Kodaira dimension and zeros of holomorphic one-forms

Abstract

We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties.

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

  • [Arapura] Go to document D. Arapura, "Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves," Bull. Amer. Math. Soc., vol. 26, iss. 2, pp. 310-314, 1992.
    @article {Arapura, MRKEY = {1129312},
      AUTHOR = {Arapura, Donu},
      TITLE = {Higgs line bundles, {G}reen-{L}azarsfeld sets, and maps of {K}ähler manifolds to curves},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {26},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {310--314},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {14C22 (14C30 14F35 14F40 14J05)},
      MRNUMBER = {1129312},
      MRREVIEWER = {Claire Voisin},
      ZBLNUMBER = {0759.14016},
      DOI = {10.1090/S0273-0979-1992-00283-5},
      }
  • [Carrell] Go to document J. B. Carrell, "Holomorphic one forms and characteristic numbers," Topology, vol. 13, pp. 225-228, 1974.
    @article {Carrell, MRKEY = {0348147},
      AUTHOR = {Carrell, James B.},
      TITLE = {Holomorphic one forms and characteristic numbers},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {13},
      YEAR = {1974},
      PAGES = {225--228},
      ISSN = {0040-9383},
      MRCLASS = {32J15 (57D20)},
      MRNUMBER = {0348147},
      MRREVIEWER = {I. Vaisman},
      ZBLNUMBER = {0305.32016},
      DOI = {10.1016/0040-9383(74)90015-9},
      }
  • [EV] Go to document H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, Basel: Birkhäuser, 1992, vol. 20.
    @book {EV, MRKEY = {1193913},
      AUTHOR = {Esnault, H{é}l{è}ne and Viehweg, Eckart},
      TITLE = {Lectures on Vanishing Theorems},
      SERIES = {DMV Seminar},
      VOLUME = {20},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1992},
      PAGES = {vi+164},
      ISBN = {3-7643-2822-3},
      MRCLASS = {14F17 (14F40 32L10 32L20)},
      MRNUMBER = {1193913},
      MRREVIEWER = {Marko Roczen},
      ZBLNUMBER = {0779.14003},
      DOI = {10.1007/978-3-0348-8600-0},
      }
  • [GreenLazarsfeld] Go to document M. Green and R. Lazarsfeld, "Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville," Invent. Math., vol. 90, iss. 2, pp. 389-407, 1987.
    @article {GreenLazarsfeld, MRKEY = {0910207},
      AUTHOR = {Green, Mark and Lazarsfeld, Robert},
      TITLE = {Deformation theory, generic vanishing theorems, and some conjectures of {E}nriques, {C}atanese and {B}eauville},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {90},
      YEAR = {1987},
      NUMBER = {2},
      PAGES = {389--407},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32G10 (14C22 14H15)},
      MRNUMBER = {0910207},
      MRREVIEWER = {Sheldon H. Katz},
      ZBLNUMBER = {0659.14007},
      DOI = {10.1007/BF01388711},
      }
  • [HK] Go to document C. D. Hacon and S. J. Kovács, "Holomorphic one-forms on varieties of general type," Ann. Sci. École Norm. Sup., vol. 38, iss. 4, pp. 599-607, 2005.
    @article {HK, MRKEY = {2172952},
      AUTHOR = {Hacon, Christopher D. and Kov{á}cs, S{á}ndor J.},
      TITLE = {Holomorphic one-forms on varieties of general type},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {38},
      YEAR = {2005},
      NUMBER = {4},
      PAGES = {599--607},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14F10 (14J99)},
      MRNUMBER = {2172952},
      MRREVIEWER = {E. Akyıldız},
      ZBLNUMBER = {1083.14015},
      DOI = {10.1016/j.ansens.2004.12.002},
      }
  • [Kawamata] Go to document Y. Kawamata, "Characterization of abelian varieties," Compositio Math., vol. 43, iss. 2, pp. 253-276, 1981.
    @article {Kawamata, MRKEY = {0622451},
      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},
      CODEN = {CMPMAF},
      MRCLASS = {14J10 (32J15)},
      MRNUMBER = {0622451},
      MRREVIEWER = {Daniel Comenetz},
      ZBLNUMBER = {0471.14022},
      URL = {http://www.numdam.org/item?id=CM_1981__43_2_253_0},
      }
  • [Kollar] J. Kollár, "Subadditivity of the Kodaira dimension: fibers of general type," in Algebraic Geometry, Amsterdam: North-Holland, 1987, vol. 10, pp. 361-398.
    @incollection {Kollar, MRKEY = {0946244},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Subadditivity of the {K}odaira dimension: fibers of general type},
      BOOKTITLE = {Algebraic Geometry},
      VENUE={{S}endai, 1985},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {10},
      PAGES = {361--398},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1987},
      MRCLASS = {14J10 (14C30 14D20)},
      MRNUMBER = {0946244},
      MRREVIEWER = {Yujiro Kawamata},
      ZBLNUMBER = {0659.14024},
      }
  • [Kovacs-smooth] S. J. Kovács, "Smooth families over rational and elliptic curves," J. Algebraic Geom., vol. 5, iss. 2, pp. 369-385, 1996.
    @article {Kovacs-smooth, MRKEY = {1374712},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Smooth families over rational and elliptic curves},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {5},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {369--385},
      ISSN = {1056-3911},
      MRCLASS = {14J10 (14J15)},
      MRNUMBER = {1374712},
      MRREVIEWER = {Marco Andreatta},
      ZBLNUMBER = {0861.14033},
      }
  • [Kovacs-nef] Go to document S. J. Kovács, "Families over a base with a birationally nef tangent bundle," Math. Ann., vol. 308, iss. 2, pp. 347-359, 1997.
    @article {Kovacs-nef, MRKEY = {1464907},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Families over a base with a birationally nef tangent bundle},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {308},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {347--359},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14J10 (14D99)},
      MRNUMBER = {1464907},
      MRREVIEWER = {Yoshiaki Fukuma},
      ZBLNUMBER = {0922.14024},
      DOI = {10.1007/s002080050079},
      }
  • [Kovacs-log] Go to document S. J. Kovács, "Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties," Compositio Math., vol. 131, iss. 3, pp. 291-317, 2002.
    @article {Kovacs-log, MRKEY = {1905025},
      AUTHOR = {Kov{á}cs, S{á}ndor J.},
      TITLE = {Logarithmic vanishing theorems and {A}rakelov-{P}arshin boundedness for singular varieties},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {131},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {291--317},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14F17 (14J10)},
      MRNUMBER = {1905025},
      MRREVIEWER = {Scott R. Nollet},
      ZBLNUMBER = {1048.14006},
      DOI = {10.1023/A:1015592420937},
      }
  • [LZ] Go to document T. Luo and Q. Zhang, "Holomorphic forms on threefolds," in Recent Progress in Arithmetic and Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 386, pp. 87-94.
    @incollection {LZ, MRKEY = {2182772},
      AUTHOR = {Luo, Tie and Zhang, Qi},
      TITLE = {Holomorphic forms on threefolds},
      BOOKTITLE = {Recent Progress in Arithmetic and Algebraic Geometry},
      SERIES = {Contemp. Math.},
      VOLUME = {386},
      PAGES = {87--94},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      MRCLASS = {14E30 (14F10 14J30)},
      MRNUMBER = {2182772},
      ZBLNUMBER = {1216.14013},
      DOI = {10.1090/conm/386/07219},
      }
  • [Migliorini] L. Migliorini, "A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial," J. Algebraic Geom., vol. 4, iss. 2, pp. 353-361, 1995.
    @article {Migliorini, MRKEY = {1311355},
      AUTHOR = {Migliorini, Luca},
      TITLE = {A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {4},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {353--361},
      ISSN = {1056-3911},
      MRCLASS = {14J29 (14J10)},
      MRNUMBER = {1311355},
      MRREVIEWER = {Xiao Tao Sun},
      ZBLNUMBER = {0834.14021},
      }
  • [PopaSchnell] Go to document M. Popa and C. Schnell, "Generic vanishing theory via mixed Hodge modules," Forum Math. Sigma, vol. 1, p. 1, 2013.
    @article {PopaSchnell, MRKEY = {3090229},
      AUTHOR = {Popa, Mihnea and Schnell, Christian},
      TITLE = {Generic vanishing theory via mixed {H}odge modules},
      JOURNAL = {Forum Math. Sigma},
      FJOURNAL = {Forum of Mathematics. Sigma},
      VOLUME = {1},
      YEAR = {2013},
      PAGES = {e1, 60pp.},
      ISSN = {2050-5094},
      MRCLASS = {14C30 (14F10 14K20 32L20 32S25)},
      MRNUMBER = {3090229},
      ZBLNUMBER = {06209103},
      DOI = {10.1017/fms.2013.1},
      }
  • [Saito] Go to document M. Saito, "Modules de Hodge polarisables," Publ. Res. Inst. Math. Sci., vol. 24, iss. 6, pp. 849-995 (1989), 1988.
    @article {Saito, MRKEY = {1000123},
      AUTHOR = {Saito, Morihiko},
      TITLE = {Modules de {H}odge polarisables},
      JOURNAL = {Publ. Res. Inst. Math. Sci.},
      FJOURNAL = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      VOLUME = {24},
      YEAR = {1988},
      NUMBER = {6},
      PAGES = {849--995 (1989)},
      ISSN = {0034-5318},
      CODEN = {KRMPBV},
      MRCLASS = {32C35 (14C30 32C38 32C42 32G99)},
      MRNUMBER = {1000123},
      MRREVIEWER = {J. H. M. Steenbrink},
      ZBLNUMBER = {0691.14007},
      DOI = {10.2977/prims/1195173930},
      }
  • [ViehwegZuo] 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 {ViehwegZuo, 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},
      }
  • [Zhang] Q. Zhang, "Global holomorphic one-forms on projective manifolds with ample canonical bundles," J. Algebraic Geom., vol. 6, iss. 4, pp. 777-787, 1997.
    @article {Zhang, MRKEY = {1487236},
      AUTHOR = {Zhang, Qi},
      TITLE = {Global holomorphic one-forms on projective manifolds with ample canonical bundles},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {6},
      YEAR = {1997},
      NUMBER = {4},
      PAGES = {777--787},
      ISSN = {1056-3911},
      MRCLASS = {14J99 (32L10)},
      MRNUMBER = {1487236},
      MRREVIEWER = {A. R. Shastri},
      ZBLNUMBER = {0922.14008},
      }

Authors

Mihnea Popa

Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607

Christian Schnell

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794