},
      NOTE = {translated from the Japanese by M. Reid},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1989},
      PAGES = {xiv+320},
      ISBN = {0-521-36764-6},
      MRCLASS = {13-01},
      MRNUMBER = {1011461},
      ZBLNUMBER = {0666.13002},
      }
  • [MR0263816] Go to document T. Matsusaka, "On canonically polarized varieties. II," Amer. J. Math., vol. 92, pp. 283-292, 1970.
    @article {MR0263816, MRKEY = {0263816},
      AUTHOR = {Matsusaka, T.},
      TITLE = {On canonically polarized varieties. {II}},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {92},
      YEAR = {1970},
      PAGES = {283--292},
      ISSN = {0002-9327},
      MRCLASS = {14.20},
      MRNUMBER = {0263816},
      MRREVIEWER = {H. Popp},
      DOI = {10.2307/2373508},
      ZBLNUMBER = {0195.22802},
      }
  • [Matsusaka72] Go to document T. Matsusaka, "Polarized varieties with a given Hilbert polynomial," Amer. J. Math., vol. 94, pp. 1027-1077, 1972.
    @article {Matsusaka72, MRKEY = {0337960},
      AUTHOR = {Matsusaka, T.},
      TITLE = {Polarized varieties with a given {H}ilbert polynomial},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {94},
      YEAR = {1972},
      PAGES = {1027--1077},
      ISSN = {0002-9327},
      MRCLASS = {14D20},
      MRNUMBER = {0337960},
      MRREVIEWER = {Daniel Comenetz},
      DOI = {10.2307/2373563},
      ZBLNUMBER = {0256.14004},
      }
  • [MVZ05] Go to document M. Möller, E. Viehweg, and K. Zuo, "Special families of curves, of abelian varieties, and of certain minimal manifolds over curves," in Global Aspects of Complex Geometry, New York: Springer-Verlag, 2006, pp. 417-450.
    @incollection {MVZ05, MRKEY = {2264111},
      AUTHOR = {M{ö}ller, Martin and Viehweg, Eckart and Zuo, Kang},
      TITLE = {Special families of curves, of abelian varieties, and of certain minimal manifolds over curves},
      BOOKTITLE = {Global Aspects of Complex Geometry},
      PAGES = {417--450},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2006},
      MRCLASS = {14H10 (14H15 14K22 14K30)},
      MRNUMBER = {2264111},
      MRREVIEWER = {Arvid Siqveland},
      DOI = {10.1007/3-540-35480-8-11},
      ZBLNUMBER = {1112.14027},
      }
  • [Mumford70] D. Mumford, Abelian Varieties, Published for the Tata Institute of Fundamental Research, Bombay, 1970, vol. 5.
    @book {Mumford70, MRKEY = {0282985},
      AUTHOR = {Mumford, David},
      TITLE = {Abelian Varieties},
      SERIES = {Tata Inst. Fund. Res. Stud. Math.},
      VOLUME={5},
      PUBLISHER = {Published for the Tata Institute of Fundamental Research, Bombay},
      YEAR = {1970},
      PAGES = {viii+242},
      MRCLASS = {14.51},
      MRNUMBER = {2514037},
      MRREVIEWER = {J. S. Milne},
      ZBLNUMBER = {0223.14022},
      }
  • [OSS] C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Mass.: Birkhäuser, 1980, vol. 3.
    @book {OSS, MRKEY = {561910},
      AUTHOR = {Okonek, Christian and Schneider, Michael and Spindler, Heinz},
      TITLE = {Vector Bundles on Complex Projective Spaces},
      SERIES = {Progr. Math.},
      VOLUME = {3},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Mass.},
      YEAR = {1980},
      PAGES = {vii+389},
      ISBN = {3-7643-3000-7},
      MRCLASS = {14-02 (14D20 14F05 32G13 81C30)},
      MRNUMBER = {561910},
      MRREVIEWER = {Werner Kleinert},
      ZBLNUMBER = {0438.32016},
      }
  • [MR2051053] Go to document M. Olsson, "A stacky semistable reduction theorem," Int. Math. Res. Not., vol. 2004, iss. 29, pp. 1497-1509, 2004.
    @article {MR2051053, MRKEY = {2051053},
      AUTHOR = {Olsson, Martin},
      TITLE = {A stacky semistable reduction theorem},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2004},
      NUMBER = {29},
      PAGES = {1497--1509},
      ISSN = {1073-7928},
      MRCLASS = {14G40 (14H10)},
      MRNUMBER = {2051053},
      MRREVIEWER = {Tohru Nakashima},
      DOI = {10.1155/S1073792804132972},
      VOLUME = {2004},
      ZBLNUMBER = {1093.14002},
      }
  • [Parshin68] A. N. Parvsin, "Algebraic curves over function fields. I," Izv. Akad. Nauk SSSR Ser. Mat., vol. 32, pp. 1191-1219, 1968.
    @article {Parshin68, MRKEY = {0257086},
      AUTHOR = {Par{š}in, A. N.},
      TITLE = {Algebraic curves over function fields. {I}},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {32},
      YEAR = {1968},
      PAGES = {1191--1219},
      ISSN = {0373-2436},
      MRCLASS = {14.20},
