ACC for log canonical thresholds

Abstract

We show that log canonical thresholds satisfy the \rm ACC.

  • [Shokurov93] Go to document V. V. Shokurov, "Three-dimensional log perestroikas," Izv. Ross. Akad. Nauk Ser. Mat., vol. 56, iss. 1, pp. 105-203, 1992.
    @article{Shokurov93,
      author = {Shokurov, V. V.},
      journal = {Izv. Ross. Akad. Nauk Ser. Mat.},
      number = {1},
      pages = {105--203},
      title = {Three-dimensional log perestroikas},
      volume = {56},
      year = {1992},
      doi = {10.1070/IM1993v040n01ABEH001862},
      issn = {0373-2436},
      }
  • [Kollar92b] Go to document J. Kollár, "Log surfaces of general type; some conjectures," in Classification of Algebraic Varieties, Amer. Math. Soc., Providence, RI, 1994, vol. 162, pp. 261-275.
    @incollection{Kollar92b,
      author = {Koll{á}r, J{á}nos},
      booktitle = {Classification of Algebraic Varieties},
      pages = {261--275},
      publisher = {Amer. Math. Soc., Providence, RI},
      series = {Contemp. Math.},
      title = {Log surfaces of general type; some conjectures},
      volume = {162},
      year = {1994},
      doi = {10.1090/conm/162/01538},
      }
  • [Kollar95] J. Kollár, "Singularities of pairs," in Algebraic Geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, vol. 62, pp. 221-287.
    @incollection{Kollar95,
      author = {Koll{á}r, J{á}nos},
      booktitle = {Algebraic Geometry---{S}anta {C}ruz 1995},
      pages = {221--287},
      publisher = {Amer. Math. Soc., Providence, RI},
      series = {Proc. Sympos. Pure Math.},
      title = {Singularities of pairs},
      volume = {62},
      year = {1997},
      }
  • [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,
      author = {Alexeev, Valery},
      journal = {Internat. J. Math.},
      number = {6},
      pages = {779--810},
      title = {Boundedness and {$K^2$} for log surfaces},
      volume = {5},
      year = {1994},
      doi = {10.1142/S0129167X94000395},
      issn = {0129-167X},
      }
  • [EFM09] Go to document T. de Fernex, L. Ein, and M. Mustactua, "Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties," Duke Math. J., vol. 152, iss. 1, pp. 93-114, 2010.
    @article{EFM09,
      author = {de Fernex, Tommaso and Ein, Lawrence and Musta{\c{t}}{\u{a}},
      Mircea},
      journal = {Duke Math. J.},
      number = {1},
      pages = {93--114},
      title = {Shokurov's {ACC} conjecture for log canonical thresholds on smooth varieties},
      volume = {152},
      year = {2010},
      doi = {10.1215/00127094-2010-008},
      issn = {0012-7094},
      }
  • [EFM11] Go to document T. de Fernex, L. Ein, and M. Mustactua, "Log canonical thresholds on varieties with bounded singularities," in Classification of Algebraic Varieties, Eur. Math. Soc., Zürich, 2011, pp. 221-257.
    @incollection{EFM11,
      author = {de Fernex, Tommaso and Ein, Lawrence and Musta{\c{t}}{\u{a}},
      Mircea},
      booktitle = {Classification of Algebraic Varieties},
      pages = {221--257},
      publisher = {Eur. Math. Soc., Zürich},
      series = {EMS Ser. Congr. Rep.},
      title = {Log canonical thresholds on varieties with bounded singularities},
      year = {2011},
      doi = {10.4171/007-1/10},
      }
  • [Kollar07] Go to document J. Kollár, "Kodaira’s canonical bundle formula and adjunction," in Flips for 3-Folds and 4-Folds, Oxford Univ. Press, Oxford, 2007, vol. 35, pp. 134-162.
    @incollection{Kollar07,
      author = {Koll{á}r, J{á}nos},
      booktitle = {Flips for 3-Folds and 4-Folds},
      pages = {134--162},
      publisher = {Oxford Univ. Press, Oxford},
      series = {Oxford Lecture Ser. Math. Appl.},
      title = {Kodaira's canonical bundle formula and adjunction},
      volume = {35},
      year = {2007},
      doi = {10.1093/acprof:oso/9780198570615.003.0008},
      }
  • [Totaro11] Go to document B. Totaro, "The ACC conjecture for log canonical thresholds (after de Fernex, Ein, Musta\c t\u a, Kollár)," in Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026, , 2011, vol. 339, p. exp. no. 1025, ix, 371-385.
    @incollection{Totaro11,
      author = {Totaro, Burt},
      booktitle = {S{é}minaire Bourbaki. Vol. 2009/2010. Expos{é}s 1012--1026},
      pages = {Exp. No. 1025, ix, 371--385},
      series = {Astérisque},
