ACC for Minimal Log Discrepancies of Exceptional Singularities

Abstract

In this paper, we study the ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds), proposed by the third author. We show the ACC conjecture holds for singularities admitting $\epsilon $-plt blow-ups. In particular, this gives the ACC for mlds for exceptional singularities. The key ingredients in the proofs of our main results are the Birkar–Borisov–Alexeev–Borisov theorem, proved by Birkar, the boundedness of complements conjecture for arbitrary DCC coefficients, proposed by the third author and proved in this paper, and the existence of uniform $\mathbb {R}$-complementary rational polytopes.

Keywords

Minimal log discrepancy / Minimal model program / Ascending chain condition / Complements / 14E30

References

[1] Alexeev V. Two two-dimensional terminations. Duke Math. J., 1993, 69(3): 527-545

[2] Ambro F. The set of toric minimal log discrepancies. Cent. Eur. J. Math., 2006, 4(3): 358-370

[3] Birkar C. Anti-pluricanonical systems on Fano varieties. Ann. Math. (2), 2019, 190(2): 345-463

[4] Birkar C. Singularities of linear systems and boundedness of Fano varieties. Ann. Math. (2), 2021, 193(2): 347-405

[5] Birkar C, Cascini P, Hacon CD, McKernan J. Existence of minimal models for varieties of log general type. J. Am. Math. Soc., 2010, 23(2): 405-468

[6] Blum H, Liu Y, Xu C. Openness of K-semistability for Fano varieties. Duke Math. J., 2022, 171(13): 2753-2797

[7] Borisov AA. Minimal discrepancies of toric singularities. Manuscr. Math., 1997, 92(1): 33-45

[8] Cheltsov I, Shramov C. On exceptional quotient singularities. Geom. Topol., 2011, 15(4): 1843-1882

[9] Cheltsov I, Shramov C. Weakly-exceptional singularities in higher dimensions. J. Reine Angew. Math., 2014, 689: 201-241

[10] Chen, G., Han, J.: Boundedness of $(\epsilon , n)$-complements for surfaces. Adv. Math. 383, 107703, 40 pp. (2021)

[11] Chen, W., Di Cerbo, G., Han, J., Jiang, C., Svaldi, R.: Birational boundedness of rationally connected Calabi–Yau $3$-folds. Adv. Math. 378, 107541, 32 pp. (2021)

[12] Hacon, C.D., Liu, J.: Existence of flips for generalized lc pairs. Camb. J. Math. 11(4), 795–828 (2023)

[13] Hacon CD , McKernan J, Xu C. ACC for log canonical thresholds. Ann. Math. (2), 2014, 180(2): 523-571

[14] Hacon CD, Witaszek J. On the rationality of Kawamata log terminal singularities in positive characteristic. Algebr. Geom., 2019, 6(5): 516-529

[15] Han, J., Liu, J., Luo, Y.: ACC for minimal log discrepancies of terminal threefolds. arXiv:2202.05287v2 (2022)

[16] Han J, Liu Y, Qi L. ACC for local volumes and boundedness of singularities. J. Algebr. Geom., 2023, 32(3): 519-583

[17] Han J, Luo Y. On boundedness of divisors computing minimal log discrepancies for surfaces. J. Inst. Math. Jussieu, 2023, 22(6): 2907-2930

[18] Hartshorne R. Algebraic Geometry, 1977, New York, Springer 52

[19] Jiang C. A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three. J. Algebr. Geom., 2021, 30(4): 759-800

[20] Kawakita M. Towards boundedness of minimal log discrepancies by the Riemann–Roch theorem. Am. J. Math., 2011, 133(5): 1299-1311

[21] Kawakita M. Discreteness of log discrepancies over log canonical triples on a fixed pair. J. Algebr. Geom., 2014, 23(4): 765-774

[22] Kawakita, M.: A connectedness theorem over the spectrum of a formal power series ring. Int. J. Math. 26(11), 1550088, 27 pp. (2015)

[23] Kawakita M. On equivalent conjectures for minimal log discrepancies on smooth threefolds. J. Algebr. Geom., 2021, 30(1): 97-149

[24] Kollár J. Singularities of the Minimal Model Program, 2013, Cambridge, Cambridge University Press 200

[25] Kollár, J., et al.: Flip and abundance for algebraic threefolds. Astérisque 211, 258 pp. (1992)

[26] Kollár J, Mori S. Birational Geometry of Algebraic Varieties, 1998, Cambridge, Cambridge University Press 134

[27] Kudryavtsev SA. Pure log terminal blow-ups. Math. Notes, 2001, 69(6): 814-819

[28] Li C. Minimizing normalized volumes of valuations. Math. Z., 2018, 289(1–2): 491-513

[29] Li C, Xu C. Stability of valuations and Kollár components. J. Eur. Math. Soc., 2020, 22(8): 2573-2627

[30] Li, Z.: A variant of the effective adjunction conjecture with applications. J. Pure Appl. Algebra 228(6), 107626, 22 pp. (2024)

[31] Liu, J.: Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies. arXiv:1809.04839v3 (2019)

[32] Liu, J., Xiao, L.: An optimal gap of minimal log discrepancies of threefold non-canonical singularities. J. Pure Appl. Algebra 225(9), 106674, 23 pp. (2021)

[33] Liu Y, Xu C, Zhuang Z. Finite generation for valuations computing stability thresholds and applications to K-stability. Ann. Math. (2), 2022, 196(2): 507-566

[34] Markushevich D, Prokhorov YG. Exceptional quotient singularities. Am. J. Math., 1999, 121(6): 1179-1189

[35] Mustaţǎ, M., Nakamura, Y.: A boundedness conjecture for minimal log discrepancies on a fixed germ. In: Local and Global Methods in Algebraic Geometry. Contemp. Math., vol. 712, pp. 287–306. American Mathematical Society, Providence, RI (2018)

[36] Nakamura Y. On minimal log discrepancies on varieties with fixed Gorenstein index. Mich. Math. J., 2016, 65(1): 165-187

[37] Nakamura Y, Shibata K. Inversion of adjunction for quotient singularities. Algebr. Geom., 2022, 9(2): 214-251

[38] Prokhorov, Y.G.: Blow-ups of canonical singularities. In: Algebra (Moscow, 1998), pp. 301–317. Walter de Gruyter & Co., Berlin (2000)

[39] Prokhorov YG, Shokurov VV. The first main theorem on complements: from global to local. Izv. Ross. Akad. Nauk Ser. Mat., 2001, 65(6): 1169-1196

[40] Prokhorov YG, Shokurov VV. Towards the second main theorem on complements. J. Algebr. Geom., 2009, 18(1): 151-199

[41] Shokurov, V.V.: Problems about Fano varieties. In: Birational Geometry of Algebraic Varieties, Open Problems. The 23rd Taniguchi International Symposium (Katata, Aug. 22–27, 1988), pp. 30–32, Division of Mathematics, The Taniguchi Foundation, Katata (1988)

[42] Shokurov VV. Three-fold log flips. Izv. Ross. Akad. Nauk Ser. Mat., 1992, 56(1): 105-203

[43] Shokurov VV. 3-fold log models. J. Math. Sci., 1996, 81(3): 2677-2699

[44] Shokurov VV. Complements on surfaces. J. Math. Sci. (N. Y.), 2000, 102(2): 3876-3932

[45] Shokurov VV. Letters of a bi-rationalist, V. Minimal log discrepancies and termination of log flips (Russian). Tr. Mat. Inst. Steklova, 2004, 246: 328-351

[46] Shokurov, V.V.: Existence and boundedness of $n$-complements. arXiv:2012.06495v1 (2020)

[47] The Stacks Project Authors. The Stacks Project. https://stacks.math.columbia.edu

[48] Xu C. Finiteness of algebraic fundamental groups. Compos. Math., 2014, 150(3): 409-414

[49] Xu C . A minimizing valuation is quasi-monomial. Ann. Math. (2), 2020, 191(3): 1003-1030

[50] Xu, C., Zhuang, Z.: Stable degenerations of singularities. arXiv:2205.10915v1 (2022)

[51] Zhuang Z. On boundedness of singularities and minimal log discrepancies of Kollár components. J. Algebr. Geom., 2024, 33(3): 521-565

Funding

National Science Fund for the Excellent Young Scientists(12322102)

RIGHTS & PERMISSIONS

Peking University