Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties

Abstract

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère 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) Kähler-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 Kähler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the Kähler-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.

arXiv comment53 pages

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Cited by in corpus (4)

• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Connecting toric manifolds by conical Kahler-Einstein metrics
• Statistical mechanics of permanents, real-Monge-Ampere equations and optimal transport
• An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons