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]
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]
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]
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]
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]
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]
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]
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},
} -
@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]
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]
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]
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]
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]
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},
}