On the birational automorphisms of varieties of general type

Pages 1077-1111 from Volume 177 (2013), Issue 3 by Christopher D. Hacon, James M$^{\textbf{c}}$Kernan, Chenyang Xu

Abstract

We show that the number of birational automorphisms of a variety of general type $X$ is bounded by $c \cdot \mathrm{vol}(X,K_X)$, where $c$ is a constant that only depends on the dimension of $X$.

  • [Alexeev94] Go to document V. Alexeev, "Boundedness and $K^2$ for log surfaces," Internat. J. Math., vol. 5, iss. 6, pp. 779-810, 1994.
    @article {Alexeev94, MRKEY = {1298994},
      AUTHOR = {Alexeev, Valery},
      TITLE = {Boundedness and {$K\sp 2$} for log surfaces},
      JOURNAL = {Internat. J. Math.},
      FJOURNAL = {International Journal of Mathematics},
      VOLUME = {5},
      YEAR = {1994},
      NUMBER = {6},
      PAGES = {779--810},
      ISSN = {0129-167X},
      MRCLASS = {14J10 (14J25)},
      MRNUMBER = {1298994},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1142/S0129167X94000395},
      ZBLNUMBER = {0838.14028},
      }
  • [AM04] Go to document V. Alexeev and S. Mori, "Bounding singular surfaces of general type," in Algebra, Arithmetic and Geometry with Applications, New York: Springer-Verlag, 2004, pp. 143-174.
    @incollection {AM04, MRKEY = {2037085},
      AUTHOR = {Alexeev, Valery and Mori, Shigefumi},
      TITLE = {Bounding singular surfaces of general type},
      BOOKTITLE = {Algebra, Arithmetic and Geometry with Applications},
      VENUE={{W}est {L}afayette, {IN},
      2000},
      PAGES = {143--174},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      MRCLASS = {14J29},
      MRNUMBER = {2037085},
      MRREVIEWER = {S{á}ndor J. Kov{á}cs},
      ZBLNUMBER = {1103.14021},
      DOI = {10.1007/978-3-642-18487-1_7},
     }
  • [Andreotti50] A. Andreotti, "Sopra le superficie algebriche che posseggono trasformazioni birazionali in sè," Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl., vol. 9, pp. 255-279, 1950.
    @article {Andreotti50, MRKEY = {0049601},
      AUTHOR = {Andreotti, Aldo},
      TITLE = {Sopra le superficie algebriche che posseggono trasformazioni birazionali in sè},
      JOURNAL = {Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl.},
      VOLUME = {9},
      YEAR = {1950},
      PAGES = {255--279},
      MRCLASS = {14.0X},
      MRNUMBER = {0049601},
      MRREVIEWER = {J. A. Todd},
      ZBLNUMBER = {0041.08502},
     }
  • [AS95] Go to document U. Angehrn and Y. T. Siu, "Effective freeness and point separation for adjoint bundles," Invent. Math., vol. 122, iss. 2, pp. 291-308, 1995.
    @article {AS95, MRKEY = {1358978},
      AUTHOR = {Angehrn, Urban and Siu, Yum Tong},
      TITLE = {Effective freeness and point separation for adjoint bundles},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {122},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {291--308},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32J25 (14C20 32L10)},
      MRNUMBER = {1358978},
      MRREVIEWER = {Jaros{\l}aw A. Wi{\'s}niewski},
      DOI = {10.1007/BF01231446},
      ZBLNUMBER = {0847.32035},
      }
  • [Ballico93] E. Ballico, "On the automorphisms of surfaces of general type in positive characteristic," Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., vol. 4, iss. 2, pp. 121-129, 1993.
    @article {Ballico93, MRKEY = {1233400},
      AUTHOR = {Ballico, Edoardo},
      TITLE = {On the automorphisms of surfaces of general type in positive characteristic},
      JOURNAL = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl.},
      FJOURNAL = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni},
      VOLUME = {4},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {121--129},
