Semipositivity theorems for moduli problems

Abstract

We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollár. This completes Kollár’s projectivity criterion for the moduli spaces of higher-dimensional stable varieties.

Note: To view the article, click on the URL link for the DOI number.

  • [abramovich-karu] Go to document D. Abramovich and B. Hassett, "Stable varieties with a twist," in Classification of Algebraic Varieties, Eur. Math. Soc., Zürich, 2011, pp. 1-38.
    @INCOLLECTION{abramovich-karu,
      author = {Abramovich, Dan and Hassett, Brendan},
      title = {Stable varieties with a twist},
      booktitle = {Classification of Algebraic Varieties},
      series = {EMS Ser. Congr. Rep.},
      pages = {1--38},
      publisher = {Eur. Math. Soc., Zürich},
      year = {2011},
      mrclass = {14D23 (14D22 14E30 14J10)},
      mrnumber = {2779465},
      mrreviewer = {Arvid Perego},
      doi = {10.4171/007-1/1},
      zblnumber = {1223.14039},
      }
  • [alexeev1] Go to document V. Alexeev, "Boundedness and $K^2$ for log surfaces," Internat. J. Math., vol. 5, iss. 6, pp. 779-810, 1994.
    @ARTICLE{alexeev1,
      author = {Alexeev, Valery},
      title = {Boundedness and {$K^2$} 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},
      }
  • [alexeev2] Go to document V. Alexeev, "Moduli spaces $M_{g,n}(W)$ for surfaces," in Higher-Dimensional Complex Varieties, de Gruyter, Berlin, 1996, pp. 1-22.
    @INCOLLECTION{alexeev2,
      author = {Alexeev, Valery},
      title = {Moduli spaces {$M_{g,n}(W)$} for surfaces},
      booktitle = {Higher-Dimensional Complex Varieties},
      venue = {{T}rento, 1994},
      pages = {1--22},
      publisher = {de Gruyter, Berlin},
      year = {1996},
      mrclass = {14D20 (14D22 14J10)},
      mrnumber = {1463171},
      mrreviewer = {Wolfgang K. Seiler},
      doi = {10.1515/9783110814736.1},
      zblnumber = {0896.14014},
      }
  • [alexeev-mori] V. Alexeev and S. Mori, "Bounding singular surfaces of general type," in Algebra, Arithmetic and Geometry with Applications, Springer, Berlin, 2004, pp. 143-174.
    @INCOLLECTION{alexeev-mori,
      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, Berlin},
      year = {2004},
      mrclass = {14J29},
      mrnumber = {2037085},
      mrreviewer = {Sándor J. Kovács},
      zblnumber = {1103.14021},
      }
  • [bierstone] E. Bierstone and F. Vera Pacheco, "Resolution of singularities of pairs preserving semi-simple normal crossings," Rev. R. Acad. Cienc. Exactas F\’\i s. Nat. Ser. A Math. RACSAM, vol. 107, iss. 1, pp. 159-188, 2013.
    @ARTICLE{bierstone,
      author = {Bierstone, Edward and Vera Pacheco, Franklin},
      title = {Resolution of singularities of pairs preserving semi-simple normal crossings},
      journal = {Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Math. RACSAM},
      fjournal = {Revista de la Real Academia de Ciencias Exactas, F\'\i sicas y Naturales. Serie A. Matematicas. RACSAM},
      volume = {107},
      year = {2013},
      number = {1},
      pages = {159--188},
      issn = {1578-7303},
      mrclass = {14E15 (32S10 32S45)},
      mrnumber = {3031268},
      mrreviewer = {Ana Bravo},
      zblnumber = {1285.14014},
      }
  • [campana] Go to document F. Campana, "Orbifolds, special varieties and classification theory," Ann. Inst. Fourier (Grenoble), vol. 54, iss. 3, pp. 499-630, 2004.
    @ARTICLE{campana,
      author = {Campana, Frédéric},
      title = {Orbifolds, special varieties and classification theory},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {54},
      year = {2004},
      number = {3},
      pages = {499--630},
      issn = {0373-0956},
      mrclass = {14E05 (14D06 14J40 32Q57 35Q15)},
      mrnumber = {2097416},
      mrreviewer = {Dan Abramovich},
      doi = {10.5802/aif.2027},
      zblnumber = {1062.14014},
      }
  • [ev] Go to document H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, Birkhäuser Verlag, Basel, 1992, vol. 20.