      title = {The {ACC} conjecture for log canonical thresholds (after de {F}ernex, {E}in, {M}usta\c t\u a, {K}ollár)},
      volume = {339},
      year = {2011},
      isbn = {978-2-85629-326-3},
      issn = {0303-1179},
      url = {http://smf4.emath.fr/Publications/Asterisque/2011/339/html/ smf_ast_339.php},
      }
  • [Birkar05] Go to document C. Birkar, "Ascending chain condition for log canonical thresholds and termination of log flips," Duke Math. J., vol. 136, iss. 1, pp. 173-180, 2007.
    @article{Birkar05,
      author = {Birkar, Caucher},
      journal = {Duke Math. J.},
      number = {1},
      pages = {173--180},
      title = {Ascending chain condition for log canonical thresholds and termination of log flips},
      volume = {136},
      year = {2007},
      doi = {10.1215/S0012-7094-07-13615-9},
      issn = {0012-7094},
      }
  • [HMX10] Go to document C. D. Hacon, J. McKernan, and C. Xu, "On the birational automorphisms of varieties of general type," Ann. of Math., vol. 177, iss. 3, pp. 1077-1111, 2013.
    @article{HMX10,
      author = {Hacon, Christopher D. and McKernan, James and Xu, Chenyang},
      journal = {Ann. of Math.},
      number = {3},
      pages = {1077--1111},
      title = {On the birational automorphisms of varieties of general type},
      volume = {177},
      year = {2013},
      doi = {10.4007/annals.2013.177.3.6},
      issn = {0003-486X},
      }
  • [Kollar10a] J. Kollár, "Moduli of varieties of general type," in Handbook of Moduli: Volume II, Somerville, MA: Internat. Press, 2013, vol. 24, pp. 115-130.
    @incollection{Kollar10a, address = {Somerville, MA},
      author = {Koll{á}r, J{á}nos},
      booktitle = {Handbook of Moduli: Volume {II}},
      pages = {115--130},
      publisher = {Internat. Press},
      series = {Adv. Lect. Math. (ALM)},
      title = {Moduli of varieties of general type},
      volume = {24},
      year = {2013},
      }
  • [Kollaretal] Go to document J. Kollár, "Flips and abundance for algebraic threefolds," in Second Summer Seminar on Algebraic Geometry, Paris: Société Mathématique de France, 1992, vol. 211, pp. 223-232.
    @incollection{Kollaretal, address = {Paris},
      author = {Koll{á}r, J{á}nos},
      booktitle = {Second Summer Seminar on Algebraic Geometry},
      pages = {223--232},
      publisher = {Société Mathématique de France},
      series = {Ast{é}risque},
      title = {Flips and abundance for algebraic threefolds},
      volume = {211},
      year = {1992},
      issn = {0303-1179},
      url = {http://smf4.emath.fr/Publications/Asterisque/1992/211/html/ smf_ast_211.html},
      }
  • [Borisov99] A. Borisov, "Boundedness of Fano threefolds with log-terminal singularities of given index," J. Math. Sci. Univ. Tokyo, vol. 8, iss. 2, pp. 329-342, 2001.
    @article{Borisov99,
      author = {Borisov, Alexandr},
      journal = {J. Math. Sci. Univ. Tokyo},
      number = {2},
      pages = {329--342},
      title = {Boundedness of {F}ano threefolds with log-terminal singularities of given index},
      volume = {8},
      year = {2001},
      issn = {1340-5705},
      }
  • [Alexeev88] Go to document V. Alexeev, "Fractional indices of log del Pezzo surfaces," Izv. Akad. Nauk SSSR Ser. Mat., vol. 52, pp. 1288-1304, 1998.
    @article{Alexeev88,
      author = {Alexeev, Valery},
      journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
      pages = {1288--1304},
      title = {Fractional indices of log del {P}ezzo surfaces},
      volume = {52},
      year = {1998},
      doi = {10.1070/IM1989v033n03ABEH000859},
      }
  • [Alexeev91] Go to document V. Alexeev, "Theorems about good divisors on log Fano varieties (case of index $r>n-2$)," in Algebraic Geometry, New York: Springer-Verlag, 1991, vol. 1479, pp. 1-9.
    @incollection{Alexeev91, address = {New York},
      author = {Alexeev, Valery},
      booktitle = {Algebraic Geometry},
      pages = {1--9},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {Theorems about good divisors on log {F}ano varieties (case of index {$r>n-2$})},
      volume = {1479},
      year = {1991},
      doi = {10.1007/BFb0086258},
      }
  • [MP04] Go to document J. McKernan and Y. Prokhorov, "Threefold thresholds," Manuscripta Math., vol. 114, iss. 3, pp. 281-304, 2004.