      MRNUMBER = {0257086},
      MRREVIEWER = {J.-E. Bertin},
      ZBLNUMBER={0181.23902},
      }
  • [MR554237] J. Serre, Local Fields, New York: Springer-Verlag, 1979, vol. 67.
    @book {MR554237, MRKEY = {554237},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Local Fields},
      SERIES = {Grad. Texts Math.},
      VOLUME = {67},
      NOTE = {translated from the French by Marvin Jay Greenberg},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1979},
      PAGES = {viii+241},
      ISBN = {0-387-90424-7},
      MRCLASS = {12Bxx},
      MRNUMBER = {554237},
      ZBLNUMBER = {0423.12016},
      }
  • [SGA3] Schémas en Groupes. I: Propriétés Générales des Schémas en Groupes, New York: Springer-Verlag, 1970, vol. 151.
    @book{SGA3, KEY={SGA3},
      TITLE={Schémas en Groupes. {\rm I}: {P}ropriétés Générales des Schémas en Groupes},
      SERIES = {Lecture Notes in Math.},
      VOLUME={151},
      NOTE={Séminaire de Géométrie Algébrique du Bois Marie 1962/64. Dirigé par M. Demazure et A. Grothendieck},
      PUBLISHER={Springer-Verlag},
      ADDRESS={New York},
      YEAR={1970},
      MRNUMBER={0274458},
      ZBLNUMBER={0207.51401},
      }
  • [SGA6] Théorie des Intersections et Théorème de Riemann-Roch, New York: Springer-Verlag, 1971, vol. 225.
    @book {SGA6, KEY={SGA6},
      MRKEY = {0354655},
      TITLE = {Théorie des Intersections et Théorème de {R}iemann-{R}och},
      SERIES = {Lecture Notes in Math.},
      VOLUME={225},
      NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1966--1967 (SGA 6), Dirig{é} par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {xii+700},
      MRCLASS = {14-06},
      MRNUMBER = {0354655},
      ZBLNUMBER={0218.14001},
      }
  • [MR1647069] Go to document M. Spivakovsky, "A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms," J. Amer. Math. Soc., vol. 12, iss. 2, pp. 381-444, 1999.
    @article {MR1647069, MRKEY = {1647069},
      AUTHOR = {Spivakovsky, Mark},
      TITLE = {A new proof of {D}. {P}opescu's theorem on smoothing of ring homomorphisms},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {12},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {381--444},
      ISSN = {0894-0347},
      MRCLASS = {13B40 (13F40 14B12 14E40)},
      MRNUMBER = {1647069},
      MRREVIEWER = {D.-M. Popescu},
      DOI = {10.1090/S0894-0347-99-00294-5},
      ZBLNUMBER = {0919.13009},
      }
  • [MR1106918] Go to document R. W. Thomason and T. Trobaugh, "Higher algebraic $K$-theory of schemes and of derived categories," in The Grothendieck Festschrift, Vol. III, Boston, MA: Birkhäuser, 1990, vol. 88, pp. 247-435.
    @incollection {MR1106918, MRKEY = {1106918},
      AUTHOR = {Thomason, R. W. and Trobaugh, Thomas},
      TITLE = {Higher algebraic {$K$}-theory of schemes and of derived categories},
      BOOKTITLE = {The {G}rothendieck {F}estschrift, {V}ol. {\rm III}},
      SERIES = {Progr. Math.},
      VOLUME = {88},
      PAGES = {247--435},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {19E08 (14C35 19D10)},
      MRNUMBER = {1106918},
      MRREVIEWER = {Charles Weibel},
      DOI = {10.1007/978-0-8176-4576-2-10},
      ZBLNUMBER = {0731.14001},
      }
  • [Viehweg83a] E. Viehweg, "Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces," in Algebraic Varieties and Analytic Varieties, Amsterdam: North-Holland, 1983, vol. 1, pp. 329-353.
    @incollection {Viehweg83a, MRKEY = {715656},
      AUTHOR = {Viehweg, Eckart},
      TITLE = {Weak positivity and the additivity of the {K}odaira dimension for certain fibre spaces},
      BOOKTITLE = {Algebraic Varieties and Analytic Varieties},
      VENUE={{T}okyo, 1981},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {1},
      PAGES = {329--353},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1983},
      MRCLASS = {14J10 (14D20 14F05)},
      MRNUMBER = {715656},
      MRREVIEWER = {Yujiro Kawamata},
      ZBLNUMBER = {0513.14019},
      }
  • [Viehweg95] E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, New York: Springer-Verlag, 1995, vol. 30.
    @book {Viehweg95, MRKEY = {1368632},
      AUTHOR = {Viehweg, Eckart},
      TITLE = {Quasi-Projective Moduli for Polarized Manifolds},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {30},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {viii+320},
      ISBN = {3-540-59255-5},
      MRCLASS = {14-02 (14D20 14D22)},
      MRNUMBER = {1368632},