      ISSN = {1120-6330},
      MRCLASS = {14J29 (14E09)},
      MRNUMBER = {1233400},
      MRREVIEWER = {Yoichi Miyaoka},
      ZBLNUMBER = {0795.14022},
      }
  • [BP10] Go to document B. Berndtsson and M. Puaun, "Quantitative extensions of pluricanonical forms and closed positive currents," Nagoya Math. J., vol. 205, pp. 25-65, 2012.
    @article {BP10, MRKEY = {2891164},
      AUTHOR = {Berndtsson, Bo and P{\u{a}}un, Mihai},
      TITLE = {Quantitative extensions of pluricanonical forms and closed positive currents},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {205},
      YEAR = {2012},
      PAGES = {25--65},
      ISSN = {0027-7630},
      MRCLASS = {32Q15 (14D07 32U40)},
      MRNUMBER = {2891164},
      MRREVIEWER = {James McKernan},
      URL = {http://projecteuclid.org/euclid.nmj/1330611001},
      ZBLNUMBER = {1248.32012},
      }
  • [BCHM06] Go to document C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, "Existence of minimal models for varieties of log general type," J. Amer. Math. Soc., vol. 23, iss. 2, pp. 405-468, 2010.
    @article {BCHM06, MRKEY = {2601039},
      AUTHOR = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
      TITLE = {Existence of minimal models for varieties of log general type},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {23},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {405--468},
      ISSN = {0894-0347},
      MRCLASS = {14E30 (14E05)},
      MRNUMBER = {2601039},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1090/S0894-0347-09-00649-3},
      ZBLNUMBER = {1210.14019},
      }
  • [Cai00] Go to document J. Cai, "Bounds of automorphisms of surfaces of general type in positive characteristic," J. Pure Appl. Algebra, vol. 149, iss. 3, pp. 241-250, 2000.
    @article {Cai00, MRKEY = {1762766},
      AUTHOR = {Cai, Jin-Xing},
      TITLE = {Bounds of automorphisms of surfaces of general type in positive characteristic},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {149},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {241--250},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {14J50 (14H37 14J29)},
      MRNUMBER = {1762766},
      MRREVIEWER = {Miguel A. Barja},
      DOI = {10.1016/S0022-4049(99)00064-X},
      ZBLNUMBER = {0968.14023},
      }
  • [CS95] Go to document F. Catanese and M. Schneider, "Polynomial bounds for abelian groups of automorphisms," Compositio Math., vol. 97, iss. 1-2, pp. 1-15, 1995.
    @article {CS95, MRKEY = {1355114},
      AUTHOR = {Catanese, Fabrizio and Schneider, Michael},
      TITLE = {Polynomial bounds for abelian groups of automorphisms},
      NOTE = {Special issue in honour of Frans Oort},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {97},
      YEAR = {1995},
      NUMBER = {1-2},
      PAGES = {1--15},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14L30 (14E09)},
      MRNUMBER = {1355114},
      MRREVIEWER = {P. Abellanas},
      URL = {http://www.numdam.org/item?id=CM_1995__97_1-2_1_0},
      ZBLNUMBER = {0858.14008},
      }
  • [Corti91] Go to document A. Corti, "Polynomial bounds for the number of automorphisms of a surface of general type," Ann. Sci. École Norm. Sup., vol. 24, iss. 1, pp. 113-137, 1991.
    @article {Corti91, MRKEY = {1088272},
      AUTHOR = {Corti, Alessio},
      TITLE = {Polynomial bounds for the number of automorphisms of a surface of general type},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {24},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {113--137},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14J50 (14J29)},
      MRNUMBER = {1088272},
      MRREVIEWER = {Jong Hae Keum},
