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

Berman, Robert J.; Berndtsson, Bo

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 649-711.

Nous montrons, grâce à une approche variationnelle directe, que le deuxième problème avec valeurs au bord pour l’équation de Monge-Ampère dans ${ℝ}^n$ avec non-linéarité exponentielle, et ensemble cible un corps convexe $P$, admet une solution si et seulement si $0$ est le barycentre de $P$. En combinant ce résultat avec de la géométrie torique, on obtient en particulier confirmation de la conjecture de Yau-Tian-Donaldson (généralisée) pour les variétés toriques log-Fano $(X,\Delta )$  ; à savoir que $(X,\Delta )$ admet une une métrique de Kähler-Einstein (singulière) si et seulement si elle est K-stable au sens algébro-géométrique. Nous obtenons donc une nouvelle démonstration, qui s’étend au cas log-Fano, du résultat fondateur de Wang-Zhou qui concerne le cas où $X$ est lisse et $\Delta$ est trivial. Nous généralisons également la formule torique de Li pour la borne inférieure de la courbure de Ricci. Plus généralement, nous obtenons des solitons de Kähler-Ricci sur toute variété (singulière) log-Fano, et montrons qu’ils apparaissent comme la limite en temps grand du flot de Kähler-Ricci. De plus, en utilisant la dualité, nous confirmons aussi une conjecture de Donaldson sur les solutions du problème de valeurs au bord d’Abreu sur le corps convexe $P$ dans le cas d’une mesure canonique donnée sur la frontière de $P$.

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 $(X,\Delta )$ 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 $P.$

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