A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three
Author:
Chen Jiang
Journal:
J. Algebraic Geom. 30 (2021), 759-800
DOI:
https://doi.org/10.1090/jag/759
Published electronically:
June 9, 2021
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We show that there exists a positive real number $\delta >0$ such that for any normal quasi-projective $\mathbb {Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all noncanonical klt Calabi–Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau $3$-folds are bounded from above.
References
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References
- Valery Alexeev, Two two-dimensional terminations, Duke Math. J. 69 (1993), no. 3, 527–545. MR 1208810, DOI 10.1215/S0012-7094-93-06922-0
- Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI 10.1142/S0129167X94000395
- Valery Alexeev and Shigefumi Mori, Bounding singular surfaces of general type, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) Springer, Berlin, 2004, pp. 143–174. MR 2037085
- Florin Ambro, The set of toric minimal log discrepancies, Cent. Eur. J. Math. 4 (2006), no. 3, 358–370. MR 2233855, DOI 10.2478/s11533-006-0013-x
- Caucher Birkar, Singularities on the base of a Fano type fibration, J. Reine Angew. Math. 715 (2016), 125–142. MR 3507921, DOI 10.1515/crelle-2014-0033
- Caucher Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math. (2) 190 (2019), no. 2, 345–463. MR 3997127, DOI 10.4007/annals.2019.190.2.1
- C. Birkar, Log Calabi–Yau fibrations, arXiv:1811.10709v2 (2018).
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- C. Birkar and V. V. Shokurov, Mld’s vs thresholds and flips, J. Reine Angew. Math. 638 (2010), 209–234. MR 2595341, DOI 10.1515/CRELLE.2010.008
- Raimund Blache, The structure of l.c. surfaces of Kodaira dimension zero. I, J. Algebraic Geom. 4 (1995), no. 1, 137–179. MR 1299007
- Alexandr Borisov, Minimal discrepancies of toric singularities, Manuscripta Math. 92 (1997), no. 1, 33–45. MR 1427666, DOI 10.1007/BF02678179
- Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), no. 2, 201–248. MR 3019449, DOI 10.1090/S1056-3911-2012-00574-8
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- Akira Fujiki, On resolutions of cyclic quotient singularities, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 293–328. MR 0385162, DOI 10.2977/prims/1195192183
- Osamu Fujino and Yoshinori Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), no. 2, 255–264. MR 2944479, DOI 10.1307/mmj/1339011526
- Yoshinori Gongyo, Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs, J. Algebraic Geom. 22 (2013), no. 3, 549–564. MR 3048544, DOI 10.1090/S1056-3911-2012-00593-1
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- Christopher D. Hacon, James McKernan, and Chenyang Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), no. 2, 523–571. MR 3224718, DOI 10.4007/annals.2014.180.2.3
- Christopher D. Hacon, James McKernan, and Chenyang Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 865–901. MR 3779687, DOI 10.4171/JEMS/778
- Christopher D. Hacon and Chenyang Xu, Existence of log canonical closures, Invent. Math. 192 (2013), no. 1, 161–195. MR 3032329, DOI 10.1007/s00222-012-0409-0
- J. Han, J. Liu, V.V. Shokurov, ACC for minimal log discrepancies of exceptional singularities, arXiv:1903.04338v1 (2019).
- Shihoko Ishii, Introduction to singularities, Springer, Tokyo, 2018. Second edition of [MR3288750]. MR 3838338, DOI 10.1007/978-4-431-56837-7
- Masayuki Kawakita, Discreteness of log discrepancies over log canonical triples on a fixed pair, J. Algebraic Geom. 23 (2014), no. 4, 765–774. MR 3263668, DOI 10.1090/S1056-3911-2014-00630-5
- M. Kawakita, On equivalent conjectures for minimal log discrepancies on smooth threefolds, J. Algebraic Geom. 30 (2021), no. 1, 97–149. MR 4233179
- Yujiro Kawamata, On the plurigenera of minimal algebraic $3$-folds with $K\equiv 0$, Math. Ann. 275 (1986), no. 4, 539–546. MR 859328, DOI 10.1007/BF01459135
- Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. MR 2426353, DOI 10.2977/prims/1210167332
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- J. Kollár et al., Flips and abundance for algebraic threefolds, A Summer Seminar on Algebraic Geometry (Salt Lake City, Utah, August 1991), Astérisque 211 (1992), 183–192.
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI 10.1007/BF01389370
- J. Liu, Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies, arXiv:1809.04839v2 (2018).
- J. Liu, L. Xiao, An optimal gap of minimal log discrepancies of threefold non-canonical singularities, J. Pure Appl. Algebra 225 (2021), no 9, 106674, 23 pp. MR 4200812
- J. Moraga, On minimal log discrepancies and Kollár components, arXiv:1810.10137v1 (2018) (to appear in Manuscripta Math.).
- Shigefumi Mori, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. MR 792770, DOI 10.1017/S0027763000021358
- David R. Morrison, A remark on: “On the plurigenera of minimal algebraic $3$-folds with $K\equiv 0$” [Math. Ann. 275 (1986), no. 4, 539–546; MR0859328 (88c:14049)] by Y. Kawamata, Math. Ann. 275 (1986), no. 4, 547–553. MR 859329, DOI 10.1007/BF01459136
- David R. Morrison and Glenn Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), no. 1, 15–20. MR 722406, DOI 10.2307/2044659
- Mircea Mustaţă and Yusuke Nakamura, A boundedness conjecture for minimal log discrepancies on a fixed germ, Local and global methods in algebraic geometry, Contemp. Math., vol. 712, Amer. Math. Soc., Providence, RI, 2018, pp. 287–306. MR 3832408, DOI 10.1090/conm/712/14351
- Yusuke Nakamura, On minimal log discrepancies on varieties with fixed Gorenstein index, Michigan Math. J. 65 (2016), no. 1, 165–187. MR 3466821
- Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980, pp. 273–310. MR 605348
- Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
- V.V. Shokurov, Problems about Fano varieties, Birational Geometry of Algebraic Varieties, Open Problems, The XXIIIrd International Symposium, Division of Mathematics, The Taniguchi Foundation, pages 30–32, August 22–27, 1988.
- V.V. Shokurov, A.c.c. in codimension $2$, 1991 (preprint).
- V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, DOI 10.1070/IM1993v040n01ABEH001862
- V. V. Shokurov, $3$-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699. Algebraic geometry, 4. MR 1420223, DOI 10.1007/BF02362335
- V. V. Shokurov, Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 328–351 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 315–336. MR 2101303
- De-Qi Zhang, Logarithmic Enriques surfaces, J. Math. Kyoto Univ. 31 (1991), no. 2, 419–466. MR 1121173, DOI 10.1215/kjm/1250519795
- De-Qi Zhang, Logarithmic Enriques surfaces. II, J. Math. Kyoto Univ. 33 (1993), no. 2, 357–397. MR 1231749, DOI 10.1215/kjm/1250519265
Additional Information
Chen Jiang
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, 2005 Songhu Road, Shanghai, 200438, People’s Republic of China
Email:
chenjiang@fudan.edu.cn
Received by editor(s):
August 26, 2019
Received by editor(s) in revised form:
November 21, 2019
Published electronically:
June 9, 2021
Additional Notes:
Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140.
Article copyright:
© Copyright 2021
University Press, Inc.