      MRREVIEWER = {P. E. Newstead},
      ZBLNUMBER = {0844.14004},
      }
  • [Viehweg01] E. Viehweg, "Positivity of direct image sheaves and applications to families of higher dimensional manifolds," in School on Vanishing Theorems and Effective Results in Algebraic Geometry, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, vol. 6, pp. 249-284.
    @incollection {Viehweg01, MRKEY = {1919460},
      AUTHOR = {Viehweg, Eckart},
      TITLE = {Positivity of direct image sheaves and applications to families of higher dimensional manifolds},
      BOOKTITLE = {School on {V}anishing {T}heorems and {E}ffective {R}esults in {A}lgebraic {G}eometry},
      VENUE={{T}rieste, 2000},
      SERIES = {{\rm ICTP} Lect. Notes},
      VOLUME = {6},
      PAGES = {249--284},
      PUBLISHER = {Abdus Salam Int. Cent. Theoret. Phys., Trieste},
      YEAR = {2001},
      MRCLASS = {14H10 (14J10)},
      MRNUMBER = {1919460},
      MRREVIEWER = {Arvid Siqveland},
      ZBLNUMBER = {1092.14044},
      }
  • [Viehweg06a] Go to document E. Viehweg, "Compactifications of smooth families and of moduli spaces of polarized manifolds," Ann. of Math., vol. 172, pp. 809-910, 2010.
    @article{Viehweg06a,
      author = {Viehweg, Eckart},
      TITLE={Compactifications of smooth families and of moduli spaces of polarized manifolds},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={809--910},
      MRNUMBER={2680483},
      ZBLNUMBER={pre05808578},
      DOI={10.4007/annals.2010.172.809},
      }
  • [Vie-Zuo01] E. Viehweg and K. Zuo, "On the isotriviality of families of projective manifolds over curves," J. Algebraic Geom., vol. 10, iss. 4, pp. 781-799, 2001.
    @article {Vie-Zuo01, MRKEY = {1838979},
      AUTHOR = {Viehweg, Eckart and Zuo, Kang},
      TITLE = {On the isotriviality of families of projective manifolds over curves},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {10},
      YEAR = {2001},
      NUMBER = {4},
      PAGES = {781--799},
      ISSN = {1056-3911},
      MRCLASS = {14D06 (14J10)},
      MRNUMBER = {1838979},
      MRREVIEWER = {S{á}ndor J. Kov{á}cs},
      ZBLNUMBER = {1079.14503},
      }
  • [Vie-Zuo02] E. Viehweg and K. Zuo, "Base spaces of non-isotrivial families of smooth minimal models," in Complex Geometry, New York: Springer-Verlag, 2002, pp. 279-328.
    @incollection {Vie-Zuo02, MRKEY = {1922109},
      AUTHOR = {Viehweg, Eckart and Zuo, Kang},
      TITLE = {Base spaces of non-isotrivial families of smooth minimal models},
      BOOKTITLE = {Complex Geometry},
      VENUE={{G}öttingen, 2000},
      PAGES = {279--328},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2002},
      MRCLASS = {14D06 (14E30)},
      MRNUMBER = {1922109},
      MRREVIEWER = {Mark Gross},
      ZBLNUMBER = {1006.14004},
      }
  • [Vie-Zuo03a] E. Viehweg and K. Zuo, "Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks," in Surveys in Differential Geometry, Vol. VIII, Internat. Press, Somerville, MA, 2003, vol. VIII, pp. 337-356.
    @incollection {Vie-Zuo03a, MRKEY = {2039995},
      AUTHOR = {Viehweg, Eckart and Zuo, Kang},
      TITLE = {Discreteness of minimal models of {K}odaira dimension zero and subvarieties of moduli stacks},
      BOOKTITLE = {Surveys in Differential Geometry, {V}ol. {\rm VIII}},
      VENUE={{B}oston, {MA},
      2002},
      SERIES = {Surv. Differ. Geom.},
      VOLUME={VIII},
      PAGES = {337--356},
      PUBLISHER = {Internat. Press, Somerville, MA},
      YEAR = {2003},
      MRCLASS = {14D20 (14E30)},
      MRNUMBER = {1922109},
      MRREVIEWER = {Thiruvalloor E. Venkata Balaji},
      ZBLNUMBER = {1085.14016},
      }
  • [MR2223406] A. Vistoli, "Grothendieck topologies, fibered categories and descent theory," in Fundamental Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 123, pp. 1-104.
    @incollection {MR2223406, MRKEY = {2223406},
      AUTHOR = {Vistoli, Angelo},
      TITLE = {Grothendieck topologies, fibered categories and descent theory},
      BOOKTITLE = {Fundamental Algebraic Geometry},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {123},
      PAGES = {1--104},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      MRCLASS = {14F20 (14A20)},
      MRNUMBER = {2223406},
      ZBLNUMBER={1085.14001},
      }

Authors

Sándor J. Kovács

Department of Mathematics
University of Washington,
Seattle, WA 98195

Max Lieblich

Department of Mathematics
University of Washington,
Seattle, WA 98195