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Envelopes of positive metrics with prescribed singularities
Julius Ross1 ; David Witt Nyström2
1 University of Cambridge, UK
2 University of Gothenburg, Sweden
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 687-727.

Nous étudions des enveloppes de métriques à courbure positive à singularités prescrites. En premier lieu, nous généralisons le travail de Berman dans ce contexte, nous prouvons la régularité C 1,1 de telles enveloppes, nous montrons que leur mesure de Monge–Ampère a pour support un certain « ensemble d’équilibre » et nous les relions aux asymptotiques de fonctions de Bergman partielles provenant d’idéaux multiplicateurs. Nous examinons comment ces enveloppes se comportent sur certains produits et leur relation avec la transformée de Legendre d’une courbe test de singularités psh dans le contexte des rayons géodésiques de l’espace des potentiels Kähler. Enfin, nous considérons la fonction d’exhaustion associée à ces ensembles d’équilibre, la reliant à la transformée de Legendre et à la géométrie du corps d’Okounkov.

We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C 1,1 regularity of such envelopes, showing their Monge–Ampère measure is supported on a certain “equilibrium set” and connecting with the asymptotics of the partial Bergman functions coming from multiplier ideals. We investigate how these envelopes behave on certain products, and how they relate to the Legendre transform of a test curve of singularity types in the context of geodesic rays in the space of Kähler potentials. Finally we consider the associated exhaustion function of these equilibrium sets, connecting it both to the Legendre transform and to the geometry of the Okounkov body.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1549

Julius Ross 1 ; David Witt Nyström 2

1 University of Cambridge, UK
2 University of Gothenburg, Sweden
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Julius Ross; David Witt Nyström. Envelopes of positive metrics with prescribed singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 687-727. doi : 10.5802/afst.1549. https://afst.centre-mersenne.org/articles/10.5802/afst.1549/

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