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

Robert Berman (Institutionen för matematiska vetenskaper) ; Bo Berndtsson (Institutionen för matematiska vetenskaper)
Annales de la faculté des sciences de Toulouse (0240-2963). Vol. 22 (2013), 4, p. 649-711.

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in \$\mathbb{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,\Delta )\$ saying that \$(X,\Delta )\$ 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 Wang-Zhou concerning the case when \$X\$ is smooth and \$\Delta \$ 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.\$

CPL Pubid: 207055

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