Download this article
 Download this article For screen
For printing
Recent Issues

Volume 29
Issue 8, 3921–4476
Issue 7, 3345–3919
Issue 6, 2783–3343
Issue 5, 2251–2782
Issue 4, 1693–2250
Issue 3, 1115–1691
Issue 2, 549–1114
Issue 1, 1–548

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties

Jiyuan Han and Chi Li

Geometry & Topology 28 (2024) 539–592
Bibliography
1 H Abban, Z Zhuang, K–stability of Fano varieties via admissible flags, Forum Math. Pi 10 (2022) MR4448177
2 K Altmann, J Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006) 557 MR2207875
3 K Altmann, J Hausen, H Süss, Gluing affine torus actions via divisorial fans, Transform. Groups 13 (2008) 215 MR2426131
4 R Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. of Math. 188 (2018) 753 MR3866886
5 R J Berman, K–polystability of –Fano varieties admitting Kähler–Einstein metrics, Invent. Math. 203 (2016) 973 MR3461370
6 R J Berman, S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. Reine Angew. Math. 751 (2019) 27 MR3956691
7 R J Berman, S Boucksom, M Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021) 605 MR4334189
8 R J Berman, D W Nystrom, Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons, preprint (2014) arXiv:1401.8264
9 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
10 H Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018) 820 MR3778195
11 H Blum, M Jonsson, Thresholds, valuations, and K–stability, Adv. Math. 365 (2020) 107062 MR4067358
12 H Blum, Y Liu, The normalized volume of a singularity is lower semicontinuous, J. Eur. Math. Soc. 23 (2021) 1225 MR4228279
13 H Blum, Y Liu, Openness of uniform K–stability in families of –Fano varieties, Ann. Sci. Éc. Norm. Supér. 55 (2022) 1 MR4411858
14 H Blum, Y Liu, C Xu, Openness of K–semistability for Fano varieties, Duke Math. J. 171 (2022) 2753 MR4505846
15 H Blum, Y Liu, C Xu, Z Zhuang, The existence of the Kähler–Ricci soliton degeneration, Forum Math. Pi 11 (2023) MR4558985
16 H Blum, Y Liu, C Zhou, Optimal destabilization of K–unstable Fano varieties via stability thresholds, Geom. Topol. 26 (2022) 2507 MR4521248
17 S Boucksom, H Chen, Okounkov bodies of filtered linear series, Compos. Math. 147 (2011) 1205 MR2822867
18 S Boucksom, D Eriksson, Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, Adv. Math. 378 (2021) 107501 MR4192993
19 S Boucksom, T Hisamoto, M Jonsson, Uniform K–stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 MR3669511
20 S Boucksom, M Jonsson, A non-Archimedean approach to K–stability, preprint (2018) arXiv:1805.11160v1
21 S Boucksom, M Jonsson, A non-Archimedean approach to K–stability, I : Metric geometry of spaces of test configurations and valuations, preprint (2021) arXiv:2107.11221
22 S Boucksom, A Küronya, C Maclean, T Szemberg, Vanishing sequences and Okounkov bodies, Math. Ann. 361 (2015) 811 MR3319549
23 M Brion, Sur l’image de l’application moment, from: "Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin" (editor M P Malliavin), Lecture Notes in Math. 1296, Springer (1987) 177 MR0932055
24 H Chen, C Maclean, Distribution of logarithmic spectra of the equilibrium energy, Manuscripta Math. 146 (2015) 365 MR3312451
25 X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, I : Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015) 183 MR3264766
26 X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, II : Limits with cone angle less than 2π, J. Amer. Math. Soc. 28 (2015) 199 MR3264767
27 X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, III : Limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc. 28 (2015) 235 MR3264768
28 X Chen, S Sun, B Wang, Kähler–Ricci flow, Kähler–Einstein metric, and K–stability, Geom. Topol. 22 (2018) 3145 MR3858762
29 X Chen, B Wang, Space of Ricci flows, II : Part B : weak compactness of the flows, J. Differential Geom. 116 (2020) 1 MR4146357
30 T Darvas, W He, Geodesic rays and Kähler–Ricci trajectories on Fano manifolds, Trans. Amer. Math. Soc. 369 (2017) 5069 MR3632560
31 R Dervan, G Székelyhidi, The Kähler–Ricci flow and optimal degenerations, J. Differential Geom. 116 (2020) 187 MR4146359
32 S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
33 S K Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005) 453 MR2192937
34 S Donaldson, S Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differential Geom. 107 (2017) 327 MR3707646
35 L Ein, R Lazarsfeld, K E Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003) 409 MR1963690
36 K Fujita, Optimal bounds for the volumes of Kähler–Einstein Fano manifolds, Amer. J. Math. 140 (2018) 391 MR3783213
37 K Fujita, A valuative criterion for uniform K–stability of –Fano varieties, J. Reine Angew. Math. 751 (2019) 309 MR3956698
38 K Fujita, Y Odaka, On the K–stability of Fano varieties and anticanonical divisors, Tohoku Math. J. 70 (2018) 511 MR3896135
39 J Han, C Li, On the Yau–Tian–Donaldson conjecture for generalized Kähler–Ricci soliton equations, Comm. Pure Appl. Math. 76 (2023) 1793 MR4612573
40 W He, Kähler–Ricci soliton and H–functional, Asian J. Math. 20 (2016) 645 MR3570456
41 T Hisamoto, Stability and coercivity for toric polarizations, preprint (2016) arXiv:1610.07998
42 T Hisamoto, Mabuchi’s soliton metric and relative D–stability, preprint (2019) arXiv:1905.05948
43 T Hisamoto, Geometric flow, multiplier ideal sheaves and optimal destabilizer for a Fano manifold, J. Geom. Anal. 33 (2023) 265 MR4602002
44 M Jonsson, M Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012) 2145 MR3060755
45 K Kaveh, A G Khovanskii, Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. 176 (2012) 925 MR2950767
46 K Kaveh, A Khovanskii, Convex bodies and multiplicities of ideals, Proc. Steklov Inst. Math. 286 (2014) 268 MR3482603
47 S J Kovács, Z Patakfalvi, Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension, J. Amer. Math. Soc. 30 (2017) 959 MR3671934
48 R Lazarsfeld, Positivity in algebraic geometry, II : Positivity for vector bundles, and multiplier ideals, 49, Springer (2004) MR2095472
49 R Lazarsfeld, M Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. 42 (2009) 783 MR2571958
50 C Li, K-semistability is equivariant volume minimization, Duke Math. J. 166 (2017) 3147 MR3715806
51 C Li, Minimizing normalized volumes of valuations, Math. Z. 289 (2018) 491 MR3803800
52 C Li, G–uniform stability and Kähler–Einstein metrics on Fano varieties, Invent. Math. 227 (2022) 661 MR4372222
53 C Li, Y Liu, Kähler–Einstein metrics and volume minimization, Adv. Math. 341 (2019) 440 MR3872852
54 C Li, Y Liu, C Xu, A guided tour to normalized volume, from: "Geometric analysis : in honor of Gang Tian’s 60th birthday" (editors J Chen, P Lu, Z Lu, Z Zhang), Progr. Math. 333, Springer (2020) 167 MR4181002
55 C Li, X Wang, C Xu, Algebraicity of the metric tangent cones and equivariant K–stability, J. Amer. Math. Soc. 34 (2021) 1175 MR4301561
56 C Li, C Xu, Special test configuration and K–stability of Fano varieties, Ann. of Math. 180 (2014) 197 MR3194814
57 C Li, C Xu, Stability of valuations: higher rational rank, Peking Math. J. 1 (2018) 1 MR4059992
58 C Li, C Xu, Stability of valuations and Kollár components, J. Eur. Math. Soc. 22 (2020) 2573 MR4118616
59 Y Li, Z Li, Equivariant –test configurations and semistable limits of –Fano group compactifications, preprint (2021) arXiv:2103.06439
60 Y Liu, The volume of singular Kähler–Einstein Fano varieties, Compos. Math. 154 (2018) 1131 MR3797604
61 D Mumford, Stability of projective varieties, Enseign. Math. 23 (1977) 39 MR0450272
62 A Okounkov, Brunn–Minkowski inequality for multiplicities, Invent. Math. 125 (1996) 405 MR1400312
63 W Rossmann, Equivariant multiplicities on complex varieties, from: "Orbites unipotentes et représentations, III" (editor M Andler), Astérisque 173–174, Soc. Math. France (1989) 313 MR1021516
64 G Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015) 451 MR3343885
65 G Székelyhidi, The partial C0–estimate along the continuity method, J. Amer. Math. Soc. 29 (2016) 537 MR3454382
66 B Teissier, Valuations, deformations, and toric geometry, from: "Valuation theory and its applications, II" (editors F V Kuhlmann, S Kuhlmann, M Marshall), Fields Inst. Commun. 33, Amer. Math. Soc. (2003) 361 MR2018565
67 G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
68 G Tian, K–stability and Kähler–Einstein metrics, Comm. Pure Appl. Math. 68 (2015) 1085 MR3352459
69 G Tian, S Zhang, Z Zhang, X Zhu, Perelman’s entropy and Kähler–Ricci flow on a Fano manifold, Trans. Amer. Math. Soc. 365 (2013) 6669 MR3105766
70 G Tian, Z Zhang, Regularity of Kähler–Ricci flows on Fano manifolds, Acta Math. 216 (2016) 127 MR3508220
71 G Tian, X Zhu, A new holomorphic invariant and uniqueness of Kähler–Ricci solitons, Comment. Math. Helv. 77 (2002) 297 MR1915043
72 G Tian, X Zhu, Convergence of Kähler–Ricci flow, J. Amer. Math. Soc. 20 (2007) 675 MR2291916
73 F Wang, X Zhu, Tian’s partial C0–estimate implies Hamilton–Tian’s conjecture, Adv. Math. 381 (2021) 107619 MR4206790
74 F Wang, X Zhu, Uniformly strong convergence of Kähler–Ricci flows on a Fano manifold, Sci. China Math. 65 (2022) 2337 MR4499520
75 X J Wang, X Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004) 87 MR2084775
76 D Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012) 1736 MR2999302
77 M Xia, On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows, Anal. PDE 14 (2021) 1951 MR4308671
78 C Xu, A minimizing valuation is quasi-monomial, Ann. of Math. 191 (2020) 1003 MR4088355
79 C Xu, Z Zhuang, On positivity of the CM line bundle on K–moduli spaces, Ann. of Math. 192 (2020) 1005 MR4172625
80 C Xu, Z Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021) 149 MR4325260
81 Y Yao, Relative Ding stability and an obstruction to the existence of Mabuchi solitons, J. Geom. Anal. 32 (2022) 105 MR4372898