      URL = {http://www.numdam.org/item?id=ASENS_1991_4_24_1_113_0},
      ZBLNUMBER = {0758.14026},
      }
  • [FH11] Go to document T. de Fernex and C. D. Hacon, "Deformations of canonical pairs and Fano varieties," J. Reine Angew. Math., vol. 651, pp. 97-126, 2011.
    @article {FH11, MRKEY = {2774312},
      AUTHOR = {de Fernex, Tommaso and Hacon, Christopher D.},
      TITLE = {Deformations of canonical pairs and {F}ano varieties},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {651},
      YEAR = {2011},
      PAGES = {97--126},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {14E30 (14J45)},
      MRNUMBER = {2774312},
      MRREVIEWER = {Anne-Sophie Kaloghiros},
      DOI = {10.1515/CRELLE.2011.010},
      ZBLNUMBER = {1220.14026},
      }
  • [Fulton93] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993, vol. 131.
    @book {Fulton93, MRKEY = {1234037},
      AUTHOR = {Fulton, William},
      TITLE = {Introduction to Toric Varieties},
      SERIES = {Annals of Math. Studies},
      VOLUME = {131},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1993},
      PAGES = {xii+157},
      ISBN = {0-691-00049-2},
      MRCLASS = {14M25 (14-02 14J30)},
      MRNUMBER = {1234037},
      MRREVIEWER = {T. Oda},
      ZBLNUMBER = {0813.14039},
      }
  • [GLS94] D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Providence, RI: Amer. Math. Soc., 1994, vol. 40.
    @book {GLS94, MRKEY = {1303592},
      AUTHOR = {Gorenstein, Daniel and Lyons, Richard and Solomon, Ronald},
      TITLE = {The Classification of the Finite Simple Groups},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {40},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1994},
      PAGES = {xiv+165},
      ISBN = {0-8218-0334-4},
      MRCLASS = {20D05 (20-02)},
      MRNUMBER = {1303592},
      MRREVIEWER = {J. L. Alperin},
      ZBLNUMBER = {0816.20016},
      }
  • [EGAIV] Go to document A. Grothendieck, "Eléménts de Géométrie Algébrique. IV," Publ. Math. IHES, vol. 28, pp. 5-255, 1966.
    @article{EGAIV, MRKEY={0217086},
      AUTHOR={Grothendieck, A.},
      TITLE={Eléménts de Géométrie Algébrique. {IV}},
      JOURNAL={Publ. Math. IHES},
      VOLUME={28},
      YEAR={1966},
      PAGES={5--255},
      ZBLNUMBER = {0144.19904},
      MRNUMBER = {0217086},
      URL = {http://www.numdam.org/item?id=PMIHES_1966__28__5_0},
     }
  • [HM05b] Go to document C. D. Hacon and J. McKernan, "Boundedness of pluricanonical maps of varieties of general type," Invent. Math., vol. 166, iss. 1, pp. 1-25, 2006.
    @article {HM05b, MRKEY = {2242631},
      AUTHOR = {Hacon, Christopher D. and McKernan, James},
      TITLE = {Boundedness of pluricanonical maps of varieties of general type},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {166},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {1--25},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14E05 (14J40)},
      MRNUMBER = {2242631},
      MRREVIEWER = {Meng Chen},
      DOI = {10.1007/s00222-006-0504-1},
      ZBLNUMBER = {1121.14011},
      }
  • [HS91] Go to document A. T. Huckleberry and M. Sauer, "On the order of the automorphism group of a surface of general type," Math. Z., vol. 205, iss. 2, pp. 321-329, 1990.
    @article {HS91, MRKEY = {1076138},
      AUTHOR = {Huckleberry, A. T. and Sauer, M.},
      TITLE = {On the order of the automorphism group of a surface of general type},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {205},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {321--329},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {14J50 (14J29)},
      MRNUMBER = {1076138},
      MRREVIEWER = {Jong Hae Keum},
      DOI = {10.1007/BF02571245},
