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Envelopes of positive metrics with prescribed singularities
Julius Ross; David Witt Nyström
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, p. 687-727

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.

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.

Received : 2016-04-22
Accepted : 2016-07-06
Published online : 2017-06-13
DOI : https://doi.org/10.5802/afst.1549
@article{AFST_2017_6_26_3_687_0,
     author = {Julius Ross and David Witt Nystr\"om},
     title = {Envelopes of positive metrics with prescribed singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {3},
     year = {2017},
     pages = {687-727},
     doi = {10.5802/afst.1549},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_3_687_0}
}
Ross, Julius; Nyström, David Witt. Envelopes of positive metrics with prescribed singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, pp. 687-727. doi : 10.5802/afst.1549. afst.centre-mersenne.org/item/AFST_2017_6_26_3_687_0/

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