Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, p. 649-711

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 extend to the log Fano setting the seminal result of Wang-Zhou concerning the case when X is smooth and Δ 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.

Nous montrons, grâce à une approche variationnelle directe, que le deuxième problème avec valeurs au bord pour l’équation de Monge-Ampère dans 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,Δ)  ; à savoir que (X,Δ) 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 Δ 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.

@article{AFST_2013_6_22_4_649_0,
     author = {Berman, Robert J. and Berndtsson, Bo},
     title = {Real Monge-Amp\`ere equations and K\"ahler-Ricci solitons on toric log Fano varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {4},
     year = {2013},
     pages = {649-711},
     doi = {10.5802/afst.1386},
     mrnumber = {3137248},
     zbl = {1283.58013},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_4_649_0}
}
Berman, Robert J.; Berndtsson, Bo. Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, pp. 649-711. doi : 10.5802/afst.1386. http://www.numdam.org/item/AFST_2013_6_22_4_649_0/

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