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 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 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.
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@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},
pages = {687--727},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {3},
year = {2017},
doi = {10.5802/afst.1549},
zbl = {1421.32032},
mrnumber = {3669969},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1549/}
}
TY - JOUR AU - Julius Ross AU - David Witt Nyström TI - Envelopes of positive metrics with prescribed singularities JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 687 EP - 727 VL - 26 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1549/ DO - 10.5802/afst.1549 LA - en ID - AFST_2017_6_26_3_687_0 ER -
%0 Journal Article %A Julius Ross %A David Witt Nyström %T Envelopes of positive metrics with prescribed singularities %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 687-727 %V 26 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1549/ %R 10.5802/afst.1549 %G en %F AFST_2017_6_26_3_687_0
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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