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Göteborgs universitets publikationer

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

Författare och institution:
Robert Berman (Institutionen för matematiska vetenskaper, Chalmers/GU); Bo Berndtsson (Institutionen för matematiska vetenskaper, Chalmers/GU)
Publicerad i:
Annales de la faculté des sciences de Toulouse, 22 ( 4 ) s. 649-711
Artikel, refereegranskad vetenskaplig
Sammanfattning (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 $P.$
Ämne (baseras på Högskoleverkets indelning av forskningsämnen):
Chalmers fundament:
Grundläggande vetenskaper
Postens nummer:
Posten skapad:
2014-12-02 12:43
Posten ändrad:
2016-09-14 16:09

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