Optimal destabilization of K–unstable Fano varieties via stability thresholds

Blum, Harold¹ ; Liu, Yuchen² ; Zhou, Chuyu³

¹ Department of Mathematics, Stony Brook University, Stony Brook, NY, United States
² Department of Mathematics, Northwestern University, Evanston, IL, United States
³ Institute of Mathematics, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland

Geometry & topology, Tome 26 (2022) no. 6, pp. 2507-2564.

Résumé

We show that for a K–unstable Fano variety, any divisorial valuation computing its stability threshold induces a nontrivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K–polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Székelyhidi and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K–stability in the sense of Dervan.

Détail

Classification : 14J10, 14J45, 32Q20
Keywords: K-stability, Fano varieties, optimal destabilization

Bibliographie

[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 | DOI

[2] F Ambro, Variation of log canonical thresholds in linear systems, Int. Math. Res. Not. 2016 (2016) 4418 | DOI

[3] K Ascher, K Devleming, Y Liu, Wall crossing for K–moduli spaces of plane curves, preprint (2019)

[4] R J Berman, K–polystability of Q–Fano varieties admitting Kähler–Einstein metrics, Invent. Math. 203 (2016) 973 | DOI

[5] R J Berman, S Boucksom, M Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021) 605 | DOI

[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 | DOI

[7] H Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018) 820 | DOI

[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) | DOI

[9] H Blum, M Jonsson, Thresholds, valuations, and K–stability, Adv. Math. 365 (2020) | DOI

[10] H Blum, Y Liu, Openness of uniform K–stability in families of Q–Fano varieties, Ann. Sci. Éc. Norm. Supér. 55 (2022) 1 | DOI

[11] H Blum, Y Liu, C Xu, Openness of K–semistability for Fano varieties, Duke Math. J. 171 (2022) 2753 | DOI

[12] H Blum, Y Liu, C Xu, Z Zhuang, The existence of the Kähler–Ricci soliton degeneration, preprint (2021)

[13] H Blum, C Xu, Uniqueness of K–polystable degenerations of Fano varieties, Ann. of Math. 190 (2019) 609 | DOI

[14] S Boucksom, T Hisamoto, M Jonsson, Uniform K–stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 | DOI

[15] S Boucksom, M Jonsson, A non-Archimedean approach to K–stability, preprint (2018)

[16] F Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. 25 (1992) 539 | DOI

[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) | DOI

[18] W Chen, Boundedness of weak Fano pairs with alpha-invariants and volumes bounded below, Publ. Res. Inst. Math. Sci. 56 (2020) 539 | DOI

[19] X Chen, S Sun, B Wang, Kähler–Ricci flow, Kähler–Einstein metric, and K–stability, Geom. Topol. 22 (2018) 3145 | DOI

[20] V Datar, G Székelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016) 975 | DOI

[21] R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728 | DOI

[22] R Dervan, K–semistability of optimal degenerations, Q. J. Math. 71 (2020) 989 | DOI

[23] R Dervan, G Székelyhidi, The Kähler–Ricci flow and optimal degenerations, J. Differential Geom. 116 (2020) 187 | DOI

[24] S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289

[25] S K Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005) 453

[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 Q–Fano varieties, J. Reine Angew. Math. 751 (2019) 309 | DOI

[28] K Fujita, K–stability of log Fano hyperplane arrangements, J. Algebraic Geom. 30 (2021) 603 | DOI

[29] K Fujita, Y Odaka, On the K–stability of Fano varieties and anticanonical divisors, Tohoku Math. J. 70 (2018) 511 | DOI

[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 | DOI

[31] J Han, C Li, Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties, preprint (2020)

[32] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[33] T Hisamoto, Geometric flow, multiplier ideal sheaves and optimal destabilizer for a Fano manifold, preprint (2019)

[34] C Jiang, Boundedness of Q–Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. École Norm. Sup. 53 (2020) 1235 | DOI

[35] G R Kempf, Instability in invariant theory, Ann. of Math. 108 (1978) 299 | DOI

[36] G Kempf, F F Knudsen, D Mumford, B Saint-Donat, Toroidal embeddings, I, 339, Springer (1973) | DOI

[37] J Kollár, Rational curves on algebraic varieties, 32, Springer (1996) | DOI

[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 | DOI

[39] J Kollár, Singularities of the minimal model program, 200, Cambridge Univ. Press (2013) | DOI

[40] J Kollár, Y Miyaoka, S Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992) 765

[41] J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) | DOI

[42] C Li, Remarks on logarithmic K–stability, Commun. Contemp. Math. 17 (2015) | DOI

[43] C Li, K–semistability is equivariant volume minimization, Duke Math. J. 166 (2017) 3147 | DOI

[44] C Li, Minimizing normalized volumes of valuations, Math. Z. 289 (2018) 491 | DOI

[45] C Li, Y Liu, Kähler–Einstein metrics and volume minimization, Adv. Math. 341 (2019) 440 | DOI

[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 | DOI

[47] C Li, X Wang, C Xu, On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties, Duke Math. J. 168 (2019) 1387 | DOI

[48] C Li, X Wang, C Xu, Algebraicity of the metric tangent cones and equivariant K–stability, J. Amer. Math. Soc. 34 (2021) 1175 | DOI

[49] C Li, C Xu, Special test configuration and K–stability of Fano varieties, Ann. of Math. 180 (2014) 197 | DOI

[50] C Li, C Xu, Stability of valuations: higher rational rank, Peking Math. J. 1 (2018) 1 | DOI

[51] C Li, C Xu, Stability of valuations and Kollár components, J. Eur. Math. Soc. 22 (2020) 2573 | DOI

[52] Y Liu, The volume of singular Kähler–Einstein Fano varieties, Compos. Math. 154 (2018) 1131 | DOI

[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 | DOI

[54] D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994) | DOI

[55] Y Odaka, S Sun, Testing log K–stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. 24 (2015) 505 | DOI

[56] J Ross, G Székelyhidi, Twisted Kähler–Einstein metrics, Pure Appl. Math. Q. 17 (2021) 1025 | DOI

[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 | DOI

[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 | DOI

[59] G Székelyhidi, Optimal test-configurations for toric varieties, J. Differential Geom. 80 (2008) 501

[60] G Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011) 319 | DOI

[61] G Tian, On stability of the tangent bundles of Fano varieties, Int. J. Math. 3 (1992) 401 | DOI

[62] G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 | DOI

[63] M Xia, On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows, Anal. PDE 14 (2021) 1951 | DOI

[64] C Xu, A minimizing valuation is quasi-monomial, Ann. of Math. 191 (2020) 1003 | DOI

[65] C Xu, Z Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021) 149 | DOI

[66] C Zhou, Z Zhuang, Some criteria for uniform K–stability, Math. Res. Lett. 28 (2021) 1613 | DOI

[67] Z Zhuang, Fano varieties with large Seshadri constants, Adv. Math. 340 (2018) 883 | DOI

[68] Z Zhuang, Optimal destabilizing centers and equivariant K–stability, Invent. Math. 226 (2021) 195 | DOI