Principal bundles on metric graphs: The GLn case

Using the notion of a root datum of a reductive group G we propose a tropical analogue of a principal G-bundle on a metric graph. We focus on the case G=GLn, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2022-12, Vol.411, Article 108775
Main Authors: Gross, Andreas, Ulirsch, Martin, Zakharov, Dmitry
Format: Article
Language:English
Subjects:
Metric graphs
Moduli spaces
Principal bundles
Tate curve
Tropical geometry
Vector bundles
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Summary:Using the notion of a root datum of a reductive group G we propose a tropical analogue of a principal G-bundle on a metric graph. We focus on the case G=GLn, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil–Riemann–Roch theorem and the Narasimhan–Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to a certain component of the moduli space of semistable tropical vector bundles on its dual metric graph.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2022.108775