Extensions of vector bundles on the Fargues-Fontaine curve

Christopher Birkbeck; Tony Feng; David Hansen; Serin Hong; Qirui Li; Anthony Wang; Lynnelle Ye

doi:10.1017/S1474748020000183

Extensions of vector bundles on the Fargues-Fontaine curve

Christopher Birkbeck, Tony Feng, David Hansen, Serin Hong, Qirui Li, Anthony Wang, Lynnelle Ye

Algebra, Number Theory, Logic, and Representations (ANTLR)

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

8 Downloads (Pure)

Abstract

We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

Original language	English
Pages (from-to)	487-532
Number of pages	46
Journal	Journal of the Institute of Mathematics of Jussieu
Volume	21
Issue number	2
Early online date	14 May 2020
DOIs	https://doi.org/10.1017/S1474748020000183
Publication status	Published - 14 Mar 2022

Keywords

Diamonds
Fargues-Fontaine
Vector bundles

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10.1017/S1474748020000183

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Cite this

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title = "Extensions of vector bundles on the Fargues-Fontaine curve",

abstract = "We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons. ",

keywords = "Diamonds, Fargues-Fontaine, Vector bundles",

author = "Christopher Birkbeck and Tony Feng and David Hansen and Serin Hong and Qirui Li and Anthony Wang and Lynnelle Ye",

note = "Acknowledgements: The authors came together around these questions at the 2017 Arizona Winter School, under the umbrella of Kiran Kedlaya's project group. The authors would like to heartily thank Kiran for giving them this opportunity. They would also like to thank the organizers of the Winter School for creating such a wonderful experience, and they gratefully acknowledge NSF support of the Winter School via grant DMS-1504537.",

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T1 - Extensions of vector bundles on the Fargues-Fontaine curve

AU - Birkbeck, Christopher

AU - Feng, Tony

AU - Hansen, David

AU - Hong, Serin

AU - Li, Qirui

AU - Wang, Anthony

AU - Ye, Lynnelle

N1 - Acknowledgements: The authors came together around these questions at the 2017 Arizona Winter School, under the umbrella of Kiran Kedlaya's project group. The authors would like to heartily thank Kiran for giving them this opportunity. They would also like to thank the organizers of the Winter School for creating such a wonderful experience, and they gratefully acknowledge NSF support of the Winter School via grant DMS-1504537.

PY - 2022/3/14

Y1 - 2022/3/14

N2 - We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

AB - We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

KW - Diamonds

KW - Fargues-Fontaine

KW - Vector bundles

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UR - https://arxiv.org/abs/1705.00710

U2 - 10.1017/S1474748020000183

DO - 10.1017/S1474748020000183

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JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

IS - 2

ER -

Extensions of vector bundles on the Fargues-Fontaine curve

Abstract

Keywords

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