      ZBLNUMBER = {0718.32022},
      }
  • [Kollar92b] Go to document J. Kollár, "Log surfaces of general type; some conjectures," in Classification of Algebraic Varieties, Providence, RI: Amer. Math. Soc., 1994, vol. 162, pp. 261-275.
    @incollection {Kollar92b, MRKEY = {1272703},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Log surfaces of general type; some conjectures},
      BOOKTITLE = {Classification of Algebraic Varieties},
      VENUE={{L}'{A}quila, 1992},
      SERIES = {Contemp. Math.},
      VOLUME = {162},
      PAGES = {261--275},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1994},
      MRCLASS = {14J29 (14E05)},
      MRNUMBER = {1272703},
      MRREVIEWER = {Gang Xiao},
      DOI = {10.1090/conm/162/01538},
      ZBLNUMBER = {0860.14014},
      }
  • [Kollar95] J. Kollár, "Singularities of pairs," in Algebraic Geometry—Santa Cruz 1995, Providence, RI: Amer. Math. Soc., 1997, vol. 62, pp. 221-287.
    @incollection {Kollar95, MRKEY = {1492525},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Singularities of pairs},
      BOOKTITLE = {Algebraic Geometry---{S}anta {C}ruz 1995},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {62},
      PAGES = {221--287},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1997},
      MRCLASS = {14E30 (14E15 32S40)},
      MRNUMBER = {1492525},
      MRREVIEWER = {Alessio Corti},
      ZBLNUMBER = {0905.14002},
      }
  • [Kollar08] Go to document J. Kollár, "Is there a topological Bogomolov-Miyaoka-Yau inequality?," Pure Appl. Math. Q., vol. 4, pp. 203-236, 2008.
    @article {Kollar08, MRKEY = {2400877},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Is there a topological {B}ogomolov-{M}iyaoka-{Y}au inequality?},
      JOURNAL = {Pure Appl. Math. Q.},
      FJOURNAL = {Pure and Applied Mathematics Quarterly},
      VOLUME = {4},
      YEAR = {2008},
      PAGES = {203--236},
      ISSN = {1558-8599},
      MRCLASS = {14J80 (14J17 57N13)},
      MRNUMBER = {2400877},
      MRREVIEWER = {Arnaud Beauville},
      ZBLNUMBER = {1145.14031},
      DOI = {10.1090/pspum/062.1/1492525},
     }
  • [KM98] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
    @book {KM98, 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},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {viii+254},
      ISBN = {0-521-63277-3},
      MRCLASS = {14E30},
      MRNUMBER = {1658959},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1017/CBO9780511662560},
      ZBLNUMBER = {0926.14003},
      }
  • [Lazarsfeld04b] R. Lazarsfeld, Positivity in Algebraic Geometry. II, New York: Springer-Verlag, 2004, vol. 49.
    @book {Lazarsfeld04b, MRKEY = {2095472},
      AUTHOR = {Lazarsfeld, Robert},
      TITLE = {Positivity in Algebraic Geometry. {II}},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {49},
      NOTE = {Positivity for vector bundles, and multiplier ideals},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      PAGES = {xviii+385},
      ISBN = {3-540-22534-X},
      MRCLASS = {14-02 (14C20 14F05 14F17)},
      MRNUMBER = {2095472},
      MRREVIEWER = {Mihnea Popa},
      ZBLNUMBER = {1093.14500},
      }
  • [Matsumura63] H. Matsumura, "On algebraic groups of birational transformations," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 34, pp. 151-155, 1963.
    @article {Matsumura63, MRKEY = {0159825},
      AUTHOR = {Matsumura, Hideyuki},
      TITLE = {On algebraic groups of birational transformations},
      JOURNAL = {Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.},
      FJOURNAL = {Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali},
      VOLUME = {34},
      YEAR = {1963},
      PAGES = {151--155},
