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Optimal destabilization of K–unstable Fano varieties via stability thresholds

Harold Blum, Yuchen Liu and Chuyu Zhou

Geometry & Topology 26 (2022) 2507–2564
DOI: 10.2140/gt.2022.26.2507
Bibliography
1 J Alper, H Blum, D Halpern-Leistner, C Xu, Reductivity of the automorphism group of K–polystable Fano varieties, Invent. Math. 222 (2020) 995 MR4169054
2 F Ambro, Variation of log canonical thresholds in linear systems, Int. Math. Res. Not. 2016 (2016) 4418 MR3556423
3 K Ascher, K DeVleming, Y Liu, Wall crossing for K–moduli spaces of plane curves, preprint (2019) arXiv:1909.04576
4 R J Berman, K–polystability of –Fano varieties admitting Kähler–Einstein metrics, Invent. Math. 203 (2016) 973 MR3461370
5 R J Berman, S Boucksom, M Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021) 605 MR4334189
6 C Birkar, P Cascini, C D Hacon, J McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010) 405 MR2601039
7 H Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018) 820 MR3778195
8 H Blum, D Halpern-Leistner, Y Liu, C Xu, On properness of K–moduli spaces and optimal degenerations of Fano varieties, Selecta Math. 27 (2021) MR4292783
9 H Blum, M Jonsson, Thresholds, valuations, and K–stability, Adv. Math. 365 (2020) MR4067358
10 H Blum, Y Liu, Openness of uniform K–stability in families of –Fano varieties, Ann. Sci. Éc. Norm. Supér. 55 (2022) 1 MR4411858
11 H Blum, Y Liu, C Xu, Openness of K–semistability for Fano varieties, Duke Math. J. 171 (2022) 2753 MR4505846
12 H Blum, Y Liu, C Xu, Z Zhuang, The existence of the Kähler–Ricci soliton degeneration, preprint (2021) arXiv:2103.15278
13 H Blum, C Xu, Uniqueness of K–polystable degenerations of Fano varieties, Ann. of Math. 190 (2019) 609 MR3997130
14 S Boucksom, T Hisamoto, M Jonsson, Uniform K–stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 MR3669511
15 S Boucksom, M Jonsson, A non-Archimedean approach to K–stability, preprint (2018) arXiv:1805.11160
16 F Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. 25 (1992) 539 MR1191735
17 I A Cheltsov, Y A Rubinstein, K Zhang, Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces, Selecta Math. 25 (2019) MR3945265
18 W Chen, Boundedness of weak Fano pairs with alpha-invariants and volumes bounded below, Publ. Res. Inst. Math. Sci. 56 (2020) 539 MR4116691
19 X Chen, S Sun, B Wang, Kähler–Ricci flow, Kähler–Einstein metric, and K–stability, Geom. Topol. 22 (2018) 3145 MR3858762
20 V Datar, G Székelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016) 975 MR3558304
21 R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728 MR3564626
22 R Dervan, K–semistability of optimal degenerations, Q. J. Math. 71 (2020) 989 MR4142719
23 R Dervan, G Székelyhidi, The Kähler–Ricci flow and optimal degenerations, J. Differential Geom. 116 (2020) 187 MR4146359
24 S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
25 S K Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005) 453 MR2192937
26 K Fujita, Toward criteria for K–stability of log Fano pairs, lecture notes (2019)
27 K Fujita, A valuative criterion for uniform K–stability of –Fano varieties, J. Reine Angew. Math. 751 (2019) 309 MR3956698
28 K Fujita, K–stability of log Fano hyperplane arrangements, J. Algebraic Geom. 30 (2021) 603 MR4372401
29 K Fujita, Y Odaka, On the K–stability of Fano varieties and anticanonical divisors, Tohoku Math. J. 70 (2018) 511 MR3896135
30 D Halpern-Leistner, Θ–stratifications, Θ–reductive stacks, and applications, from: "Algebraic geometry" (editors T de Fernex, B Hassett, M Mustaţă, M Olsson, M Popa, R Thomas), Proc. Sympos. Pure Math. 97, Amer. Math. Soc. (2018) 349 MR3821155
31 J Han, C Li, Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties, preprint (2020) arXiv:2009.01010
32 R Hartshorne, Algebraic geometry, 52, Springer (1977) MR0463157
33 T Hisamoto, Geometric flow, multiplier ideal sheaves and optimal destabilizer for a Fano manifold, preprint (2019) arXiv:1901.08480
34 C Jiang, Boundedness of –Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. École Norm. Sup. 53 (2020) 1235 MR4174851
35 G R Kempf, Instability in invariant theory, Ann. of Math. 108 (1978) 299 MR506989
36 G Kempf, F F Knudsen, D Mumford, B Saint-Donat, Toroidal embeddings, I, 339, Springer (1973) MR0335518
37 J Kollár, Rational curves on algebraic varieties, 32, Springer (1996) MR1440180
38 J Kollár, Singularities of pairs, from: "Algebraic geometry" (editors J Kollár, R Lazarsfeld, D R Morrison), Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 221 MR1492525
39 J Kollár, Singularities of the minimal model program, 200, Cambridge Univ. Press (2013) MR3057950
40 J Kollár, Y Miyaoka, S Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992) 765 MR1189503
41 J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) MR1658959
42 C Li, Remarks on logarithmic K–stability, Commun. Contemp. Math. 17 (2015) MR3313212
43 C Li, K–semistability is equivariant volume minimization, Duke Math. J. 166 (2017) 3147 MR3715806
44 C Li, Minimizing normalized volumes of valuations, Math. Z. 289 (2018) 491 MR3803800
45 C Li, Y Liu, Kähler–Einstein metrics and volume minimization, Adv. Math. 341 (2019) 440 MR3872852
46 C Li, Y Liu, C Xu, A guided tour to normalized volume, from: "Geometric analysis" (editors J Chen, P Lu, Z Lu, Z Zhang), Progr. Math. 333, Birkhäuser (2020) 167 MR4181002
47 C Li, X Wang, C Xu, On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties, Duke Math. J. 168 (2019) 1387 MR3959862
48 C Li, X Wang, C Xu, Algebraicity of the metric tangent cones and equivariant K–stability, J. Amer. Math. Soc. 34 (2021) 1175 MR4301561
49 C Li, C Xu, Special test configuration and K–stability of Fano varieties, Ann. of Math. 180 (2014) 197 MR3194814
50 C Li, C Xu, Stability of valuations: higher rational rank, Peking Math. J. 1 (2018) 1 MR4059992
51 C Li, C Xu, Stability of valuations and Kollár components, J. Eur. Math. Soc. 22 (2020) 2573 MR4118616
52 Y Liu, The volume of singular Kähler–Einstein Fano varieties, Compos. Math. 154 (2018) 1131 MR3797604
53 Y Liu, C Xu, Z Zhuang, Finite generation for valuations computing stability thresholds and applications to K–stability, Ann. of Math. 196 (2022) 507 MR4445441
54 D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994) MR1304906
55 Y Odaka, S Sun, Testing log K–stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. 24 (2015) 505 MR3403730
56 J Ross, G Székelyhidi, Twisted Kähler–Einstein metrics, Pure Appl. Math. Q. 17 (2021) 1025 MR4278957
57 Y A Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218 (2008) 1526 MR2419932
58 Y A Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009) 5839 MR2529916
59 G Székelyhidi, Optimal test-configurations for toric varieties, J. Differential Geom. 80 (2008) 501 MR2472481
60 G Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011) 319 MR2771134
61 G Tian, On stability of the tangent bundles of Fano varieties, Int. J. Math. 3 (1992) 401 MR1163733
62 G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
63 M Xia, On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows, Anal. PDE 14 (2021) 1951 MR4308671
64 C Xu, A minimizing valuation is quasi-monomial, Ann. of Math. 191 (2020) 1003 MR4088355
65 C Xu, Z Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021) 149 MR4325260
66 C Zhou, Z Zhuang, Some criteria for uniform K–stability, Math. Res. Lett. 28 (2021) 1613 MR4471722
67 Z Zhuang, Fano varieties with large Seshadri constants, Adv. Math. 340 (2018) 883 MR3886183
68 Z Zhuang, Optimal destabilizing centers and equivariant K–stability, Invent. Math. 226 (2021) 195 MR4309493