Anti-pluricanonical systems on Fano varieties

Abstract

In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with “good singularities” for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon >0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon $ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov’s conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

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      title = {Pluricanonical systems on minimal algebraic varieties},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {79},
      year = {1985},
      number = {3},
      pages = {567--588},
      issn = {0020-9910},
      mrclass = {14C20 (14J10)},
      mrnumber = {0782236},
      mrreviewer = {Rick Miranda},
      doi = {10.1007/BF01388524},
      url = {https://doi.org/10.1007/BF01388524},
      zblnumber = {0593.14010},
      }
  • [KMM-mmp] Go to document Y. Kawamata, K. Matsuda, and K. Matsuki, "Introduction to the minimal model problem," in Algebraic Geometry, Sendai, 1985, North-Holland, Amsterdam, 1987, vol. 10, pp. 283-360.
    @INCOLLECTION{KMM-mmp,
      author = {Kawamata, Yujiro and Matsuda, Katsumi and Matsuki, Kenji},
      title = {Introduction to the minimal model problem},
      booktitle = {Algebraic Geometry, {S}endai, 1985},
      series = {Adv. Stud. Pure Math.},
      volume = {10},
      pages = {283--360},
      publisher = {North-Holland, Amsterdam},
      year = {1987},
      mrclass = {14E30 (14E05 14J40)},
      mrnumber = {0946243},
      mrreviewer = {David R. Morrison},
      zblnumber = {0672.14006},
      doi = {10.2969/aspm/01010283},
     }
  • [kollar-rational-curves] Go to document J. Kollár, Rational Curves on Algebraic Varieties, Springer-Verlag, Berlin, 1996, vol. 32.
    @BOOK{kollar-rational-curves,
      author = {Koll\'{a}r, J\'{a}nos},
      title = {Rational Curves on Algebraic Varieties},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {32},
      publisher = {Springer-Verlag, Berlin},
      year = {1996},
      pages = {viii+320},
      isbn = {3-540-60168-6},
      mrclass = {14-02 (14C05 14E05 14F17 14J45)},
      mrnumber = {1440180},
      mrreviewer = {Yuri G. Prokhorov},
      doi = {10.1007/978-3-662-03276-3},
      url = {https://doi.org/10.1007/978-3-662-03276-3},
      zblnumber = {0877.14012},
      }
  • [kollar-sing-pairs] J. Kollár, "Singularities of pairs," in Algebraic Geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, vol. 62, pp. 221-287.
    @INCOLLECTION{kollar-sing-pairs,
      author = {Koll\'{a}r, J\'{a}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., Providence, RI},
      year = {1997},
      mrclass = {14E30 (14E15 32S40)},
      mrnumber = {1492525},
      mrreviewer = {Alessio Corti},
      zblnumber = {0905.14002},
      }
  • [kollar-ebpf] Go to document J. Kollár, "Effective base point freeness," Math. Ann., vol. 296, iss. 4, pp. 595-605, 1993.
    @ARTICLE{kollar-ebpf,
      author = {Koll\'{a}r, J\'{a}nos},
      title = {Effective base point freeness},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {296},
      year = {1993},
      number = {4},
      pages = {595--605},
      issn = {0025-5831},
      mrclass = {14C20 (14J10)},
      mrnumber = {1233485},
      mrreviewer = {Marino Palleschi},
      doi = {10.1007/BF01445123},
      url = {https://doi.org/10.1007/BF01445123},
      zblnumber = {0818.14002},
      }
  • [Kollar-flip-abundance] J. Kollár, Flips and Abundance for Algebraic Threefolds, Soc. Math. de France, Paris, 1992, vol. 211.
    @BOOK{Kollar-flip-abundance,
      author = {Koll\'{a}r, J\'{a}nos},
      note = {(with 14 co-authors; papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991)},
      title = {Flips and Abundance for {A}lgebraic {T}hreefolds},
      series = {Astérisque},
      publisher={Soc. Math. de France, Paris},
      volume = {211},
      issn = {0303-1179},
      mrclass = {14E30 (14E35 14M10)},
      mrnumber = {1225842},
      mrreviewer = {Mark Gross},
      year = {1992},
      zblnumber = {0782.00075},
      }
  • [KMM-smooth-fano] Go to document J. Kollár, Y. Miyaoka, and S. Mori, "Rational connectedness and boundedness of Fano manifolds," J. Differential Geom., vol. 36, iss. 3, pp. 765-779, 1992.
    @ARTICLE{KMM-smooth-fano,
      author = {Koll\'{a}r, J\'{a}nos and Miyaoka, Yoichi and Mori, Shigefumi},
      title = {Rational connectedness and boundedness of {F}ano manifolds},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {36},
      year = {1992},
      number = {3},
      pages = {765--779},
      issn = {0022-040X},
      mrclass = {14J45},
      mrnumber = {1189503},
      mrreviewer = {Yuri G. Prokhorov},
      doi = {10.4310/jdg/1214453188},
      url = {https://doi.org/10.4310/jdg/1214453188},
      zblnumber = {0759.14032},
      }
  • [KMMT-can-3-folds] Go to document J. Kollár, Y. Miyaoka, S. Mori, and H. Takagi, "Boundedness of canonical $\Bbb Q$-Fano 3-folds," Proc. Japan Acad. Ser. A Math. Sci., vol. 76, iss. 5, pp. 73-77, 2000.
    @ARTICLE{KMMT-can-3-folds,
      author = {Koll\'{a}r, J\'{a}nos and Miyaoka, Yoichi and Mori, Shigefumi and Takagi, Hiromichi},
      title = {Boundedness of canonical {$\bold Q$}-{F}ano 3-folds},
      journal = {Proc. Japan Acad. Ser. A Math. Sci.},
      fjournal = {Japan Academy. Proceedings. Series A. Mathematical Sciences},
