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

Robert J. Berman; Bo Berndtsson

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 4, page 649-711
  • ISSN: 0240-2963

Abstract

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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 , 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 .

How to cite

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Berman, Robert J., and Berndtsson, Bo. "Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties." Annales de la faculté des sciences de Toulouse Mathématiques 22.4 (2013): 649-711. <http://eudml.org/doc/275385>.

@article{Berman2013,
abstract = {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 $P.$},
author = {Berman, Robert J., Berndtsson, Bo},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {second boundary value problem; Monge-Ampère equation; toric log Fano varieties; Kähler-Ricci solitons; Kähler-Ricci flow},
language = {eng},
month = {6},
number = {4},
pages = {649-711},
publisher = {Université Paul Sabatier, Toulouse},
title = {Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties},
url = {http://eudml.org/doc/275385},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Berman, Robert J.
AU - Berndtsson, Bo
TI - Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 4
SP - 649
EP - 711
AB - 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 $P.$
LA - eng
KW - second boundary value problem; Monge-Ampère equation; toric log Fano varieties; Kähler-Ricci solitons; Kähler-Ricci flow
UR - http://eudml.org/doc/275385
ER -

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