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EXTENSIONS OF VECTOR BUNDLES ON THE FARGUES-FONTAINE CURVE

Part of: Families, fibrations

Published online by Cambridge University Press:  14 May 2020

Christopher Birkbeck Open the ORCID record for Christopher Birkbeck [Opens in a new window] ,
Tony Feng Open the ORCID record for Tony Feng [Opens in a new window] ,
David Hansen ,
Serin Hong Open the ORCID record for Serin Hong [Opens in a new window] ,
Qirui Li ,
Anthony Wang  and
Lynnelle Ye
Christopher Birkbeck
Affiliation:
Department of Mathematics, University College London, Gower street, WC1E 6BT (c.birkbeck@ucl.ac.uk)
Tony Feng
Affiliation:
MIT Department of Mathematics, 182 Memorial Dr., Cambridge, MA02142 (fengt@mit.edu)
David Hansen
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111Bonn, Germany (dhansen@mpim-bonn.mpg.de)
Serin Hong
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI48109, USA (serinh@umich.edu)
Qirui Li
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, 10 NY10027, USA (qiruili@math.columbia.edu)
Anthony Wang
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL60637, USA (anthonyw@math.uchicago.edu)
Lynnelle Ye
Affiliation:
Department of Mathematics, Building 380, Stanford, California 94305 (lynnelle@stanford.edu)
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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

Fargues–FontaineDiamondsVector bundles

MSC classification

Primary: 14D24: Geometric Langlands program
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 2 , March 2022 , pp. 487 - 532
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

DH is grateful to Christian Johansson for some useful conversations about the material in § 3.2, and Peter Scholze for providing early access to the manuscript [13] and for some helpful conversations about the results therein. The project group students (CB, TF, SH, QL, AW, and LY) thank DH and Kiran Kedlaya for suggesting the problem. TF gratefully acknowledges the support of an NSF Graduate Fellowship. LY gratefully acknowledges the support of the National Defense Science and Engineering Graduate Fellowship. We would also like to thank David Linus Hamann and the referee for their valuable feedback on the first version of this paper.

References

Caraiani, A. and Scholze, P., On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2).Google Scholar
Colmez, P., Espaces de Banach de dimension finie, J. Inst. Math. Jussieu 1(3) (2002), 331–439. MR 1956055.CrossRefGoogle Scholar
Fargues, L., Simple connexité des fibres d’une application d’Abel-Jacobi et corps de classe local, Preprint, https://webusers.imj-prg.fr/∼laurent.fargues/cdc.pdf. To appear in Annales scientifiques de l’École normale supérieure.Google Scholar
Fargues, L. and Fontaine, J.-M., Courbes et fibrés vectoriels en théorie de Hodge p-adique, Astérisque (406) (2018), xiii+382. ISSN 0303-1179.Google Scholar
Fargues, L. and Fontaine, J.-M., Vector bundles on curves and p-adic Hodge theory, in Automorphic forms and Galois representations. vol. 2, London Mathematical Society Lecture Note Series, Volumne 415, pp. 17–104 (Cambridge University Press, Cambridge, 2014). MR 3444231.CrossRefGoogle Scholar
Hansen, D., Degenerating vector bundles in $p$ -adic Hodge theory, Preprint, http://www.math.columbia.edu/∼hansen/degen.pdf.Google Scholar
Huber, R., étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30 (Friedr. Vieweg & Sohn, Braunschweig, 1996). MR 1734903.10.1007/978-3-663-09991-8CrossRefGoogle Scholar
Kedlaya, K., Sheaves, stacks, and shtukas, Preprint, http://swc.math.arizona.edu/aws/2017/2017KedlayaNotes.pdf.Google Scholar
Kedlaya, K. S., Slope filtrations for relative Frobenius, Astérisque I(319) (2008), 259–301. Représentations $p$ -adiques de groupes $p$ -adiques. I. Représentations galoisiennes et $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D6E4})$ -modules. MR 2493220.Google Scholar
Kedlaya, K. S., Noetherian properties of Fargues-Fontaine curves, Int. Math. Res. Not. IMRN 2016(8) (2016), 2544–2567. MR 3519123.CrossRefGoogle Scholar
Kedlaya, K. S. and Liu, R., Relative p-adic Hodge theory: foundations, Astérisque (371) (2015), 239. MR 3379653.Google Scholar
Le Bras, A.-C., Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine, Duke Math. J. 167(18) (2018), 3455–3532.Google Scholar
Scholze, P., Étale cohomology of diamonds, Preprint, http://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf.Google Scholar
The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, (2018).Google Scholar
Scholze, P. and Weinstein, J., Lectures on p-adic geometry, Annals of Mathematics Studies (Princeton University Press). ISBN 9780691202099.Google Scholar
Scholze, P. and Weinstein, J., Moduli of p-divisible groups, Camb. J. Math. 1(2) (2013), 145–237. MR 3272049.CrossRefGoogle Scholar
Weinstein, J., Gal(Qp/Qp) as a geometric fundamental group, International Mathematics Research Notices 2017(10) (2016), 2964–2997. 1073-7928.Google Scholar