Generalized Kähler-Ricci flow on toric Fano varieties

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

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Gorenstein toric Fano varieties

manuscripta mathematica ◽

10.1007/s00229-004-0532-3 ◽

2005 ◽

Vol 116 (2) ◽

pp. 183-210 ◽

Cited By ~ 16

Author(s):

Benjamin Nill

Keyword(s):

Fano Varieties ◽

Toric Fano Varieties

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Toward the classification of higher-dimensional toric Fano varieties

Tohoku Mathematical Journal ◽

10.2748/tmj/1178207820 ◽

2000 ◽

Vol 52 (3) ◽

pp. 383-413 ◽

Cited By ~ 24

Author(s):

Hiroshi Sato

Keyword(s):

Fano Varieties ◽

Higher Dimensional ◽

Toric Fano Varieties

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On toric geometry and K-stability of Fano varieties

Transactions of the American Mathematical Society Series B ◽

10.1090/btran/82 ◽

2021 ◽

Vol 8 (19) ◽

pp. 548-577

Author(s):

Anne-Sophie Kaloghiros ◽

Andrea Petracci

Keyword(s):

Moduli Spaces ◽

Deformation Theory ◽

The Other ◽

Fano Varieties ◽

Toric Geometry ◽

Content Type ◽

Closed Point ◽

Toric Fano Varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3 3 -fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3 3 -fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3 3 -fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3 3 -folds by building degenerations to K-polystable toric Fano 3 3 -folds.

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