Abstract
We prove a version of Jonsson-Mustaţǎ’s Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li’s conjecture that a minimizer of the normalized volume function is always quasi-monomial.
Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with previous works by many people, we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space, whose geometric points parametrize K-polystable klt Fano varieties.
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@MISC{ABHX-reductivity,
author = {Alper, Jarod and Blum, Harold and Halpern-Leistner, Daniel and Xu, Chenyang},
title = {Reductivity of the automorphism group of {K}-polystable {F}ano varieties},
year = {2019},
arxiv = {1906.03122},
zblnumber = {},
} -
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@INCOLLECTION{Berndtsson-openness,
author = {Berndtsson, Bo},
title = {The openness conjecture and complex {B}runn-{M}inkowski inequalities},
booktitle = {Complex Geometry and Dynamics},
series = {Abel Symp.},
volume = {10},
pages = {29--44},
publisher = {Springer, Cham},
year = {2015},
mrclass = {32U05 (28A75 31C10 52A40)},
mrnumber = {3587460},
mrreviewer = {Steven George Krantz},
zblnumber = {1337.32001},
doi = {10.1007/978-3-319-20337-9_2},
url = {https://doi.org/10.1007/978-3-319-20337-9_2},
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author = {Birkar, Caucher},
title = {Anti-pluricanonical systems on {F}ano varieties},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {190},
year = {2019},
number = {2},
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mrclass = {14J45 (14C20 14E05 14E30)},
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doi = {10.4007/annals.2019.190.2.1},
url = {https://doi.org/10.4007/annals.2019.190.2.1},
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author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
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author = {Blum, Harold and Liu, Yuchen},
title = {The normalized volume of a singularity is lower semicontinuous},
year = {2018},
note = {to appear in {\em J. Euro. Math. Soc.}},
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} -
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author = {de Fernex, Tommaso and Koll\'{a}r, J\'{a}nos and Xu, Chenyang},
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[]
C. Li and Y. Liu, "Kähler-Einstein metrics and volume minimization," Adv. Math., vol. 341, pp. 440-492, 2019.
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author={Li, Chi},
title={Minimizing normalized volumes of valuations},
journal={Math. Z.},
volume={289},
number={1-2},
pages={491--513},
mrnumber={3803800},
doi = {10.1007/s00209-017-1963-3},
url = {https://doi.org/10.1007/s00209-017-1963-3},
zblnumber = {1423.14025},
year={2018},
} -
[LLX-Tiansurvey] C. Li, Y. Liu, and C. Xu, A guided tour to normalized volume, 2017.
@MISC{LLX-Tiansurvey,
author = {Li, Chi and Liu, Yuchen and Xu, Chenyang},
title = {A guided tour to normalized volume},
year = {2017},
arxiv = {1806.07112},
zblnumber = {},
} -
[LX-specialTC]
C. Li and C. Xu, "Special test configuration and K-stability of Fano varieties," Ann. of Math. (2), vol. 180, iss. 1, pp. 197-232, 2014.
@ARTICLE{LX-specialTC,
author = {Li, Chi and Xu, Chenyang},
title = {Special test configuration and {K}-stability of {F}ano varieties},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {180},
year = {2014},
number = {1},
pages = {197--232},
issn = {0003-486X},
mrclass = {14J45 (14E30 14J10 14J80)},
mrnumber = {3194814},
mrreviewer = {Anne-Sophie Kaloghiros},
doi = {10.4007/annals.2014.180.1.4},
url = {https://doi.org/10.4007/annals.2014.180.1.4},
zblnumber = {1301.14026},
} -
[LX-SDCI] C. Li and C. Xu, Stability of valuations and Kollár components, 2016.
@MISC{LX-SDCI,
author = {Li, Chi and Xu, Chenyang},
title = {Stability of valuations and {K}ollár components},
year = {2016},
note = {to appear in {\em J. Euro. Math. Soc.}},
arxiv = {1604.05398},
zblnumber = {},
} -
[LX-SDCII]
C. Li and C. Xu, "Stability of Valuations: Higher Rational Rank," Peking Math. J., vol. 1, iss. 1, pp. 1-79, 2018.
@ARTICLE{LX-SDCII,
author = {Li, Chi and Xu, Chenyang},
title = {Stability of {V}aluations: {H}igher {R}ational {R}ank},
journal = {Peking Math. J.},
fjournal = {Peking Mathematical Journal},
volume = {1},
year = {2018},
number = {1},
pages = {1--79},
issn = {2096-6075},
mrclass = {14 (32)},
mrnumber = {4059992},
doi = {10.1007/s42543-018-0001-7},
url = {https://doi.org/10.1007/s42543-018-0001-7},
zblnumber = {1423.14262},
} -
[LX-cubic]
Y. Liu and C. Xu, "K-stability of cubic threefolds," Duke Math. J., vol. 168, iss. 11, pp. 2029-2073, 2019.
@ARTICLE{LX-cubic,
author = {Liu, Yuchen and Xu, Chenyang},
title = {K-stability of cubic threefolds},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {168},
year = {2019},
number = {11},
pages = {2029--2073},
issn = {0012-7094},
mrclass = {14L24 (14E30 14J30 32Q20)},
mrnumber = {3992032},
doi = {10.1215/00127094-2019-0006},
url = {https://doi.org/10.1215/00127094-2019-0006},
zblnumber = {07114913},
} -
[Naka-zariski] N. Nakayama, Zariski-Decomposition and Abundance, Math. Soc. of Japan, Tokyo, 2004, vol. 14.
@BOOK{Naka-zariski,
author = {Nakayama, Noboru},
title = {Zariski-Decomposition and Abundance},
series = {MSJ Memoirs},
volume = {14},
publisher = {Math. Soc. of Japan, Tokyo},
year = {2004},
pages = {xiv+277},
isbn = {4-931469-31-0},
mrclass = {14C20 (14E15 14E30 14J10)},
mrnumber = {2104208},
mrreviewer = {Tommaso De Fernex},
zblnumber = {1061.14018},
} -
[Prok-plt]
Y. G. Prokhorov, "Blow-ups of Canonical Singularities," in Algebra, de Gruyter, Berlin, 2000, pp. 301-317.
@INCOLLECTION{Prok-plt,
author = {Prokhorov, Yu. G.},
title = {Blow-ups of Canonical Singularities},
booktitle = {Algebra},
venue = {{M}oscow, 1998},
pages = {301--317},
publisher = {de Gruyter, Berlin},
year = {2000},
mrclass = {14E30 (14B05 14E05 14J30)},
mrnumber = {1754677},
mrreviewer = {Massimiliano Mella},
zblnumber = {1003.14005},
doi = {10.1515/9783110805697},
url = {https://doi.org/10.1515/9783110805697},
} -
[Shokurov-threedim]
V. V. Shokurov, "Three-dimensional log perestroikas," Izv. Ross. Akad. Nauk Ser. Mat., vol. 56, iss. 1, pp. 105-203, 1992.
@ARTICLE{Shokurov-threedim,
author = {Shokurov, V. V.},
title = {Three-dimensional log perestroikas},
journal = {Izv. Ross. Akad. Nauk Ser. Mat.},
fjournal = {Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya},
note={with an appendix in English by Yujiro Kawamata; translation in \emph{Russian Acad. Sci. Izv. Math.} {\bf 40} (1993), no. 1, 95--2020},
volume = {56},
year = {1992},
number = {1},
pages = {105--203},
issn = {1607-0046},
mrclass = {14E05 (14E35)},
mrnumber = {1162635},
mrreviewer = {A. S. Tikhomirov},
doi = {10.1070/IM1993v040n01ABEH001862},
url = {https://doi.org/10.1070/IM1993v040n01ABEH001862},
zblnumber = {0785.14023},
} -
[Xu-kollar]
C. Xu, "Finiteness of algebraic fundamental groups," Compos. Math., vol. 150, iss. 3, pp. 409-414, 2014.
@ARTICLE{Xu-kollar,
author = {Xu, Chenyang},
title = {Finiteness of algebraic fundamental groups},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {150},
year = {2014},
number = {3},
pages = {409--414},
issn = {0010-437X},
mrclass = {14J17 (14J45)},
mrnumber = {3187625},
mrreviewer = {Andrzej Kozlowski},
doi = {10.1112/S0010437X13007562},
url = {https://doi.org/10.1112/S0010437X13007562},
zblnumber = {1291.14057},
}