    @BOOK{ev,
      author = {Esnault, Hélène and Viehweg, Eckart},
      title = {Lectures on Vanishing Theorems},
      series = {DMV Seminar},
      volume = {20},
      publisher = {Birkhäuser Verlag, Basel},
      year = {1992},
      pages = {vi+164},
      isbn = {3-7643-2822-3},
      mrclass = {14F17 (14F40 32L10 32L20)},
      mrnumber = {1193913},
      mrreviewer = {Marko Roczen},
      doi = {10.1007/978-3-0348-8600-0},
      zblnumber = {0779.14003},
      }
  • [fujino-high] Go to document O. Fujino, "Higher direct images of log canonical divisors," J. Differential Geom., vol. 66, iss. 3, pp. 453-479, 2004.
    @ARTICLE{fujino-high, key={Fn04},
      author = {Fujino, Osamu},
      title = {Higher direct images of log canonical divisors},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {66},
      year = {2004},
      number = {3},
      pages = {453--479},
      issn = {0022-040X},
      mrclass = {14E30 (14C20 14F05)},
      mrnumber = {2106473},
      mrreviewer = {Feng-Wen An},
      doi = {10.4310/jdg/1098137840},
      zblnumber = {1072.14019},
      }
  • [fujino-fund] 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{fujino-fund,
      author = {Fujino, Osamu},
      key={Fn11a},
      title = {Fundamental theorems for the log minimal model program},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Publications of the Research Institute for Mathematical Sciences},
      volume = {47},
      year = {2011},
      number = {3},
      pages = {727--789},
      issn = {0034-5318},
      mrclass = {14E30 (14C30 14F17)},
      mrnumber = {2832805},
      mrreviewer = {Vladimir Lazić},
      doi = {10.2977/PRIMS/50},
      zblnumber = {1234.14013},
      }
  • [fujino-sugaku] O. Fujino, "Recent developments in minimal model theory," Sugaku Expositions, vol. 24, iss. 2, pp. 205-237, 2011.
    @ARTICLE{fujino-sugaku,
      author = {Fujino, Osamu},
      key={Fn11b},
      title = {Recent developments in minimal model theory},
      journal = {Sugaku Expositions},
      fjournal = {Sugaku Expositions},
      volume = {24},
      year = {2011},
      number = {2},
      pages = {205--237},
      issn = {0898-9583},
      mrclass = {14E30},
      mrnumber = {2882844},
      zblnumber = {},
      }
  • [fujino] Go to document O. Fujino, "Fundamental theorems for semi log canonical pairs," Algebr. Geom., vol. 1, iss. 2, pp. 194-228, 2014.
    @ARTICLE{fujino,
      author = {Fujino, Osamu},
      key={Fn14},
      title = {Fundamental theorems for semi log canonical pairs},
      journal = {Algebr. Geom.},
      fjournal = {Algebraic Geometry},
      volume = {1},
      year = {2014},
      number = {2},
      pages = {194--228},
      issn = {2214-2584},
      mrclass = {14E30 (14F17)},
      mrnumber = {3238112},
      mrreviewer = {Patrick Graf},
      doi = {10.14231/AG-2014-011},
      zblnumber = {1296.14014},
      }
  • [fujino-weak] O. Fujino, "Notes on the weak positivity theorems," in Algebraic Varieties and Automorphism Groups, Math. Soc. Japan, Tokyo, 2017, vol. 75, pp. 73-118.
    @INCOLLECTION{fujino-weak,
      author = {Fujino, Osamu},
      key={Fn17b},
      title = {Notes on the weak positivity theorems},
      booktitle = {Algebraic Varieties and Automorphism Groups},
      pages = {73--118},
      series = {Adv. Stud. Pure Math.},
      volume = {75},
      publisher = {Math. Soc. Japan, Tokyo},
      year = {2017},
      zblnumber = {},
      }
  • [fujino-foundation] Go to document O. Fujino, Foundations of the Minimal Model Program, Mathematical Society of Japan, Tokyo, 2017, vol. 35.
