# The asymptotic Fubini-Study operator over general non-Archimedean fields

The asymptotic Fubini-Study operator over general non-Archimedean fields Given an ample line bundle L over a projective K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {K}}}$$\end{document}-variety X, with K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {K}}}$$\end{document} a non-Archimedean field, we study limits of non-Archimedean metrics on L associated to submultiplicative sequences of norms on the graded pieces of the section ring R(X, L). We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied in the trivially valued case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

# The asymptotic Fubini-Study operator over general non-Archimedean fields

, Volume OnlineFirst – May 18, 2021
38 pages

/lp/springer-journals/the-asymptotic-fubini-study-operator-over-general-non-archimedean-EAGNyPPBxU
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-021-02770-2
Publisher site
See Article on Publisher Site

### Abstract

Given an ample line bundle L over a projective K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {K}}}$$\end{document}-variety X, with K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {K}}}$$\end{document} a non-Archimedean field, we study limits of non-Archimedean metrics on L associated to submultiplicative sequences of norms on the graded pieces of the section ring R(X, L). We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied in the trivially valued case.

### Journal

Mathematische ZeitschriftSpringer Journals

Published: May 18, 2021