The GIT stability of polarized varieties via discrepancy

Pages 645-661 from Volume 177 (2013), Issue 2 by Yuji Odaka

Abstract

We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.

  • [Ale96] V. Alexeev, Log canonical singularities and complete moduli of stable pairs, 1996.
    @misc{Ale96,
      author={Alexeev, V.},
      TITLE={Log canonical singularities and complete moduli of stable pairs},
      ARXIV={alg-geom/960813},
      YEAR={1996},
     }
  • [BCHM09] 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 {BCHM09, MRKEY = {2601039},
      AUTHOR = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
      TITLE = {Existence of minimal models for varieties of log general type},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {23},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {405--468},
      ISSN = {0894-0347},
      MRCLASS = {14E30 (14E05)},
      MRNUMBER = {2601039},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1090/S0894-0347-09-00649-3},
      ZBLNUMBER = {1210.14019},
      }
  • [Don02] Go to document S. K. Donaldson, "Scalar curvature and stability of toric varieties," J. Differential Geom., vol. 62, iss. 2, pp. 289-349, 2002.
    @article {Don02, MRKEY = {1988506},
      AUTHOR = {Donaldson, S. K.},
      TITLE = {Scalar curvature and stability of toric varieties},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {62},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {289--349},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {32Q15 (14M25 53C21 53C55)},
      MRNUMBER = {1988506},
      URL = {http://projecteuclid.org/euclid.jdg/1090950195},
      ZBLNUMBER = {1074.53059},
      }
  • [Don05] Go to document S. K. Donaldson, "Lower bounds on the Calabi functional," J. Differential Geom., vol. 70, iss. 3, pp. 453-472, 2005.
    @article {Don05, MRKEY = {2192937},
      AUTHOR = {Donaldson, S. K.},
      TITLE = {Lower bounds on the {C}alabi functional},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {70},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {453--472},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {32Q15 (53C55)},
      MRNUMBER = {2192937},
      MRREVIEWER = {Julius Ross},
      URL = {http://projecteuclid.org/euclid.jdg/1143642909},
      ZBLNUMBER = {1149.53042},
      }
  • [Don10] S. K. Donaldson, Stability, birational transformations and the Kähler-Einstein problem, 2010.
    @misc{Don10,
      author={Donaldson, S. K.},
      TITLE = {Stability, birational transformations and the {K}ähler-{E}instein problem},
      YEAR={2010},
      ARXIV={1007.4220},
     }
  • [DM69] Go to document P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Publ. Math. I.H.E.S., vol. 36, pp. 75-109, 1969.
    @article{DM69, MRKEY = {0262240},
      AUTHOR={Deligne, P. and Mumford, D.},
      TITLE={The irreducibility of the space of curves of given genus},
      JOURNAL={Publ. Math. I.H.E.S.},
      VOLUME={36},
      YEAR={1969},
      PAGES = {75--109},
      MRNUMBER = {0262240},
      DOI = {10.1007/BF02684599},
      ZBLNUMBER = {0181.48803},
      }
  • [Ful84] W. Fulton, Intersection Theory, New York: Springer-Verlag, 1984, vol. 2.
    @book {Ful84, MRKEY = {0732620},
      AUTHOR = {Fulton, William},
      TITLE = {Intersection Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {2},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1984},
      PAGES = {xi+470},
      ISBN = {3-540-12176-5},
      MRCLASS = {14C17 (14-02 14C40)},
      MRNUMBER = {0732620},
      MRREVIEWER = {Werner Kleinert},
      ZBLNUMBER = {0541.14005},
      }
  • [Gie82] D. Gieseker, Lectures on Moduli of Curves, Bombay: Tata Institute of Fundamental Research, 1982, vol. 69.
    @book {Gie82, MRKEY = {0691308},
      AUTHOR = {Gieseker, D.},
      TITLE = {Lectures on Moduli of Curves},
      SERIES = {Tata Inst. Fund. Res. Lect. Math. Phys.},
      VOLUME = {69},
      PUBLISHER = {Tata Institute of Fundamental Research},
      ADDRESS={ Bombay},
      YEAR = {1982},
      PAGES = {iii+99},
      ISBN = {3-540-11953-1},
      MRCLASS = {14H10 (14D25 32J05)},
      MRNUMBER = {0691308},
      MRREVIEWER = {Ciro Ciliberto},
      ZBLNUMBER = {0534.14012},
      }
  • [Har77] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @book {Har77, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      NOTE = {Graduate Texts Math.},
      VOLUME={ 52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {0463157},
      MRREVIEWER = {Robert Speiser},
      ZBLNUMBER = {0367.14001},
      }
  • [KM76] Go to document F. F. Knudsen and D. Mumford, "The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”," Math. Scand., vol. 39, iss. 1, pp. 19-55, 1976.
