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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 5 (2015)

Vestislav Apostolov, David M. J. Calderbank, Paul Gauduchon
Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
Annales scientifiques de l'ENS 48, fascicule 5 (2015), 1075-1112

Télécharger cet article : Fichier PDF

Résumé :
Géométrie ambitorique II : surfaces toriques complexes extrémales et orbifolds d'Einstein de dimension 4
Nous donnons une solution complète et explicite du problème d'existence de métriques kählériennes extrémales sur un orbifold torique M de dimension réelle 4, dont le nombre de Betti b _2 (M) est égal à 2. Nous montrons plus précisément que M admet de telles métriques si et seulement si son polytope de Delzant rationnel — qui est alors un quadrilatère étiqueté — est K-polystable, suivant la théorie générale développée dans le cas torique par S. K. Donaldson, E. Legendre, G. Székelyhidi et al., et que ces métriques sont alors ambitoriques, donc complètement explicites d'après la classification figurant dans la première partie de ce travail. Notre approche donne de surcroît une façon effective de tester la stabilité des quadrilatères étiquetés. Parmi les métriques kählériennes construites dans cet article figurent celles dont le tenseur de Bach est nul, qui sont à la fois extrémales et conformément Einstein. Nous obtenons ainsi, en dimension 4, de nouveaux exemples explicites d'orbifolds d'Einstein compacts ou de variétés d'Einstein non-compactes, complètes et lisses.

Mots-clefs : Métriques kählériennes extrémales, géométrie torique, orbifolds d'Einstein de dimension 4.

Abstract:
We provide an explicit resolution of the existence problem for extremal Kähler metrics on toric 4-orbifolds M with second Betti number b_2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Székelyhidi et al.). Furthermore, in this case, the extremal Kähler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kähler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kähler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kähler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.

Keywords: Extremal Kähler metrics, toric geometry, Einstein 4-orbifolds.

Class. math. : 53C55, 53C25.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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