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Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds. (English) Zbl 1439.53047

Summary: We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E11 Critical metrics
14J45 Fano varieties
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References:

[1] M. Abreu, Kähler geometry of toric varieties and extremal metrics, Internat. J. Math. 9 (1998), 641-651. · Zbl 0932.53043
[2] R. Berman, K-polystability of \(\mathbb{Q} \)-Fano varieties admitting Kähler-Einstein metrics, Invent. Math. 203 (2016), no. 3, 973-1025. · Zbl 1353.14051
[3] R. Berman and B. Berndtsson, Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 649-711. · Zbl 1283.58013
[4] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability, Duistermaat-Heckman Measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743-841. · Zbl 1391.14090
[5] R. Dervan, Uniform stability of twisted constant scalar curvature Kähler metric, Int. Math. Res. Not. IMRN 2016, no. 15, 4728-4783. · Zbl 1405.32032
[6] S. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom (2) 62 (2002), 289-349. · Zbl 1074.53059
[7] K. Fujita, Optimal Bounds for the volumes of Kähler-Einstein Fano manifolds, Amer. J. Math. 140 (2018), no. 2, 391-414. · Zbl 1400.14105
[8] T. Hisamoto, Stability and coersivity for toric polarizations, arXiv:1610.07998v1.
[9] T. Mabuchi, Kähler Einstein metrics for manifolds with non vanishing Futaki character, Tohoku Math. J. (2) 53 (2001), 171-182. · Zbl 1040.53084
[10] G. Tian, Canonical Metrics on Kähler Manifolds, Birkhäuser, 1999.
[11] Y. Yao, Mabuchi metrics and relative Ding stability of toric Fano varieties, arXiv:1701.04016v2.
[12] B. Zhou and X. Zhu, Relative K-stability and modified K-energy on toric manifolds, Adv. Math. 219 (2008), no. 4, 1327-1362. · Zbl 1153.53030
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