    @article {KM76, MRKEY = {0437541},
      AUTHOR = {Knudsen, Finn Faye and Mumford, David},
      TITLE = {The projectivity of the moduli space of stable curves. {I}. {P}reliminaries on ``det'' and ``{D}iv''},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {39},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {19--55},
      ISSN = {0025-5521},
      MRCLASS = {14H10 (14F05 14C05)},
      MRNUMBER = {0437541},
      MRREVIEWER = {P. E. Newstead},
      URL={http://www.mscand.dk/issue.php?year=1976&volume=39&issue= },
      ZBLNUMBER = {0343.14008},
      }
  • [KM98] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
    @book {KM98, MRKEY = {1658959},
      AUTHOR = {Koll{á}r, J{á}nos and Mori, Shigefumi},
      TITLE = {Birational Geometry of Algebraic Varieties},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {134},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {viii+254},
      ISBN = {0-521-63277-3},
      MRCLASS = {14E30},
      MRNUMBER = {1658959},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1017/CBO9780511662560},
      ZBLNUMBER = {0926.14003},
      }
  • [KSB88] 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 {KSB88, MRKEY = {0922803},
      AUTHOR = {Koll{á}r, J{á}nos 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},
      CODEN = {INVMBH},
      MRCLASS = {14J10 (14D20 14J30 32G10 32G13)},
      MRNUMBER = {0922803},
      MRREVIEWER = {Yujiro Kawamata},
      DOI = {10.1007/BF01389370},
      ZBLNUMBER = {0642.14008},
      }
  • [Koletc92] Flips and Abundance for Algebraic Threefolds, Kollár, J., Ed., Paris: Soc. Math. France, 1992, vol. 211.
    @book{Koletc92, EDITOR = {Koll{á}r, J{á}nos},
      TITLE={Flips and Abundance for Algebraic Threefolds},
      TITLENOTE={(A summer seminar at the University of ÊUtah, Salt Lake City, 1991)},
      SERIES={Astérisque},
      VOLUME={211},
      YEAR={1992},
      ZBLNUMBER={0782.00075},
      PUBLISHER={Soc. Math. France},
      ADDRESS = {Paris},
      }
  • [Kol] Go to document J. Kollár, Book on Moduli of Surfaces.
    @misc{Kol,
      author = {Koll{á}r, J{á}nos},
      TITLE={Book on {M}oduli of {S}urfaces},
      NOTE={to appear; Chapters 1, 2, 3 available on-line},
      URL={http://www.math.princeton.edu/~kollar/},
     }
  • [LX11] C. Li and C. Xu, Special test configurations and ${K}$-stability of $\mathbb{Q}$-Fano varieties, 2011.
    @misc{LX11,
      author={Li, C. and Xu, C.},
      TITLE={Special test configurations and ${K}$-stability of {$\mathbb{Q}$}-{F}ano varieties},
      YEAR={2011},
      ARXIV={1111.5398},
     }
  • [Mor82] Go to document S. Mori, "Threefolds whose canonical bundles are not numerically effective," Ann. of Math., vol. 116, iss. 1, pp. 133-176, 1982.
    @article {Mor82, MRKEY = {0662120},
      AUTHOR = {Mori, Shigefumi},
      TITLE = {Threefolds whose canonical bundles are not numerically effective},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {116},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {133--176},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14J30 (14E35)},
      MRNUMBER = {0662120},
      MRREVIEWER = {Ulf Persson},
      DOI = {10.2307/2007050},
      ZBLNUMBER = {0557.14021},
      }
  • [Mum65] D. Mumford, Geometric Invariant Theory, New York: Springer-Verlag, 1965.
    @book {Mum65, MRKEY = {0214602},
      AUTHOR = {Mumford, David},
      TITLE = {Geometric Invariant Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1965},
      PAGES = {vi+145},
      MRCLASS = {14.55},
      MRNUMBER = {0214602},
      MRREVIEWER = {F. Oort},
      ZBLNUMBER = {0147.39304},
      }
  • [Mum77] D. Mumford, "Stability of Projective Varieties," Enseign. Math. (2), vol. 23, pp. 39-110, 1977.
    @article{Mum77, MRKEY = {0450273},
      AUTHOR = {Mumford, David},
      TITLE = {Stability of Projective Varieties},
      JOURNAL = {Enseign. Math. (2)},
      VOLUME={23},
      YEAR = {1977},
      PAGES = {39--110},
      MRCLASS = {14D20},
      MRNUMBER = {0450273},
      ZBLNUMBER = {0363.14003},
      }
  • [Od09] Y. Odaka, A generalization of the Ross-Thomas slope theory.
    @misc{Od09,
      author={Odaka, Y.},
      TITLE={A generalization of the {R}oss-{T}homas slope theory},
      NOTE={to appear in {\it Osaka J. Math.}},
     }
  • [Od11] Go to document Y. Odaka, "The Calabi conjecture and K-stability," Internat. Math. Res. Not., vol. 2012, p. no. 10, 2272-2288.
    @article{Od11, KEY={Oda12a},
      AUTHOR={Odaka, Y.},
      TITLE={The {C}alabi conjecture and {K}-stability},
      JOURNAL={Internat. Math. Res. Not.},
      VOLUME={2012},
      PAGES={no. 10, 2272--2288},
      SORTYEAR={2012},
      MRNUMBER={2923166},