      volume = {76},
      year = {2000},
      number = {5},
      pages = {73--77},
      issn = {0386-2194},
      mrclass = {14J45 (14E30)},
      mrnumber = {1771144},
      mrreviewer = {Yuri G. Prokhorov},
      doi = {10.3792/pjaa.76.73},
      zblnumber = {0981.14016},
      }
  • [kollar-mori] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Univ. Press, Cambridge, 1998, vol. 134.
    @BOOK{kollar-mori,
      author = {Koll\'{a}r, J\'{a}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, Cambridge},
      year = {1998},
      pages = {viii+254},
      isbn = {0-521-63277-3},
      mrclass = {14E30},
      mrnumber = {1658959},
      mrreviewer = {Mark Gross},
      doi = {10.1017/CBO9780511662560},
      url = {https://doi.org/10.1017/CBO9780511662560},
      zblnumber = {0926.14003},
      }
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    @ARTICLE{Lai,
      author = {Lai, Ching-Jui},
      title = {Bounding volumes of singular {F}ano threefolds},
      journal = {Nagoya Math. J.},
      fjournal = {Nagoya Mathematical Journal},
      volume = {224},
      year = {2016},
      number = {1},
      pages = {37--73},
      issn = {0027-7630},
      mrclass = {14E30 (14J45)},
      mrnumber = {3572749},
      mrreviewer = {Andrea Fanelli},
      doi = {10.1017/nmj.2016.21},
      url = {https://doi.org/10.1017/nmj.2016.21},
      zblnumber = {1390.14053},
      }
  • [Nakayama] Go to document N. Nakayama, Zariski-Decomposition and Abundance, Math. Soc. of Japan, Tokyo, 2004, vol. 14.
    @BOOK{Nakayama,
      author = {Nakayama, Noboru},
      title = {Zariski-Decomposition and Abundance},
      series = {MSJ Memoirs},
      volume = {14},
      publisher = {Math. Soc. of Japan, 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},
      doi = {10.2969/msjmemoirs/014010000},
     }
  • [PSh-II] Go to document Y. G. Prokhorov and V. V. Shokurov, "Towards the second main theorem on complements," J. Algebraic Geom., vol. 18, iss. 1, pp. 151-199, 2009.
    @ARTICLE{PSh-II,
      author = {Prokhorov, Yu. G. and Shokurov, V. V.},
      title = {Towards the second main theorem on complements},
      journal = {J. Algebraic Geom.},
      fjournal = {Journal of Algebraic Geometry},
      volume = {18},
      year = {2009},
      number = {1},
      pages = {151--199},
      issn = {1056-3911},
      mrclass = {14C20 (14J45)},
      mrnumber = {2448282},
      mrreviewer = {Alexandr V. Pukhlikov},
      doi = {10.1090/S1056-3911-08-00498-0},
      url = {https://doi.org/10.1090/S1056-3911-08-00498-0},
      zblnumber = {1159.14020},
      }
  • [PSh-I] Go to document Y. G. Prokhorov and V. V. Shokurov, "The first fundamental theorem on complements: from global to local," Izv. Ross. Akad. Nauk Ser. Mat., vol. 65, iss. 6, pp. 99-128, 2001.
    @ARTICLE{PSh-I,
      author = {Prokhorov, Yu. G. and Shokurov, V. V.},
      title = {The first fundamental theorem on complements: from global to local},
      journal = {Izv. Ross. Akad. Nauk Ser. Mat.},
      fjournal = {Rossiĭskaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya},
      volume = {65},
      year = {2001},
      number = {6},
      pages = {99--128},
      issn = {1607-0046},
      mrclass = {14E30 (14J40)},
      mrnumber = {1892905},
      mrreviewer = {Adrian Langer},
      doi = {10.1070/IM2001v065n06ABEH000366},
      url = {https://doi.org/10.1070/IM2001v065n06ABEH000366},
      zblnumber = {1068.14018},
      note = {translation in {\em Izv. Math.} {\bf 65} (2001), 1169--1196},
     }
  • [Shokurov-adjunction] V. V. Shokurov, Log Adjunction: effectiveness and positivity, 2013.
    @MISC{Shokurov-adjunction,
      author = {Shokurov, V. V.},
      title = {Log Adjunction: effectiveness and positivity},
      arxiv = {1308.5160},
      year = {2013},
      zblnumber = {},
      }
  • [shokurov-surf-comp] Go to document V. V. Shokurov, "Complements on surfaces," J. Math. Sci. (New York), vol. 102, iss. 2, pp. 3876-3932, 2000.
    @ARTICLE{shokurov-surf-comp,
      author = {Shokurov, V. V.},
      title = {Complements on surfaces},
      note = {Algebraic geometry, 10},
      journal = {J. Math. Sci. (New York)},
      fjournal = {Journal of Mathematical Sciences (New York)},
      volume = {102},
      year = {2000},
      number = {2},
      pages = {3876--3932},
      issn = {1072-3374},
      mrclass = {14E30 (14J26 14J30)},
      mrnumber = {1794169},
      mrreviewer = {I. Dolgachev},
      doi = {10.1007/BF02984106},
      url = {https://doi.org/10.1007/BF02984106},
      zblnumber = {1177.14078},
      }
  • [Shokurov-log-flips] Go to document V. V. Shokurov, "3-fold log flips," Russian Acad. Sci. Izv. Math., vol. 40, iss. 1, pp. 95-202, 1993.
    @ARTICLE{Shokurov-log-flips,
      author = {V. V. {Shokurov}},
      Title = {3-fold log flips},
      note={(with an appendix by Y. Kawamata: The minimal discrepancy of a 3-fold terminal singularity)},
      journal = {Russian Acad. Sci. Izv. Math.},
      fjournal = {Russian Academy of Sciences. Izvestiya Mathematics},
      year = {1993},
      volume = {40},
      number = {1},
      pages = {95--202},
      publisher = {{IOP} Publishing},
      mrnumber = {1162635},
      zblnumber = {0785.14023},
      doi= {10.1070/im1993v040n01abeh001862},
      url= {https://doi.org/10.1070/im1993v040n01abeh001862},
      },
     
  • [Shokurov-anticanonical] Go to document V. V. Shokurov, "Smoothness of the general anti-canonical divisor on a Fano 3-fold," Izv. Akad. Nauk SSSR Ser. Mat., vol. 43, iss. 2, pp. 430-441, 1979.
    @ARTICLE{Shokurov-anticanonical,
      author = {Shokurov, V. V.},
      title = {Smoothness of the general anti-canonical divisor on a Fano 3-fold},
      journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
      fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      volume = {43},
      year = {1979},
      number = {2},
      pages = {430--441},
      issn = {0373-2436},
      mrclass = {14J30},
      mrnumber = {0534602},
      mrreviewer = {Werner Kleinert},
      note = {[(English translation in {\em Math. USSR Izv.} {\bf 14} (1980), 305--405). \zbl{0429.14012}]},
      doi = {10.1070/IM1980v014n02ABEH001123},
      url = {https://doi.org/10.1070/IM1980v014n02ABEH001123},
      zblnumber={0407.14017},
     }
  • [Takayama] Go to document S. Takayama, "Pluricanonical systems on algebraic varieties of general type," Invent. Math., vol. 165, iss. 3, pp. 551-587, 2006.
    @ARTICLE{Takayama,
      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},
      mrclass = {14E05 (14C20 14J40)},
      mrnumber = {2242627},
      mrreviewer = {Tomasz Szemberg},
      doi = {10.1007/s00222-006-0503-2},
      url = {https://doi.org/10.1007/s00222-006-0503-2},
      zblnumber = {1108.14031},
      }
  • [Zhang-rat-connected] Go to document Q. Zhang, "Rational connectedness of log ${\bf Q}$-Fano varieties," J. Reine Angew. Math., vol. 590, pp. 131-142, 2006.
    @ARTICLE{Zhang-rat-connected,
      author = {Zhang, Qi},
      title = {Rational connectedness of log {${\bf Q}$}-{F}ano varieties},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {590},
      year = {2006},
      pages = {131--142},
      issn = {0075-4102},
      mrclass = {14E30 (14J45)},
      mrnumber = {2208131},
      mrreviewer = {Stefan Kebekus},
      doi = {10.1515/CRELLE.2006.006},
      url = {https://doi.org/10.1515/CRELLE.2006.006},
      zblnumber = {1093.14059},
      }

Authors

Caucher Birkar

DPMMS, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK