Volume 23, issue 3 (2019)

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

Volume 23
Issue 5, 2165–2700
Issue 4, 1621–2164
Issue 3, 1085–1619
Issue 2, 541–1084
Issue 1, 1–540

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
Editorial Board
Subscriptions
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Sasaki–Einstein metrics and K–stability

Tristan C Collins and Gábor Székelyhidi

Geometry & Topology 23 (2019) 1339–1413
Bibliography
1 K Altmann, J Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006) 557 MR2207875
2 K Altmann, N O Ilten, L Petersen, H Süß, R Vollmert, The geometry of 𝕋–varieties, from: "Contributions to algebraic geometry" (editor P Pragacz), Eur. Math. Soc. (2012) 17 MR2975658
3 M T Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990) 429 MR1074481
4 T Aubin, Réduction du cas positif de l’équation de Monge–Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité, J. Funct. Anal. 57 (1984) 143 MR749521
5 S Bando, T Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, from: "Algebraic geometry" (editor T Oda), Adv. Stud. Pure Math. 10, North-Holland (1987) 11 MR946233
6 R J Berman, K–polystability of –Fano varieties admitting Kähler–Einstein metrics, Invent. Math. 203 (2016) 973 MR3461370
7 R J Berman, S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, (2011) arXiv:1111.7158
8 R J Berman, D Witt Nyström, Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons, preprint (2014) arXiv:1401.8264
9 B Berndtsson, An introduction to things , from: "Analytic and algebraic geometry" (editors J McNeal, M Mustaţă), IAS/Park City Math. Ser. 17, Amer. Math. Soc. (2010) 7 MR2743815
10 B Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math. 200 (2015) 149 MR3323577
11 C Böhm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998) 145 MR1646591
12 C P Boyer, K Galicki, New Einstein metrics in dimension five, J. Differential Geom. 57 (2001) 443 MR1882664
13 C P Boyer, K Galicki, New Einstein metrics on 8#(S2 × S3), Differential Geom. Appl. 19 (2003) 245 MR2002662
14 C P Boyer, K Galicki, Sasakian geometry, Oxford Univ. Press (2008) MR2382957
15 C P Boyer, K Galicki, J Kollár, Einstein metrics on spheres, Ann. of Math. 162 (2005) 557 MR2178969
16 C P Boyer, K Galicki, J Kollár, E Thomas, Einstein metrics on exotic spheres in dimensions 7, 11, and 15, Experiment. Math. 14 (2005) 59 MR2146519
17 C P Boyer, M Nakamaye, On Sasaki–Einstein manifolds in dimension five, Geom. Dedicata 144 (2010) 141 MR2580423
18 W Bruns, J Herzog, Cohen–Macaulay rings, 39, Cambridge Univ. Press (1993) MR1251956
19 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888
20 J Cheeger, T H Colding, G Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002) 873 MR1937830
21 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
22 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
23 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
24 X Chen, B Wang, Space of Ricci flows, II, preprint (2014) arXiv:1405.6797
25 K Cho, A Futaki, H Ono, Uniqueness and examples of compact toric Sasaki–Einstein metrics, Comm. Math. Phys. 277 (2008) 439 MR2358291
26 C van Coevering, Monge–Ampère operators, energy functionals, and uniqueness of Sasaki-extremal metrics, preprint (2015) arXiv:1511.09167
27 T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 MR1454700
28 T C Collins, G Székelyhidi, K–semistability for irregular Sasakian manifolds, J. Differential Geom. 109 (2018) 81 MR3798716
29 T C Collins, D Xie, S T Yau, K stability and stability of chiral ring, preprint (2016) arXiv:1606.09260
30 C B Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. 13 (1980) 419 MR608287
31 V Datar, G Székelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016) 975 MR3558304
32 J P Demailly, Estimations L2 pour l’opérateur d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. 15 (1982) 457 MR690650
33 J P Demailly, Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, 19, Soc. Math. France (1985) 124 MR813252
34 J P Demailly, Complex analytic and differential geometry, book project (2012)
35 J P Demailly, J Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. 34 (2001) 525 MR1852009
36 R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728 MR3564626
37 W Y Ding, G Tian, Kähler–Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992) 315 MR1185586
38 S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
39 S K Donaldson, Stability, birational transformations and the Kahler–Einstein problem, from: "Surveys in differential geometry" (editors H D Cao, S T Yau), Surv. Differ. Geom. 17, International (2012) 203 MR3076062
40 S Donaldson, S Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math. 213 (2014) 63 MR3261011
41 S Donaldson, S Sun, Gromov–Hausdorff limits of Kahler manifolds and algebraic geometry, II, preprint (2015) arXiv:1507.05082
42 A El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990) 57 MR1042454
43 P Eyssidieux, V Guedj, A Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607 MR2505296
44 A Futaki, H Ono, G Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds, J. Differential Geom. 83 (2009) 585 MR2581358
45 J P Gauntlett, D Martelli, J Sparks, D Waldram, Sasaki–Einstein metrics on S2×S3, Adv. Theor. Math. Phys. 8 (2004) 711 MR2141499
46 J P Gauntlett, D Martelli, J Sparks, S T Yau, Obstructions to the existence of Sasaki–Einstein metrics, Comm. Math. Phys. 273 (2007) 803 MR2318866
47 A Ghigi, J Kollár, Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007) 877 MR2341843
48 D R Grayson, M E Stillman, Macaulay2, a software system for research in algebraic geometry
49 P Guan, X Zhang, A geodesic equation in the space of Sasakian metrics, from: "Geometry and analysis, I" (editor L Ji), Adv. Lect. Math. 17, International (2011) 303 MR2882427
50 R Hartshorne, Algebraic geometry, 52, Springer (1977) MR0463157
51 L Hörmander, L2 estimates and existence theorems for the  operator, Acta Math. 113 (1965) 89 MR0179443
52 N Ilten, H Süss, K–stability for Fano manifolds with torus action of complexity 1, Duke Math. J. 166 (2017) 177 MR3592691
53 W Jiang, Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture, J. Reine Angew. Math. 717 (2016) 195 MR3530538
54 J M Johnson, J Kollár, Kähler–Einstein metrics on log del Pezzo surfaces in weighted projective 3–spaces, Ann. Inst. Fourier (Grenoble) 51 (2001) 69 MR1821068
55 I R Klebanov, E Witten, Superconformal field theory on threebranes at a Calabi–Yau singularity, Nuclear Phys. B 536 (1999) 199 MR1666725
56 J Kollár, Einstein metrics on five-dimensional Seifert bundles, J. Geom. Anal. 15 (2005) 445 MR2190241
57 J Kollár, Circle actions on simply connected 5–manifolds, Topology 45 (2006) 643 MR2218760
58 J Kollár, Einstein metrics on connected sums of S2 × S3, J. Differential Geom. 75 (2007) 259 MR2286822
59 J Kollár, Positive Sasakian structures on 5–manifolds, from: "Riemannian topology and geometric structures on manifolds" (editors K Galicki, S R Simanca), Progr. Math. 271, Birkhäuser (2009) 93 MR2494170
60 A Laface, A Liendo, J Moraga, On the topology of rational T–varieties of complexity one, preprint (2015) arXiv:1503.06023
61 C Li, S Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 (2014) 927 MR3248054
62 D Luna, Slices étales, from: "Sur les groupes algébriques", Mém. Soc. Math. France 33, Soc. Math. France (1973) 81 MR0342523
63 J Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 MR1633016
64 D Martelli, J Sparks, S T Yau, The geometric dual of a–maximisation for toric Sasaki–Einstein manifolds, Comm. Math. Phys. 268 (2006) 39 MR2249795
65 D Martelli, J Sparks, S T Yau, Sasaki–Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008) 611 MR2399609
66 D H Phong, J Song, J Sturm, Degeneration of Kähler–Ricci solitons on Fano manifolds, Univ. Iagel. Acta Math. (2015) 29 MR3438282
67 J Ross, R Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 (2011) 109 MR2819757
68 L Saper, L2–cohomology of Kähler varieties with isolated singularities, J. Differential Geom. 36 (1992) 89 MR1168983
69 R Schoen, K Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982) 307 MR664498
70 B Shiffman, S Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (1999) 661 MR1675133
71 J Sparks, Sasaki–Einstein manifolds, from: "Surveys in differential geometry, XVI : Geometry of special holonomy and related topics" (editors N C Leung, S T Yau), Surv. Differ. Geom. 16, International (2011) 265 MR2893680
72 R P Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978) 57 MR0485835
73 G Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011) 319 MR2771134
74 G Székelyhidi, The partial C0–estimate along the continuity method, preprint (2013) arXiv:1310.8471
75 G Székelyhidi, Extremal Kähler metrics, from: "Proceedings of the International Congress of Mathematicians" (editors S Y Jang, Y R Kim, D W Lee, I Ye), Kyung Moon Sa (2014) 1017 MR3728650
76 G Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015) 451 MR3343885
77 G Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990) 101 MR1055713
78 G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
79 D Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012) 1736 MR2999302
80 S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339 MR480350
81 S T Yau, Survey on partial differential equations in differential geometry, from: "Seminar on differential geometry" (editor S T Yau), Ann. of Math. Stud. 102, Princeton Univ. Press (1982) 3 MR645729
82 S T Yau, Open problems in geometry, from: "Differential geometry: partial differential equations on manifolds" (editor R Greene), Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 1 MR1216573
83 S S T Yau, Y Yu, Classification of 3–dimensional isolated rational hypersurface singularities with –action, Rocky Mountain J. Math. 35 (2005) 1795 MR2206037
84 L Yi, A Bando–Mabuchi uniqueness theorem, preprint (2013) arXiv:1301.2847