Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds

P. Åhag, U. Cegrell, S. Kołodziej, H. H. Phạm and A. Zeriahi, Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Adv. Math. 222 (2009), 2036–2058. 10.1016/j.aim.2009.07.002

T. Aubin, Réduction du cas positif de Monge–Ampère sur les variétés kähleriennes compactes à la démonstration d’une inégalité, Funct. Anal. 57 (1984), 143–153. 10.1016/0022-1236(84)90093-4

S. Bando, The K-energy map, almost Einstein Kähler metrics and an inequality of the Miyaoka–Yau type, Tohuku Math. J. 39 (1987), 231–235. 10.2748/tmj/1178228326

S. Bando and T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry (Sendai 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam (1987), 11–40.

R. J. Berman, K-polystability of Q-Fano varieties admitting Kahler–Einstein metrics, preprint (2012), http://arxiv.org/abs/1205.6214.

R. J. Berman, A thermodynamic formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics, Adv. Math. 248 (2013), 1254–1297. 10.1016/j.aim.2013.08.024

R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, preprint (2011), http://arxiv.org/abs/1111.7158.

R. J. Berman, S. Boucksom, V. Guedj and A. Zeriahi, A variational approach to complex Monge–Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245. 10.1007/s10240-012-0046-6

B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, Ann. Math. 169 (2009), no. 2, 531–560. 10.4007/annals.2009.169.531

B. Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and the Bando–Mabuchi uniqueness theorem, preprint (2011), http://arxiv.org/abs/1103.0923.

B. Berndtsson and M. Paun, Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), no. 2, 341–378. 10.1215/00127094-2008-054

J. Cheeger, T. Colding and G. Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002), 873–914. 10.1007/PL00012649

X. Chen, Space of Kähler metrics (IV): On the lower bound of the K-energy, preprint (2008), http://arxiv.org/abs/0809.4081.

X. Chen, S. Donaldson and S. Sun, Kahler–Einstein metrics on Fano manifolds, II: Limits with cone angle less than 2⁢π2\pi, preprint (2012), http://arxiv.org/abs/1212.4714.

X. Chen, S. Donaldson and S. Sun, Kahler–Einstein metrics on Fano manifolds, III: Limits as cone angle approaches 2⁢π2\pi and completion of the main proof, preprint (2013), http://arxiv.org/abs/1302.0282.

J.-P. Demailly, Complex analytic and differential geometry, (2012), https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf.

J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 525–556. 10.1016/S0012-9593(01)01069-2

W. Ding, Remarks on the existence problem of positive Kähler–Einstein metrics, Math. Ann. 282 (1988), 463–471. 10.1007/BF01460045

W. Ding and G. Tian, Kähler–Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), no. 2, 315–335. 10.1007/BF01231335

W. Ding and G. Tian, The generalized Moser–Trudinger inequality, Proceedings of Nankai international conference on nonlinear analysis, World Scientific, Singapore (1993), 57–70.

S. Donaldson, Lower bounds on the Calabi functional, J. Differential Geometry 70 (2005), 453–472. 10.4310/jdg/1143642909

S. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, International Press, Somerville 2008.

S. Donaldson, Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications, Springer, Heidelberg (2012), 49–79.

T. D. Jeffres, R. Mazzeo and Y. A. Rubinstein, Kähler–Einstein metrics with edge singularities, preprint (2011), arxiv.org/abs/1105.5216.

J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge 1998.

C. Li, Kähler–Einstein metrics and K-stability, PhD thesis, Princeton, 2012, www.math.sunysb.edu/~chili/papers/PhDthesis.pdf.

C. Li, Remarks on logarithmic K-stability, Commun. Contemp. Math. 17 (2015), no. 2.

C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 (2014), no. 3, 927–973. 10.1007/s00220-014-2123-9

C. Li and C. Xu, Special test configurations and K-stability of Fano varieties, Ann. of Math. 180 (2014), 197–232. 10.4007/annals.2014.180.1.4

H. Li, On the lower bound of the K-energy and F-functional, Osaka J. Math. 45 (2008), no. 1, 253–264.

T. Mabuchi, K-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), no. 4, 575–593. 10.2748/tmj/1178228410

S. Mukai and H. Umemura, Minimal rational threefolds, Algebraic geometry, Lecture Notes in Math. 1016, Springer, Berlin (1983), 490–518.

S. T. Paul and G. Tian, CM stability and the generalized Futaki invariant II, Astérisque 328 (2009), 339–354.

D. H. Phong, J. Ross and J. Sturm, Deligne pairing and the Knudsen–Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475–496. 10.4310/jdg/1207834553

D. H. Phong and J. Sturm, Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, Ann. of Math. 152 (2000), 277–329. 10.2307/2661384

D. H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge–Ampere equations, Comm. Anal. Geom. 18 (2010), no. 1, 145–170. 10.4310/CAG.2010.v18.n1.a6

X. Rong and Y. Zhang, Continuity of extremal transitions and flops for Calabi–Yau manifolds, with an appendix by M. Gross, J. Differential Geom. 89 (2011), no. 2, 233–269.

Y. Rubinstein, On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood, J. Funct. Anal. 255 (2008), 2641–2660. 10.1016/j.jfa.2007.10.009

J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality, preprint (2012), http://arxiv.org/abs/1207.4839.

C. Spotti, S. Sun and C. Yao, Existence and deformations of Kahler–Einstein metrics on smoothable ℚ\mathbb{Q}-Fano varieties, preprint (2014), http://arxiv.org/abs/1411.1725.

H. Süß, Kähler–Einstein metrics on symmetric Fano T-varieties, Adv. Math. 246 (2013), 100–113. 10.1016/j.aim.2013.06.023

G. Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compositio Math. 147 (2011), 319–331. 10.1112/S0010437X10004938

G. Székelyhidi, A remark on conical Kähler–Einstein metrics, preprint (2012), http://arxiv.org/abs/1211.2725.

G. Tian, On Kähler–Einstein metrics on certain Kähler manifolds with c1⁢(M)>0c_{1}(M)>0, Invent. Math. 89 (1987), 225–246.

G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101–172. 10.1007/BF01231499

G. Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992), 401–413. 10.1142/S0129167X92000175

G. Tian, The K-energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994), no. 2, 239–265. 10.4310/CAG.1994.v2.n2.a4

G. Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 137 (1997), 1–37.

G. Tian, Canonical metrics on Kähler manifolds, Birkhäuser, Basel 2000.

G. Tian, Existence of Einstein metrics on Fano manifolds, Metric and differential geometry, Progress Math. 297, Springer, Berlin (2012), 119–159.

G. Tian, K-stability and Kähler–Einstein metrics, preprint (2012), http://arxiv.org/abs/1211.4669.

G. Tian and B. Wang, On the structure of almost Einstein manifolds, preprint (2012), http://arxiv.org/abs/1202.2912,

G. Tian, S. Zhang, Z. Zhang and X. Zhu, Supremum of Perelman’s entropy and Kähler–Ricci flow on a Fano manifold, preprint (2011), http://arxiv.org/abs/1107.4018.

Y. Xing, Continuity of the complex Monge–Ampère operator, Proc. Amer. Math. Soc. 124 (1996), 457–467. 10.1090/S0002-9939-96-03316-3

S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equations. I, Comm. Pure Appl. Math 31 (1978), 339–411. 10.1002/cpa.3160310304