Geometry & Topology Volume 20, issue 1 (2016)

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The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality

Jian Song and Xiaowei Wang

Geometry & Topology 20 (2016) 49–102

DOI: 10.2140/gt.2016.20.49

Bibliography

1	M Atiyah, C Lebrun, Curvature, cones and characteristic numbers, Math. Proc. Cambridge Philos. Soc. 155 (2013) 13 MR3065256
2	R J Berman, A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics, (2012) arXiv:1011.3976
3	R J Berman, K–polystability of ℚ–Fano varieties admitting Kähler–Einstein metrics, preprint (2015) arXiv:1205.6214
4	R J Berman, S Boucksom, V Guedj, A Zeriahi, A variational approach to complex Monge–Ampère equations, preprint (2009) arXiv:0907.4490
5	S Brendle, Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. 2013 (2013) 5727 MR3144178
6	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
7	V Datar, B Guo, J Song, X Wang, Connecting toric manifolds by conical Kähler–Einstein metrics, preprint (2013) arXiv:1308.6781
8	J P Demailly, On Tian’s invariant and log canonical thresholds, (2008) MR2484031
9	W Y Ding, Remarks on the existence problem of positive Kähler–Einstein metrics, Math. Ann. 282 (1988) 463 MR967024
10	S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
11	S K Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008) 389 MR2433928
12	S K Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, from: "Handbook of geometric analysis" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 7, Int. Press (2008) 29 MR2483362
13	S K Donaldson, Kähler metrics with cone singularities along a divisor, from: "Essays in mathematics and its applications" (editors P M Pardalos, T M Rassias), Springer (2012) 49 MR2975584
14	P Eyssidieux, V Guedj, A Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607 MR2505296
15	V Guillemin, Kähler structures on toric varieties, J. Differential Geom. 40 (1994) 285 MR1293656
16	T Jeffres, R Mazzeo, Y A Rubinstein, Kähler–Einstein metrics with edge singularities, Ann. Math. 183 (2016) 95 MR3432582
17	S Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998) 69 MR1618325
18	C Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds, Adv. Math. 226 (2011) 4921 MR2775890
19	C Li, On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold, Compos. Math. 148 (2012) 1985 MR2999312
20	C Li, Remarks on logarithmic K–stability, Commun. Contemp. Math. 17 (2015) 1450020, 17 MR3313212
21	C Li, S Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 (2014) 927 MR3248054
22	H Li, On the lower bound of the K–energy and F–functional, Osaka J. Math. 45 (2008) 253 MR2416659
23	F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119 MR1137227
24	T Mabuchi, K–energy maps integrating Futaki invariants, Tohoku Math. J. 38 (1986) 575 MR867064
25	R C McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222 MR938672
26	Y Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977) 225 MR0460343
27	O Munteanu, G Székelyhidi, On convergence of the Kähler–Ricci flow, Comm. Anal. Geom. 19 (2011) 887 MR2886711
28	Y Odaka, S Sun, Testing log K–stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. 24 (2015) 505 MR3403730
29	D H Phong, N Sesum, J Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613 MR2379807
30	D H Phong, J Song, J Sturm, B Weinkove, The Moser–Trudinger inequality on Kähler–Einstein manifolds, Amer. J. Math. 130 (2008) 1067 MR2427008
31	Y A Rubinstein, On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood, J. Funct. Anal. 255 (2008) 2641 MR2473271
32	J Song, The α–invariant on certain surfaces with symmetry groups, Trans. Amer. Math. Soc. 357 (2005) 45 MR2098086
33	S Sun, Note on K–stability of pairs, Math. Ann. 355 (2013) 259 MR3004583
34	G Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011) 319 MR2771134
35	G Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C₁(M) > 0, Invent. Math. 89 (1987) 225 MR894378
36	G Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992) 401 MR1163733
37	G Tian, Kähler–Einstein metrics on algebraic manifolds, from: "Transcendental methods in algebraic geometry", Lecture Notes in Math. 1646, Springer (1996) 143 MR1603624
38	G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
39	G Tian, Canonical metrics in Kähler geometry, Birkhäuser (2000) MR1787650
40	G Tian, B Wang, On the structure of almost Einstein manifolds, J. Amer. Math. Soc. 28 (2015) 1169 MR3369910
41	G Tian, S T Yau, Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, from: "Mathematical aspects of string theory" (editor S T Yau), Adv. Ser. Math. Phys. 1, World Scientific (1987) 574 MR915840
42	G Tian, X Zhu, A nonlinear inequality of Moser–Trudinger type, Calc. Var. Partial Differential Equations 10 (2000) 349 MR1767718
43	M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793 MR1005085
44	H Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology 27 (1988) 429 MR976585
45	X J Wang, X Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004) 87 MR2084775
46	S T Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977) 1798 MR0451180
47	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
48	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
49	Y Zhang, Miyaoka–Yau inequality for minimal projective manifolds of general type, Proc. Amer. Math. Soc. 137 (2009) 2749 MR2497488
50	Z Zhang, On degenerate Monge–Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006 (2006) MR2233716