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Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. BermanBo Berndtsson — 2013

Annales de la faculté des sciences de Toulouse Mathématiques

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in 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 , Δ ) saying that ( X , Δ ) admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

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