    @BOOK{fujino-foundation,
      author = {Fujino, Osamu},
      key={Fn17a},
      title = {Foundations of the Minimal Model Program},
      series = {MSJ Memoirs},
      volume = {35},
      publisher = {Mathematical Society of Japan, Tokyo},
      year = {2017},
      pages = {xv+289},
      isbn = {978-4-86497-045-7},
      mrclass = {14E30 (14F17)},
      mrnumber = {3643725},
      zblnumber = {06726987},
      doi = {10.2969/msjmemoirs/035010000},
      }
  • [fujino-zucker65] O. Fujino, "On Semipositivity, Injectivity, and Vanishing Theorems," in Hodge Theory and $L^2$-Analysis, Int. Press, Somerville, MA, 2017, vol. 39, pp. 245-282.
    @INCOLLECTION{fujino-zucker65,
      author = {Fujino, Osamu},
      key={Fn17c},
      title = {On Semipositivity, Injectivity, and Vanishing Theorems},
      booktitle = {Hodge Theory and {$L^2$}-Analysis},
      pages = {245--282},
      series = {Adv. Lect. Math.},
      volume = {39},
      publisher = {Int. Press, Somerville, MA},
      year = {2017},
      zblnumber = {06817379},
      }
  • [fujino-new] O. Fujino, Vanishing and semipositivity theorems for semi-log canonical pairs, 2015.
    @misc{fujino-new, key={Fn15},
      author = {Fujino, Osamu},
      title = {Vanishing and semipositivity theorems for semi-log canonical pairs},
      year={2015},
      arxiv={1509.00531},
     }
  • [fujino-fujisawa] Go to document O. Fujino and T. Fujisawa, "Variations of mixed Hodge structure and semipositivity theorems," Publ. Res. Inst. Math. Sci., vol. 50, iss. 4, pp. 589-661, 2014.
    @ARTICLE{fujino-fujisawa,
      author = {Fujino, Osamu and Fujisawa, Taro},
      title = {Variations of mixed {H}odge structure and semipositivity theorems},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Publications of the Research Institute for Mathematical Sciences},
      volume = {50},
      year = {2014},
      number = {4},
      pages = {589--661},
      issn = {0034-5318},
      mrclass = {14D07 (14C30 32G20)},
      mrnumber = {3273305},
      mrreviewer = {Jan Nagel},
      doi = {10.4171/PRIMS/145},
      zblnumber = {1305.14004},
      }
  • [fujino-fujisawa2] O. Fujino and T. Fujisawa, On semipositivity theorems, 2017.
    @MISC{fujino-fujisawa2,
      author = {Fujino, Osamu and Fujisawa, Taro},
      title = {On semipositivity theorems},
      note = {preprint},
      year = {2017},
      arXiv = {1701.02039},
      }
  • [ffs] Go to document O. Fujino, T. Fujisawa, and M. Saito, "Some remarks on the semipositivity theorems," Publ. Res. Inst. Math. Sci., vol. 50, iss. 1, pp. 85-112, 2014.
    @ARTICLE{ffs,
      author = {Fujino, Osamu and Fujisawa, Taro and Saito, Morihiko},
      title = {Some remarks on the semipositivity theorems},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Publications of the Research Institute for Mathematical Sciences},
      volume = {50},
      year = {2014},
      number = {1},
      pages = {85--112},
      issn = {0034-5318},
      mrclass = {14D07 (32G20)},
      mrnumber = {3167580},
      mrreviewer = {Christian Schnell},
      doi = {10.4171/PRIMS/125},
      zblnumber = {1326.14018},
      }
  • [fujisawa] T. Fujisawa, A remark on semipositivity theorems, 2017.
    @MISC{fujisawa, key={Fs17},
      author = {Fujisawa, Takao},
      title = {A remark on semipositivity theorems},
      arxiv={1710.01008},
      year = {2017},
      }
  • [fujita] Go to document T. Fujita, "On Kähler fiber spaces over curves," J. Math. Soc. Japan, vol. 30, iss. 4, pp. 779-794, 1978.
    @ARTICLE{fujita, key={Ft78},
      author = {Fujita, Takao},
      title = {On {K}ähler fiber spaces over curves},
      journal = {J. Math. Soc. Japan},
      fjournal = {Journal of the Mathematical Society of Japan},
      volume = {30},
      year = {1978},
      number = {4},
      pages = {779--794},
      issn = {0025-5645},
      mrclass = {32G13 (14E99 32J99)},
      mrnumber = {0513085},
      doi = {10.2969/jmsj/03040779},
      zblnumber = {0393.14006},
      }
  • [fp] Go to document W. Fulton and R. Pandharipande, "Notes on stable maps and quantum cohomology," in Algebraic Geometry, Amer. Math. Soc., Providence, RI, 1997, vol. 62, pp. 45-96.
    @INCOLLECTION{fp,
      author = {Fulton, W. and Pandharipande, R.},
      title = {Notes on stable maps and quantum cohomology},
      booktitle = {Algebraic Geometry},
      venue = {{S}anta {C}ruz 1995},
      series = {Proc. Sympos. Pure Math.},
      volume = {62},
      pages = {45--96},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1997},
      mrclass = {14H10 (14E99 14N10)},
      mrnumber = {1492534},
      mrreviewer = {Alexandre I. Kabanov},
      doi = {10.1090/pspum/062.2/1492534},
      zblnumber = {0898.14018},
      }
  • [hmx] C. D. Hacon, J. McKernan, and C. Xu, Boundedness of moduli of varieties of general type, 2014.
    @misc{hmx,
      author = {Hacon, Christopher D. and McKernan, James and Xu, C.},
      title = {Boundedness of moduli of varieties of general type},
      note={to appear in \emph{J. Eur. Math. Soc. (JEMS)}},
      arxiv={1412.1186},
      year={2014},
     }
  • [hacon-xu] 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{hacon-xu,
      author = {Hacon, Christopher D. and Xu, Chenyang},
      title = {Existence of log canonical closures},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {192},
      year = {2013},
      number = {1},
      pages = {161--195},
      issn = {0020-9910},
      mrclass = {14E30},
      mrnumber = {3032329},
      mrreviewer = {Ilya Karzhemanov},
      doi = {10.1007/s00222-012-0409-0},
      zblnumber = {1282.14027},
      }
  • [hassett] Go to document B. Hassett, "Moduli spaces of weighted pointed stable curves," Adv. Math., vol. 173, iss. 2, pp. 316-352, 2003.
    @ARTICLE{hassett,
      author = {Hassett, Brendan},
      title = {Moduli spaces of weighted pointed stable curves},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {173},
      year = {2003},
      number = {2},
      pages = {316--352},
      issn = {0001-8708},
      mrclass = {14H10 (14D22 14E30)},
      mrnumber = {1957831},
      mrreviewer = {Ivan S. Kausz},
      doi = {10.1016/S0001-8708(02)00058-0},
      zblnumber = {1072.14014},
      }
  • [karu] K. Karu, "Minimal models and boundedness of stable varieties," J. Algebraic Geom., vol. 9, iss. 1, pp. 93-109, 2000.
    @ARTICLE{karu,
      author = {Karu, Kalle},
      title = {Minimal models and boundedness of stable varieties},
      journal = {J. Algebraic Geom.},
      fjournal = {Journal of Algebraic Geometry},
      volume = {9},
      year = {2000},
      number = {1},
      pages = {93--109},
      issn = {1056-3911},
      mrclass = {14J10 (14D20 14E30 14J17)},
      mrnumber = {1713521},
      mrreviewer = {Alessio Corti},
      zblnumber = {0980.14008},
      }
  • [kawakita] Go to document M. Kawakita, "Inversion of adjunction on log canonicity," Invent. Math., vol. 167, iss. 1, pp. 129-133, 2007.
    @ARTICLE{kawakita,
      author = {Kawakita, Masayuki},
      title = {Inversion of adjunction on log canonicity},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {167},
      year = {2007},
      number = {1},
      pages = {129--133},
      issn = {0020-9910},
      mrclass = {14E30 (14N30)},
      mrnumber = {2264806},
      mrreviewer = {Carla Novelli},
      doi = {10.1007/s00222-006-0008-z},
      zblnumber = {1114.14009},
      }
  • [kawamata-abel] Go to document Y. Kawamata, "Characterization of abelian varieties," Compositio Math., vol. 43, iss. 2, pp. 253-276, 1981.
    @ARTICLE{kawamata-abel,
      author = {Kawamata, Yujiro},
      title = {Characterization of abelian varieties},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {43},
      year = {1981},
      number = {2},
      pages = {253--276},
      issn = {0010-437X},
      mrclass = {14J10 (32J15)},
      mrnumber = {0622451},
      mrreviewer = {Daniel Comenetz},
      url = {http://www.numdam.org/item?id=CM_1981__43_2_253_0},
      zblnumber = {0471.14022},
      }
  • [kawamata] Go to document Y. Kawamata, "Deformations of canonical singularities," J. Amer. Math. Soc., vol. 12, iss. 1, pp. 85-92, 1999.
    @ARTICLE{kawamata,
      author = {Kawamata, Yujiro},
      title = {Deformations of canonical singularities},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {12},
      year = {1999},
      number = {1},
      pages = {85--92},
      issn = {0894-0347},
      mrclass = {14B07 (14F17)},
      mrnumber = {1631527},
      mrreviewer = {Gerhard Pfister},
      doi = {10.1090/S0894-0347-99-00285-4},
      zblnumber = {0906.14001},
      }
  • [kmm] Y. Kawamata, K. Matsuda, and K. Matsuki, "Introduction to the minimal model problem," in Algebraic Geometry, North-Holland, Amsterdam, 1987, vol. 10, pp. 283-360.
    @INCOLLECTION{kmm,
      author = {Kawamata, Yujiro and Matsuda, Katsumi and Matsuki, Kenji},
      title = {Introduction to the minimal model problem},
      booktitle = {Algebraic Geometry},
      venue = {{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},
      }
  • [keel-mori] Go to document S. Keel and S. Mori, "Quotients by groupoids," Ann. of Math. (2), vol. 145, iss. 1, pp. 193-213, 1997.
    @ARTICLE{keel-mori,
      author = {Keel, Seán and Mori, Shigefumi},
      title = {Quotients by groupoids},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {145},
      year = {1997},
      number = {1},
      pages = {193--213},
      issn = {0003-486X},
      mrclass = {14D25 (14L30)},
      mrnumber = {1432041},
      mrreviewer = {Andrzej Bia\l ynicki-Birula},
      doi = {10.2307/2951828},
      zblnumber = {0881.14018},
      }
  • [kollar] Go to document J. Kollár, "Projectivity of complete moduli," J. Differential Geom., vol. 32, iss. 1, pp. 235-268, 1990.
    @ARTICLE{kollar,
      author = {Kollár, János},
      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},
      mrclass = {14D22 (14H10 14J10)},
      mrnumber = {1064874},
      mrreviewer = {Autorreferat},
      doi = {10.4310/jdg/1214445046},
      zblnumber = {0684.14002},
      }
  • [kollar-ann] Go to document J. Kollár, "Non-quasi-projective moduli spaces," Ann. of Math. (2), vol. 164, iss. 3, pp. 1077-1096, 2006.
    @ARTICLE{kollar-ann,
      author = {Kollár, János},
      title = {Non-quasi-projective moduli spaces},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {164},
      year = {2006},
      number = {3},
      pages = {1077--1096},
      issn = {0003-486X},
      mrclass = {14D20 (14J15 14L30 32G05)},
      mrnumber = {2259254},
      mrreviewer = {Vasile Br\^\i nz\u anescu},
      doi = {10.4007/annals.2006.164.1077},
      zblnumber = {1140.14011},
      }
  • [kollar-two] Go to document J. Kollár, "Two examples of surfaces with normal crossing singularities," Sci. China Math., vol. 54, iss. 8, pp. 1707-1712, 2011.
    @ARTICLE{kollar-two,
      author = {Kollár, János},
      title = {Two examples of surfaces with normal crossing singularities},
      journal = {Sci. China Math.},
      fjournal = {Science China. Mathematics},
      volume = {54},
      year = {2011},
      number = {8},
      pages = {1707--1712},
      issn = {1674-7283},
      mrclass = {14J10 (14C20 14J29)},
      mrnumber = {2824967},
      mrreviewer = {Michael A. van Opstall},
      doi = {10.1007/s11425-010-4161-x},
      zblnumber = {1241.14013},
      }
  • [kollar-hull] J. Kollár, Hulls and husks, 2008.
    @MISC{kollar-hull,
      author = {Kollár, János},
      title = {Hulls and husks},
      arxiv={0805.0576},
      year = {2008},
      }
  • [kollar2] J. Kollár, "Moduli of varieties of general type," in Handbook of Moduli. Vol. II, Int. Press, Somerville, MA, 2013, vol. 25, pp. 131-157.
    @INCOLLECTION{kollar2,
      author = {Kollár, János},
      title = {Moduli of varieties of general type},
      booktitle = {Handbook of Moduli. {V}ol. {II}},
      series = {Adv. Lect. Math.},
      volume = {25},
      pages = {131--157},
      publisher = {Int. Press, Somerville, MA},
      year = {2013},
      mrclass = {14D20 (14D22)},
      mrnumber = {3184176},
      mrreviewer = {Nicolae Manolache},
      zblnumber = {1322.14006},
      }
  • [kollar-s] J. Kollár, "Sources of log canonical centers," in Minimal Models and Extremal Rays, Math. Soc. Japan, [Tokyo], 2016, vol. 70, pp. 29-48.
    @INCOLLECTION{kollar-s,
      author = {Kollár, János},
      title = {Sources of log canonical centers},
      booktitle = {Minimal Models and Extremal Rays},
      venue = {{K}yoto, 2011},
      series = {Adv. Stud. Pure Math.},
      volume = {70},
      pages = {29--48},
      publisher = {Math. Soc. Japan, [Tokyo]},
      year = {2016},
      mrclass = {14E30 (14C20 14J10)},
      mrnumber = {3617777},
      mrreviewer = {Paul A. Hacking},
      zblnumber = {1369.14013},
      }
  • [kollar-book] Go to document J. Kollár, Singularities of the Minimal Model Program, Cambridge University Press, Cambridge, 2013, vol. 200.
    @BOOK{kollar-book,
      author = {Kollár, János},
      title = {Singularities of the Minimal Model Program},
      series = {Cambridge Tracts in Math.},
      volume = {200},
      note = {With a collaboration of Sándor Kovács},
      publisher = {Cambridge University Press, Cambridge},
      year = {2013},
      pages = {x+370},
      isbn = {978-1-107-03534-8},
      mrclass = {14E30 (14B05)},
      mrnumber = {3057950},
      mrreviewer = {Tommaso De Fernex},
      doi = {10.1017/CBO9781139547895},
      zblnumber = {1282.14028},
      }
  • [kollar-ongoing] J. Kollár, Families of varieties of general type.
    @MISC{kollar-ongoing,
      author = {Kollár, János},
      title = {Families of varieties of general type},
      note = {in preparation},
      }
  • [ksb] Go to document J. Kollár and N. I. Shepherd-Barron, "Threefolds and deformations of surface singularities," Invent. Math., vol. 91, iss. 2, pp. 299-338, 1988.
    @ARTICLE{ksb,
      author = {Kollár, J. and Shepherd-Barron, N. I.},
      title = {Threefolds and deformations of surface singularities},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {91},
      year = {1988},
      number = {2},
      pages = {299--338},
      issn = {0020-9910},
      mrclass = {14J10 (14D20 14J30 32G10 32G13)},
      mrnumber = {0922803},
      mrreviewer = {Yujiro Kawamata},
      doi = {10.1007/BF01389370},
      zblnumber = {0642.14008},
      }
  • [kovacs] Go to document S. J. Kovács, "Young person’s guide to moduli of higher dimensional varieties," in Algebraic Geometry. Part 2, Amer. Math. Soc., Providence, RI, 2009, vol. 80, pp. 711-743.
    @INCOLLECTION{kovacs,
      author = {Kovács, Sándor J.},
      title = {Young person's guide to moduli of higher dimensional varieties},
      booktitle = {Algebraic Geometry. {P}art 2},
      venue = {{S}eattle 2005},
      series = {Proc. Sympos. Pure Math.},
      volume = {80},
      pages = {711--743},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2009},
      mrclass = {14J10 (14-02 14D20 14E30)},
      mrnumber = {2483953},
      mrreviewer = {James McKernan},
      doi = {10.1090/pspum/080.2/2483953},
      zblnumber = {1182.14034},
      }
  • [kp] Go to document S. J. Kovács and Z. Patakfalvi, "Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension," J. Amer. Math. Soc., vol. 30, iss. 4, pp. 959-1021, 2017.
    @ARTICLE{kp,
      author = {Kovács, Sándor J. and Patakfalvi, Zsolt},
      title = {Projectivity of the moduli space of stable log-varieties and subadditivity of log-{K}odaira dimension},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {30},
      year = {2017},
      number = {4},
      pages = {959--1021},
      issn = {0894-0347},
      mrclass = {14J10},
      mrnumber = {3671934},
      doi = {10.1090/jams/871},
      zblnumber = {06750372},
      }
  • [patakfalvi] Go to document Z. Patakfalvi, "Fibered stable varieties," Trans. Amer. Math. Soc., vol. 368, iss. 3, pp. 1837-1869, 2016.
    @ARTICLE{patakfalvi,
      author = {Patakfalvi, Zsolt},
      title = {Fibered stable varieties},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {368},
      year = {2016},
      number = {3},
      pages = {1837--1869},
      issn = {0002-9947},
      mrclass = {14J10 (14J40)},
      mrnumber = {3449226},
      mrreviewer = {Grzegorz Kapustka},
      doi = {10.1090/tran/6386},
      zblnumber = {1375.14118},
      }
  • [px] Go to document Z. Patakfalvi and C. Xu, "Ampleness of the CM line bundle on the moduli space of canonically polarized varieties," Algebr. Geom., vol. 4, iss. 1, pp. 29-39, 2017.
    @ARTICLE{px,
      author = {Patakfalvi, Zsolt and Xu, Chenyang},
      title = {Ampleness of the {CM} line bundle on the moduli space of canonically polarized varieties},
      journal = {Algebr. Geom.},
      fjournal = {Algebraic Geometry},
      volume = {4},
      year = {2017},
      number = {1},
      pages = {29--39},
      issn = {2214-2584},
      mrclass = {14J10 (14E30 14J15 32Q05)},
      mrnumber = {3592464},
      mrreviewer = {Enrica Floris},
      doi = {10.14231/AG-2017-002},
      zblnumber = {06741285},
      }
  • [viehweg1] E. Viehweg, "Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces," in Algebraic Varieties and Analytic Varieties, North-Holland, Amsterdam, 1983, vol. 1, pp. 329-353.
    @INCOLLECTION{viehweg1,
      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, Amsterdam},
      year = {1983},
      mrclass = {14J10 (14D20 14F05)},
      mrnumber = {0715656},
      mrreviewer = {Yujiro Kawamata},
      zblnumber = {0513.14019},
      }
  • [viehweg2] Go to document E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, Springer-Verlag, Berlin, 1995, vol. 30.
    @BOOK{viehweg2,
      author = {Viehweg, Eckart},
      title = {Quasi-Projective Moduli for Polarized Manifolds},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {30},
      publisher = {Springer-Verlag, Berlin},
      year = {1995},
      pages = {viii+320},
      isbn = {3-540-59255-5},
      mrclass = {14-02 (14D20 14D22)},
      mrnumber = {1368632},
      mrreviewer = {P. E. Newstead},
      doi = {10.1007/978-3-642-79745-3},
      zblnumber = {0844.14004},
      }
  • [viehweg3] Go to document E. Viehweg, "Compactifications of smooth families and of moduli spaces of polarized manifolds," Ann. of Math. (2), vol. 172, iss. 2, pp. 809-910, 2010.
    @ARTICLE{viehweg3,
      author = {Viehweg, Eckart},
      title = {Compactifications of smooth families and of moduli spaces of polarized manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {172},
      year = {2010},
      number = {2},
      pages = {809--910},
      issn = {0003-486X},
      mrclass = {14J10 (14D20 14J15 14M27)},
      mrnumber = {2680483},
      mrreviewer = {Paul A. Hacking},
      doi = {10.4007/annals.2010.172.809},
      zblnumber = {1238.14009},
      }
  • [zucker] Go to document S. Zucker, "Remarks on a theorem of Fujita," J. Math. Soc. Japan, vol. 34, iss. 1, pp. 47-54, 1982.
    @ARTICLE{zucker,
      author = {Zucker, Steven},
      title = {Remarks on a theorem of {F}ujita},
      journal = {J. Math. Soc. Japan},
      fjournal = {Journal of the Mathematical Society of Japan},
      volume = {34},
      year = {1982},
      number = {1},
      pages = {47--54},
      issn = {0025-5645},
      mrclass = {14C30 (32G99 32L15)},
      mrnumber = {0639804},
      mrreviewer = {James A. Carlson},
      doi = {10.2969/jmsj/03410047},
      zblnumber = {0503.14002},
      }

Authors

Osamu Fujino

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan