Abstract
We show that the base complex manifold of an effectively parametrized holomorphic family of compact canonically polarized complex manifolds admits a smooth invariant Finsler metric whose holomorphic sectional curvature is bounded above by a negative constant. As a consequence, we show that such a base manifold is Kobayashi hyperbolic.
-
[Ah1]
L. V. Ahlfors, "Some remarks on Teichmüller’s space of Riemann surfaces," Ann. of Math., vol. 74, pp. 171-191, 1961.@article{Ah1, MRKEY = {0204641},
AUTHOR = {Ahlfors, Lars V.},
TITLE = {Some remarks on {T}eichmüller's space of {R}iemann surfaces},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {74},
YEAR = {1961},
PAGES = {171--191},
ISSN = {0003-486X},
MRCLASS = {30.45 (32.40)},
MRNUMBER = {0204641},
MRREVIEWER = {R. Osserman},
DOI = {10.2307/1970309},
ZBLNUMBER = {0146.30602},
} -
[Ah2] L. V. Ahlfors, "Curvature properties of Teichmüller’s space," J. Analyse Math., vol. 9, pp. 161-176, 1961/1962.
@article{Ah2, MRKEY = {0136730},
AUTHOR = {Ahlfors, Lars V.},
TITLE = {Curvature properties of {T}eichmüller's space},
JOURNAL = {J. Analyse Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {9},
YEAR = {1961/1962},
PAGES = {161--176},
ISSN = {0021-7670},
MRCLASS = {30.45 (32.44)},
MRNUMBER = {0136730},
MRREVIEWER = {R. Osserman},
DOI = {10.1007/BF02795342},
ZBLNUMBER = {0148.31201},
} -
[Ah3] L. V. Ahlfors, "The theory of meromorphic curves," Acta Soc. Sci. Fennicae. Nova Ser. A., vol. 3, iss. 4, p. 31, 1941.
@article{Ah3, MRKEY = {0004309},
AUTHOR = {Ahlfors, Lars V.},
TITLE = {The theory of meromorphic curves},
JOURNAL = {Acta Soc. Sci. Fennicae. Nova Ser. A.},
VOLUME = {3},
YEAR = {1941},
NUMBER = {4},
PAGES = {31},
MRCLASS = {30.0X},
MRNUMBER = {0004309},
MRREVIEWER = {M. H. Heins},
ZBLNUMBER = {0061.15206},
} -
[Au] T. Aubin, "Équations du type Monge-Ampère sur les variétés kähleriennes compactes," C. R. Acad. Sci. Paris Sér. A-B, vol. 283, iss. 3, p. aiii, a119-a121, 1976.
@article{Au, MRKEY = {0433520},
AUTHOR = {Aubin, Thierry},
TITLE = {\'{E}quations du type {M}onge-{A}mpère sur les variétés kähleriennes compactes},
JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
VOLUME = {283},
YEAR = {1976},
NUMBER = {3},
PAGES = {Aiii, A119--A121},
MRCLASS = {58G99 (32C10 35J60)},
MRNUMBER = {0433520},
MRREVIEWER = {J. L. Kazdan},
ZBLNUMBER = {0333.53040},
} -
[Gri] Topics in Transcendental Algebraic GeometryPrinceton, NJ: Princeton Univ. Press, 1984.
@proceedings{Gri, MRKEY = {0756842},
TITLE = {Topics in {T}ranscendental {A}lgebraic {G}eometry},
SERIES = {Ann. of Math. Studies},
VOLUME = {106},
BOOKTITLE = {Proceedings of a Seminar Held at the {I}nstitute for {A}dvanced {S}tudy, {P}rinceton, {N}.{J}., during the academic year 1981/1982},
EDITOR = {Phillip Griffiths},
PUBLISHER = {Princeton Univ. Press},
ADDRESS={Princeton, NJ},
YEAR = {1984},
PAGES = {viii+316},
ISBN = {0-691-08335-5; 0-691-08339-8},
MRCLASS = {14C30 (14-02 14D05 32J25)},
MRNUMBER = {0756842},
MRREVIEWER = {David Ortland},
ZBLNUMBER = {0528.00004},
} -
[Kob] S. Kobayashi, Hyperbolic Complex Spaces, New York: Springer-Verlag, 1998, vol. 318.
@book{Kob, MRKEY = {1635983},
AUTHOR = {Kobayashi, Shoshichi},
TITLE = {Hyperbolic Complex Spaces},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {318},
PUBLISHER = {Springer-Verlag},
YEAR = {1998},
PAGES = {xiv+471},
ISBN = {3-540-63534-3},
MRCLASS = {32H20 (32H15 32H25 32H30)},
MRNUMBER = {1635983},
MRREVIEWER = {William A. Cherry},
ADDRESS = {New York},
ZBLNUMBER = {0917.32019},
DOI = {10.1007/978-3-662-03582-5},
} -
[Koi] N. Koiso, "Einstein metrics and complex structures," Invent. Math., vol. 73, iss. 1, pp. 71-106, 1983.
@article{Koi, MRKEY = {0707349},
AUTHOR = {Koiso, N.},
TITLE = {Einstein metrics and complex structures},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {73},
YEAR = {1983},
NUMBER = {1},
PAGES = {71--106},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58D17 (53C25 53C55)},
MRNUMBER = {0707349},
MRREVIEWER = {N. J. Hitchin},
DOI = {10.1007/BF01393826},
ZBLNUMBER = {0515.53040},
} -
[Kov1] S. J. Kovács, "Families over a base with a birationally nef tangent bundle," Math. Ann., vol. 308, iss. 2, pp. 347-359, 1997.
@article{Kov1, 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},
DOI = {10.1007/s002080050079},
ZBLNUMBER = {0922.14024},
} -
[Kov2] S. J. Kovács, "Algebraic hyperbolicity of fine moduli spaces," J. Algebraic Geom., vol. 9, iss. 1, pp. 165-174, 2000.
@article{Kov2, MRKEY = {1713524},
AUTHOR = {Kov{á}cs, S{á}ndor J.},
TITLE = {Algebraic hyperbolicity of fine moduli spaces},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {9},
YEAR = {2000},
NUMBER = {1},
PAGES = {165--174},
ISSN = {1056-3911},
MRCLASS = {14D22 (14J10)},
MRNUMBER = {1713524},
MRREVIEWER = {Jean-Marc Drezet},
ZBLNUMBER = {0970.14008},
} -
[Mi] 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{Mi, 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},
} -
[NS] M. S. Narasimhan and R. R. Simha, "Manifolds with ample canonical class," Invent. Math., vol. 5, pp. 120-128, 1968.
@article{NS, MRKEY = {0236960},
AUTHOR = {Narasimhan, M. S. and Simha, R. R.},
TITLE = {Manifolds with ample canonical class},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {5},
YEAR = {1968},
PAGES = {120--128},
ISSN = {0020-9910},
MRCLASS = {57.60},
MRNUMBER = {0236960},
MRREVIEWER = {W. Kaup},
DOI = {10.1007/BF01425543},
ZBLNUMBER = {0159.37902},
} -
[R] H. L. Royden, "Intrinsic metrics on Teichmüller space," in Proceedings of the International Congress of Mathematicians, Montreal, Que., 1975, pp. 217-221.
@inproceedings{R, MRKEY = {0447636},
AUTHOR = {Royden, H. L.},
TITLE = {Intrinsic metrics on {T}eichmüller space},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians},
VENUE={{V}ancouver, {B}. {C}., 1974, {V}ol. 2},
PAGES = {217--221},
PUBLISHER = {Canad. Math. Congress},
ADDRESS={Montreal, Que.},
YEAR = {1975},
MRCLASS = {32G15 (32H15)},
MRNUMBER = {0447636},
MRREVIEWER = {L. Keen},
ZBLNUMBER = {0344.32005},
} -
[S2] Y. T. Siu, "Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class," in Contributions to Several Complex Variables, Vieweg, Braunschweig, 1986, vol. E9, pp. 261-298.
@incollection{S2, MRKEY = {0859202},
AUTHOR = {Siu, Yum Tong},
TITLE = {Curvature of the {W}eil-{P}etersson metric in the moduli space of compact {K}ähler-{E}instein manifolds of negative first {C}hern class},
BOOKTITLE = {Contributions to Several Complex Variables},
SERIES = {Aspects Math.},
VOLUME={E9},
PAGES = {261--298},
PUBLISHER = {Vieweg, Braunschweig},
YEAR = {1986},
MRCLASS = {32G15 (53C25 53C55 58E20)},
MRNUMBER = {0859202},
MRREVIEWER = {Akito Futaki},
ZBLNUMBER = {0598.32020},
} -
[Sch1] G. Schumacher, "The curvature of the Petersson-Weil metric on the moduli space of Kähler-Einstein manifolds," in Complex Analysis and Geometry, New York: Plenum, 1993, pp. 339-354.
@incollection{Sch1, MRKEY = {1211890},
AUTHOR = {Schumacher, Georg},
TITLE = {The curvature of the {P}etersson-{W}eil metric on the moduli space of {K}ähler-{E}instein manifolds},
BOOKTITLE = {Complex Analysis and Geometry},
SERIES = {Univ. Ser. Math.},
PAGES = {339--354},
PUBLISHER = {Plenum} , ADDRESS={New York},
YEAR = {1993},
MRCLASS = {32L07 (32G13)},
MRNUMBER = {1211890},
MRREVIEWER = {Steven B. Bradlow},
ZBLNUMBER = {0804.53094},
} -
[Sch2] G. Schumacher, "Positivity of relative canonical bundles and applications," Invent. Math., vol. 190, iss. 1, pp. 1-56, 2012.