      MRCLASS = {14.15},
      MRNUMBER = {0159825},
      MRREVIEWER = {P. Cartier},
      ZBLNUMBER = {0134.16601},
      }
  • [Nakayama04] N. Nakayama, Zariski-Decomposition and Abundance, Tokyo: Mathematical Society of Japan, 2004, vol. 14.
    @book {Nakayama04, MRKEY = {2104208},
      AUTHOR = {Nakayama, Noboru},
      TITLE = {Zariski-Decomposition and Abundance},
      SERIES = {MSJ Memoirs},
      VOLUME = {14},
      PUBLISHER = {Mathematical Society of Japan},
      ADDRESS = {Tokyo},
      YEAR = {2004},
      PAGES = {xiv+277},
      ISBN = {4-931469-31-0},
      MRCLASS = {14C20 (14E15 14E30 14J10)},
      MRNUMBER = {2104208},
      MRREVIEWER = {Tommaso De Fernex},
      ZBLNUMBER = {1061.14018},
      }
  • [Siu98] Go to document Y. Siu, "Invariance of plurigenera," Invent. Math., vol. 134, iss. 3, pp. 661-673, 1998.
    @article {Siu98, MRKEY = {1660941},
      AUTHOR = {Siu, Yum-Tong},
      TITLE = {Invariance of plurigenera},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {134},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {661--673},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32L10 (32G13 32J18)},
      MRNUMBER = {1660941},
      MRREVIEWER = {J{á}nos Koll{á}r},
      DOI = {10.1007/s002220050276},
      ZBLNUMBER = {0955.32017},
      }
  • [Stichtenoth73] H. Stichtenoth, "Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe," Arch. Math. $($Basel$)$, vol. 24, pp. 527-544, 1973.
    @article {Stichtenoth73, MRKEY = {0337980},
      AUTHOR = {Stichtenoth, Henning},
      TITLE = {Über die {A}utomorphismengruppe eines algebraischen {F}unktionenkörpers von {P}rimzahlcharakteristik. {I}. {E}ine {A}bschätzung der {O}rdnung der {A}utomorphismengruppe},
      JOURNAL = {Arch. Math. $($Basel$)$},
      FJOURNAL = {Archiv der Mathematik},
      VOLUME = {24},
      YEAR = {1973},
      PAGES = {527--544},
      ISSN = {0003-889X},
      MRCLASS = {14H05},
      MRNUMBER = {0337980},
      MRREVIEWER = {W.-D. Geyer},
      ZBLNUMBER = {0282.14007},
     }
  • [Szabo96] Go to document E. Szabó, "Bounding automorphism groups," Math. Ann., vol. 304, iss. 4, pp. 801-811, 1996.
    @article {Szabo96, MRKEY = {1380456},
      AUTHOR = {Szab{ó},
      Endre},
      TITLE = {Bounding automorphism groups},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {304},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {801--811},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14J50 (14E09)},
      MRNUMBER = {1380456},
      MRREVIEWER = {Alessio Corti},
      DOI = {10.1007/BF01446320},
      ZBLNUMBER = {0845.14009},
      }
  • [Takayama06] Go to document S. Takayama, "Pluricanonical systems on algebraic varieties of general type," Invent. Math., vol. 165, iss. 3, pp. 551-587, 2006.
    @article {Takayama06, MRKEY = {2242627},
      AUTHOR = {Takayama, Shigeharu},
      TITLE = {Pluricanonical systems on algebraic varieties of general type},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {165},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {551--587},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14E05 (14C20 14J40)},
      MRNUMBER = {2242627},
      MRREVIEWER = {Tomasz Szemberg},
      DOI = {10.1007/s00222-006-0503-2},
      ZBLNUMBER = {1108.14031},
      }
  • [Tsuji00] H. Tsuji, Bound of automorphisms of projective varieties of general type.
    @misc{Tsuji00,
      author = {Tsuji, Hajime},
      TITLE = {Bound of automorphisms of projective varieties of general type},
      ARXIV = {math/0004138},
      SORTYEAR={2014},
      }
  • [Tsuji07] Go to document H. Tsuji, "Pluricanonical systems of projective varieties of general type. II," Osaka J. Math., vol. 44, iss. 3, pp. 723-764, 2007.
    @article {Tsuji07, MRKEY = {2360948},
      AUTHOR = {Tsuji, Hajime},
      TITLE = {Pluricanonical systems of projective varieties of general type. {II}},
      JOURNAL = {Osaka J. Math.},
      FJOURNAL = {Osaka Journal of Mathematics},
      VOLUME = {44},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {723--764},
      ISSN = {0030-6126},
      CODEN = {OJMAA7},
      MRCLASS = {14E30 (14J40 32J18 32L10)},
      MRNUMBER = {2360948},
      MRREVIEWER = {Gianluca Occhetta},
      URL = {http://projecteuclid.org/euclid.ojm/1189717430},
      ZBLNUMBER = {1186.14043},
      }
  • [Xiao94] Go to document G. Xiao, "Bound of automorphisms of surfaces of general type. I," Ann. of Math., vol. 139, iss. 1, pp. 51-77, 1994.
    @article {Xiao94, MRKEY = {1259364},
      AUTHOR = {Xiao, Gang},
      TITLE = {Bound of automorphisms of surfaces of general type. {I}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {139},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {51--77},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14J50 (14J29)},
      MRNUMBER = {1259364},
      MRREVIEWER = {Sonny Tendian},
      DOI = {10.2307/2946627},
      ZBLNUMBER = {0811.14011},
      }
  • [Xiao95] G. Xiao, "Bound of automorphisms of surfaces of general type. II," J. Algebraic Geom., vol. 4, iss. 4, pp. 701-793, 1995.
    @article {Xiao95, MRKEY = {1339845},
      AUTHOR = {Xiao, Gang},
      TITLE = {Bound of automorphisms of surfaces of general type. {II}},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {4},
      YEAR = {1995},
      NUMBER = {4},
      PAGES = {701--793},
      ISSN = {1056-3911},
      MRCLASS = {14J50 (14J29)},
      MRNUMBER = {1339845},
      MRREVIEWER = {Kang Zuo},
      ZBLNUMBER = {0841.14011},
      }
  • [Xiao96] Go to document G. Xiao, "Linear bound for abelian automorphisms of varieties of general type," J. Reine Angew. Math., vol. 476, pp. 201-207, 1996.
    @article {Xiao96, MRKEY = {1401701},
      AUTHOR = {Xiao, Gang},
      TITLE = {Linear bound for abelian automorphisms of varieties of general type},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {476},
      YEAR = {1996},
      PAGES = {201--207},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {14J50},
      MRNUMBER = {1401701},
      MRREVIEWER = {P. Abellanas},
      DOI = {10.1515/crll.1996.476.201},
      ZBLNUMBER = {0870.14006},
      }
  • [Zhang07] Go to document D. Zhang, "Small bound for birational automorphism groups of algebraic varieties," Math. Ann., vol. 339, iss. 4, pp. 957-975, 2007.
    @article {Zhang07, MRKEY = {2341909},
      AUTHOR = {Zhang, De-Qi},
      TITLE = {Small bound for birational automorphism groups of algebraic varieties},
      NOTE = {with an appendix by Yujiro Kawamata},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {339},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {957--975},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14J50 (14E07)},
      MRNUMBER = {2341909},
      MRREVIEWER = {Massimiliano Mella},
      DOI = {10.1007/s00208-007-0162-7},
      ZBLNUMBER = {1128.14029},
      }

Keywords
,
DOI
https://doi.org/10.4007/annals.2013.177.3.6
MR
3034294
zbMATH
06176988
Milestones
Received: 8 November 2010
Revised: 26 April 2012
Accepted: 18 June 2012
Published online: 1 May 2013

Authors

Christopher D. Hacon

Department of Mathematics, The University of Utah, 155 S 1400 E, Salt Lake City, UT 84112-0090

James M$^{\textbf{c}}$Kernan

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307

Chenyang Xu

Beijing International Center of Mathematics Research, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China P.R.