    @article{MP04,
      author = {McKernan, James and Prokhorov, Yuri},
      journal = {Manuscripta Math.},
      number = {3},
      pages = {281--304},
      title = {Threefold thresholds},
      volume = {114},
      year = {2004},
      doi = {10.1007/s00229-004-0457-x},
      issn = {0025-2611},
      }
  • [Kollar08c] J. Kollár, Which powers of holomorphic functions are integrable?.
    @misc{Kollar08c,
      author = {Koll{á}r, J{á}nos},
      title = {Which powers of holomorphic functions are integrable?},
      }
  • [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,
      author = {Angehrn, Urban and Siu, Yum Tong},
      journal = {Invent. Math.},
      number = {2},
      pages = {291--308},
      title = {Effective freeness and point separation for adjoint bundles},
      volume = {122},
      year = {1995},
      doi = {10.1007/BF01231446},
      issn = {0020-9910},
      }
  • [KM99] Go to document S. Keel and J. McKernan, "Rational curves on quasi-projective surfaces," Mem. Amer. Math. Soc., vol. 140, iss. 669, p. viii, 1999.
    @article{KM99,
      author = {Keel, Se{á}n and McKernan, James},
      journal = {Mem. Amer. Math. Soc.},
      number = {669},
      pages = {viii+153},
      title = {Rational curves on quasi-projective surfaces},
      volume = {140},
      year = {1999},
      doi = {10.1090/memo/0669},
      issn = {0065-9266},
      }
  • [Lazarsfeld04b] R. Lazarsfeld, Positivity in Algebraic Geometry. II, New York: Springer-Verlag, 2004, vol. 49.
    @book{Lazarsfeld04b, address = {New York},
      author = {Lazarsfeld, Robert},
      note = {Positivity for vector bundles, and multiplier ideals},
      pages = {xviii+385},
      publisher = {Springer-Verlag},
      series = {Ergeb. Math. Grenzgeb.},
      title = {Positivity in Algebraic Geometry. {II}},
      volume = {49},
      year = {2004},
      isbn = {3-540-22534-X},
      }
  • [Lazarsfeld04a] R. Lazarsfeld, Positivity in algebraic geometry. I, New York: Springer-Verlag, 2004, vol. 48.
    @book{Lazarsfeld04a, address = {New York},
      author = {Lazarsfeld, Robert},
      note = {Classical setting: line bundles and linear series},
      pages = {xviii+387},
      publisher = {Springer-Verlag},
      series = {Ergeb. Math. Grenzgeb.},
      title = {Positivity in algebraic geometry. {I}},
      volume = {48},
      year = {2004},
      isbn = {3-540-22533-1},
      }
  • [Fujino09] Go to document O. Fujino, "Fundamental theorems for the log minimal model program," Publ. Res. Inst. Math. Sci., vol. 47, iss. 3, pp. 727-789, 2011.
    @article{Fujino09,
      author = {Fujino, Osamu},
      journal = {Publ. Res. Inst. Math. Sci.},
      number = {3},
      pages = {727--789},
      title = {Fundamental theorems for the log minimal model program},
      volume = {47},
      year = {2011},
      doi = {10.2977/PRIMS/50},
      issn = {0034-5318},
      }
  • [KK09] Go to document J. Kollár and S. J. Kovács, "Log canonical singularities are Du Bois," J. Amer. Math. Soc., vol. 23, iss. 3, pp. 791-813, 2010.
    @article{KK09,
      author = {Koll{á}r, J{á}nos and Kov{á}cs, S{á}ndor J.},
      journal = {J. Amer. Math. Soc.},
      number = {3},
      pages = {791--813},
      title = {Log canonical singularities are {D}u {B}ois},
      volume = {23},
      year = {2010},
      doi = {10.1090/S0894-0347-10-00663-6},
      issn = {0894-0347},
      }
  • [AH11] Go to document V. Alexeev and C. D. Hacon, "Non-rational centers of log canonical singularities," J. Algebra, vol. 369, pp. 1-15, 2012.
    @article{AH11,
      author = {Alexeev, Valery and Hacon, Christopher D.},
      journal = {J. Algebra},
      pages = {1--15},
      title = {Non-rational centers of log canonical singularities},
      volume = {369},
      year = {2012},
      doi = {10.1016/j.jalgebra.2012.06.015},
      issn = {0021-8693},
      }
  • [Birkar12] Go to document C. Birkar, "Existence of log canonical flips and a special LMMP," Publ. Math. Inst. Hautes Études Sci., vol. 115, pp. 325-368, 2012.
    @article{Birkar12,
      author = {Birkar, Caucher},
      journal = {Publ. Math. Inst. Hautes Études Sci.},
      pages = {325--368},
      title = {Existence of log canonical flips and a special {LMMP}},
      volume = {115},
      year = {2012},