@article{Sch2, MRKEY = {2969273},
AUTHOR = {Schumacher, Georg},
TITLE = {{P}ositivity of relative canonical bundles and applications},
NOTE={[Erratum: {\it Invent. Math.} {\bf 192} (2013), 253--255. \mr{3032331}. \zbl{1266.32025}. \doi{10.1007/s00222-013-0458-z}]},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {190},
YEAR = {2012},
NUMBER = {1},
PAGES = {1--56},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32Q99 (14D20 32G99 32L99)},
MRNUMBER = {2969273},
ZBLNUMBER = {1258.32005},
DOI = {10.1007/s00222-012-0374-7},
} -
[SY1] Y. Siu and S. Yeung, "Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane," Invent. Math., vol. 124, iss. 1-3, pp. 573-618, 1996.
@article{SY1, MRKEY = {1369429},
AUTHOR = {Siu, Yum-Tong and Yeung, Sai-Kee},
TITLE = {Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {124},
YEAR = {1996},
NUMBER = {1-3},
PAGES = {573--618},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32H25 (32H20)},
MRNUMBER = {1369429},
MRREVIEWER = {Evgenii Nochka},
DOI = {10.1007/s002220050064},
ZBLNUMBER = {0856.32017},
} -
[SY2] Y. Siu and S. Yeung, "Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees," Amer. J. Math., vol. 119, iss. 5, pp. 1139-1172, 1997.
@article{SY2, MRKEY = {1473072},
AUTHOR = {Siu, Yum-Tong and Yeung, Sai-Kee},
TITLE = {Defects for ample divisors of abelian varieties, {S}chwarz lemma, and hyperbolic hypersurfaces of low degrees},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {119},
YEAR = {1997},
NUMBER = {5},
PAGES = {1139--1172},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {32H20 (32H30)},
MRNUMBER = {1473072},
MRREVIEWER = {Min Ru},
DOI = {10.1353/ajm.1997.0033},
} -
[V] E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, New York: Springer-Verlag, 1995, vol. 30.
@book{V, MRKEY = {1368632},
AUTHOR = {Viehweg, Eckart},
TITLE = {Quasi-Projective Moduli for Polarized Manifolds},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {30},
PUBLISHER = {Springer-Verlag},
YEAR = {1995},
PAGES = {viii+320},
ISBN = {3-540-59255-5},
MRCLASS = {14-02 (14D20 14D22)},
MRNUMBER = {1368632},
MRREVIEWER = {P. E. Newstead},
ADDRESS = {New York},
ZBLNUMBER = {0844.14004},
DOI = {10.1007/978-3-642-79745-3},
} -
[VZ] E. Viehweg and K. Zuo, "On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds," Duke Math. J., vol. 118, iss. 1, pp. 103-150, 2003.
@article{VZ, MRKEY = {1978884},
AUTHOR = {Viehweg, Eckart and Zuo, Kang},
TITLE = {On the {B}rody hyperbolicity of moduli spaces for canonically polarized manifolds},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {118},
YEAR = {2003},
NUMBER = {1},
PAGES = {103--150},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14J15 (14J10 32G05 32Q45)},
MRNUMBER = {1978884},
MRREVIEWER = {S{á}ndor J. Kov{á}cs},
DOI = {10.1215/S0012-7094-03-11815-3},
ZBLNUMBER = {1042.14010},
} -
[Wo] S. A. Wolpert, "Chern forms and the Riemann tensor for the moduli space of curves," Invent. Math., vol. 85, iss. 1, pp. 119-145, 1986.
@article{Wo, MRKEY = {0842050},
AUTHOR = {Wolpert, Scott A.},
TITLE = {Chern forms and the {R}iemann tensor for the moduli space of curves},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {85},
YEAR = {1986},
NUMBER = {1},
PAGES = {119--145},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32G15 (14H15 53C20)},
MRNUMBER = {0842050},
MRREVIEWER = {C. Earle},
DOI = {10.1007/BF01388794},
ZBLNUMBER = {0595.32031},
} -
[Y] S. T. Yau, "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I," Comm. Pure Appl. Math., vol. 31, iss. 3, pp. 339-411, 1978.
@article{Y, MRKEY = {0480350},
AUTHOR = {Yau, Shing Tung},
TITLE = {On the {R}icci curvature of a compact {K}ähler manifold and the complex {M}onge-{A}mpère equation. {I}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {31},
YEAR = {1978},
NUMBER = {3},
PAGES = {339--411},
ISSN = {0010-3640},
CODEN = {CPAMAT},
MRCLASS = {53C55 (32C10 35J60)},
MRNUMBER = {0480350},
MRREVIEWER = {Robert E. Greene},
DOI = {10.1002/cpa.3160310304},
ZBLNUMBER = {0369.53059},
}
Full Article