Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
By Robert J. Berman, B. Berndtsson. 2012
135 citations recorded by the source corpus.
We show, using a direct variational approach, that the second boundary value problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that (X,D) admits a (singular) K\"ahler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Zhou-Wang concerning the case when X is smooth and D is trivial. Li's toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain K\"ahler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the K\"ahler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu's boundary value problem on the convex body P. in the case of a given canonical measure on the boundary of P.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d66181f7839b8ced709cc9dcdcf4fc13eda9f3f7
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “The existence problem in the singular case is still open in general, except for the toric case [ BB13 ]; cf also [ OSS16 ] for a related problem in the case of singular Fano surfaces.”
- Kähler–Einstein metrics and volume minimization (2016, background): “So by [38,4] there is a conical KE metric on the pair (C, D) where D = ∑”
- Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “The point is that in this situation finite energy corresponds to finite cost (see [10, 9]).”
- Stein kernels and moment maps (2018, background): “The results of [19] assume C∞ smoothness, relying on a result of [4] to deduce C∞ smoothness of φ.”
- Quantitative stability of optimal transport maps under variations of the target measure (2021, unknown): “The next proposition, found in [6], shows that a sufficient condition is that μ admits a finite moment of order p > d.”
- K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, background): “metric on the toric variety X implies that X is K-polystable for general test configurations was left open in [2].”