      DOI={10.1093/imrn/rnr107},
      }
  • [Od12] Y. Odaka, On parametrization, optimization and triviality of test configurations, 2012.
    @misc{Od12, KEY={Oda12b},
      AUTHOR={Odaka, Y.},
      TITLE={On parametrization, optimization and triviality of test configurations},
      YEAR={2012},
      ARXIV={1201.0692},
     }
  • [OX11] Y. Odaka and C. Xu, Log-canonical models of singular pairs and its applications.
    @misc{OX11,
      author={Odaka, Y. and Xu, C.},
      TITLE={Log-canonical models of singular pairs and its applications},
      NOTE={to appear in {\it Math. Res. Lett.}},
     }
  • [RT07] Go to document J. Ross and R. Thomas, "A study of the Hilbert-Mumford criterion for the stability of projective varieties," J. Algebraic Geom., vol. 16, iss. 2, pp. 201-255, 2007.
    @article {RT07, MRKEY = {2274514},
      AUTHOR = {Ross, Julius and Thomas, Richard},
      TITLE = {A study of the {H}ilbert-{M}umford criterion for the stability of projective varieties},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {16},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {201--255},
      ISSN = {1056-3911},
      MRCLASS = {14L24 (14L30 32L10 32Q20 53C25)},
      MRNUMBER = {2274514},
      MRREVIEWER = {Julien Keller},
      DOI = {10.1090/S1056-3911-06-00461-9},
      ZBLNUMBER = {1200.14095},
      }
  • [Sha81] Go to document J. Shah, "Stability of two-dimensional local rings. I," Invent. Math., vol. 64, iss. 2, pp. 297-343, 1981.
    @article {Sha81, MRKEY = {0629473},
      AUTHOR = {Shah, Jayant},
      TITLE = {Stability of two-dimensional local rings. {I}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {64},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {297--343},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14J15 (13H10)},
      MRNUMBER = {0629473},
      MRREVIEWER = {Werner Kleinert},
      DOI = {10.1007/BF01389171},
      ZBLNUMBER = {0475.14028},
      }
  • [She83] N. I. Shepherd-Barron, "Degenerations with numerically effective canonical divisor," in The Birational Geometry of Degenerations, Boston, MA: Birkhäuser, 1983, vol. 29, pp. 33-84.
    @incollection {She83, MRKEY = {0690263},
      AUTHOR = {Shepherd-Barron, N. I.},
      TITLE = {Degenerations with numerically effective canonical divisor},
      BOOKTITLE = {The Birational Geometry of Degenerations},
      VENUE={{C}ambridge, {M}ass., 1981},
      SERIES = {Progr. Math.},
      VOLUME = {29},
      PAGES = {33--84},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1983},
      MRCLASS = {14J10 (14D20)},
      MRNUMBER = {0690263},
      ZBLNUMBER = {0506.14028},
      }
  • [Stp11] J. Stoppa, A note on the definition of K-stability, 2011.
    @misc{Stp11,
      author={Stoppa, J.},
      TITLE={A note on the definition of {K}-stability},
      YEAR={2011},
      ARXIV={1111.5826},
     }
  • [Tia97] Go to document G. Tian, "Kähler-Einstein metrics with positive scalar curvature," Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997.
    @article {Tia97, MRKEY = {1471884},
      AUTHOR = {Tian, Gang},
      TITLE = {Kähler-{E}instein metrics with positive scalar curvature},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {130},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {1--37},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {53C25 (32L07 53C55)},
      MRNUMBER = {1471884},
      MRREVIEWER = {Thalia D. Jeffres},
      DOI = {10.1007/s002220050176},
      ZBLNUMBER = {0892.53027},
     }
  • [Vas05] W. Vasconcelos, Integral Closure: Rees Algebras, Multiplicities, Algorithms, New York: Springer-Verlag, 2005.
    @book {Vas05, MRKEY = {2153889},
      AUTHOR = {Vasconcelos, Wolmer},
      TITLE = {Integral Closure: Rees Algebras, Multiplicities, Algorithms},
      SERIES = {Springer Monogr. Math.},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {xii+519},
      ISBN = {978-3-540-25540-6; 3-540-25540-0},
      MRCLASS = {13A30 (13-02 13B22 13D40 13H15)},
      MRNUMBER = {2153889},
      MRREVIEWER = {Ng{ô} Vi{ê}t Trung},
      ZBLNUMBER = {1082.13006},
      }
  • [Wan08] X. Wang, "Heights and GIT weights," Math. Res. Lett., vol. 19, p. 18, 2012.
    @article{Wan08,
      author={Wang, X.},
      TITLE={Heights and {GIT} weights},
      JOURNAL = {Math. Res. Lett.},
      VOLUME = {19},
      PAGES={18pp},
      YEAR={2012},
     }

Keywords
, ,
DOI
https://doi.org/10.4007/annals.2013.177.2.6
MR
3010808
zbMATH
1271.14067
Milestones
Received: 15 April 2011
Revised: 26 March 2012
Accepted: 11 July 2012
Published online: 1 March 2013

Authors

Yuji Odaka

Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kyoto, Japan

Current address:

Imperial College London, London, England