      doi = {10.1007/s10240-012-0039-5},
      issn = {0073-8301},
      }
  • [HX11] Go to document C. D. Hacon and C. Xu, "Existence of log canonical closures," Invent. Math., vol. 192, iss. 1, pp. 161-195, 2013.
    @article{HX11,
      author = {Hacon, Christopher D. and Xu, Chenyang},
      journal = {Invent. Math.},
      number = {1},
      pages = {161--195},
      title = {Existence of log canonical closures},
      volume = {192},
      year = {2013},
      doi = {10.1007/s00222-012-0409-0},
      issn = {0020-9910},
      }
  • [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,
      author = {Hacon, Christopher D. and McKernan, James},
      journal = {Invent. Math.},
      number = {1},
      pages = {1--25},
      title = {Boundedness of pluricanonical maps of varieties of general type},
      volume = {166},
      year = {2006},
      doi = {10.1007/s00222-006-0504-1},
      issn = {0020-9910},
      }
  • [Kollar96] J. Kollár, Rational Curves on Algebraic Varieties, New York: Springer-Verlag, 1996, vol. 32.
    @book{Kollar96, address = {New York},
      author = {Koll{á}r, J{á}nos},
      pages = {viii+320},
      publisher = {Springer-Verlag},
      series = {Ergeb. Math. Grenzgeb.},
      title = {Rational Curves on Algebraic Varieties},
      volume = {32},
      year = {1996},
      isbn = {3-540-60168-6},
      }
  • [KM98] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
    @book{KM98, address = {Cambridge},
      author = {Koll{á}r, J{á}nos and Mori, Shigefumi},
      pages = {viii+254},
      publisher = {Cambridge Univ. Press},
      series = {Cambridge Tracts in Math.},
      title = {Birational Geometry of Algebraic Varieties},
      volume = {134},
      year = {1998},
      doi = {10.1017/CBO9780511662560},
      isbn = {0-521-63277-3},
      }
  • [Kawamata98] Go to document Y. Kawamata, "Subadjunction of log canonical divisors. II," Amer. J. Math., vol. 120, iss. 5, pp. 893-899, 1998.
    @article{Kawamata98,
      author = {Kawamata, Yujiro},
      journal = {Amer. J. Math.},
      number = {5},
      pages = {893--899},
      title = {Subadjunction of log canonical divisors. {II}},
      volume = {120},
      year = {1998},
      doi = {10.1353/ajm.1998.0038},
      issn = {0002-9327},
      }
  • [Kawakita05] Go to document M. Kawakita, "Inversion of adjunction on log canonicity," Invent. Math., vol. 167, iss. 1, pp. 129-133, 2007.
    @article{Kawakita05,
      author = {Kawakita, Masayuki},
      journal = {Invent. Math.},
      number = {1},
      pages = {129--133},
      title = {Inversion of adjunction on log canonicity},
      volume = {167},
      year = {2007},
      doi = {10.1007/s00222-006-0008-z},
      issn = {0020-9910},
      }
  • [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,
      author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
      journal = {J. Amer. Math. Soc.},
      number = {2},
      pages = {405--468},
      title = {Existence of minimal models for varieties of log general type},
      volume = {23},
      year = {2010},
      doi = {10.1090/S0894-0347-09-00649-3},
      issn = {0894-0347},
      }
  • [Kollar93b] Go to document J. Kollár, "Effective base point freeness," Math. Ann., vol. 296, iss. 4, pp. 595-605, 1993.
    @article{Kollar93b,
      author = {Koll{á}r, J{á}nos},
      journal = {Math. Ann.},
      number = {4},
      pages = {595--605},
      title = {Effective base point freeness},
      volume = {296},
      year = {1993},
      doi = {10.1007/BF01445123},
      issn = {0025-5831},
      }
  • [Fujino09a] Go to document O. Fujino, "Effective base point free theorem for log canonical pairs—Kollár type theorem," Tohoku Math. J., vol. 61, iss. 4, pp. 475-481, 2009.
    @article{Fujino09a,
      author = {Fujino, Osamu},
      journal = {Tohoku Math. J.},
      number = {4},
      pages = {475--481},
      title = {Effective base point free theorem for log canonical pairs---{K}ollár type theorem},
      volume = {61},
      year = {2009},
      doi = {10.2748/tmj/1264084495},
      issn = {0040-8735},
      }

Authors

Christopher D. Hacon

University of Utah, Salt Lake City, UT

James McKernan

University of California, San Diego
La Jolla, CA

Chenyang Xu

Beijing International Center of Mathematics Research, Beijing, China