Families of Kähler spaces / Familles d'espaces kählériens

Collection Families of Kähler spaces / Familles d'espaces kählériens

Organizer(s) Cadorel, Benoît ; Cao, Junyan ; Deng, Ya ; Floris, Enrica ; Guenancia, Henri

Date(s) 21/04/2025 - 25/04/2025

linked URL https://conferences.cirm-math.fr/3221.html

00:00:00 / 00:00:00

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Higgs bundles for Hecke eigensheaves in rank 2 and genus 2

By Carlos Simpson

We'll discuss our joint work with Ron Donagi and Tony Pantev on the construction of the Higgs bundles associated to Hecke eigensheaves for the geometric Langlands program in the case of rank 2 local systems on a curve of genus 2 . Recall that the moduli space of bundles in this case has two connected components: $\mathbb{P}^3$ and the intersection of two quadrics in $\mathbb{P}^5$. We look for Higgs bundles on these spaces with parabolic structure and logarithmic poles along the wobbly locus. This leads to the study of the geometry of the wobbly locus and its singularities, and the use of our Dolbeault higher direct image construction for the calculation of Hecke operators.

Information about the video

Date of recording 21/04/2025
Date of publication 13/05/2025
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20343503
Cite this video Simpson, Carlos (21/04/2025). Higgs bundles for Hecke eigensheaves in rank 2 and genus 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20343503
URL https://dx.doi.org/10.24350/CIRM.V.20343503

Domain(s)

Bibliography

DONAGI, Ron, PANTEV, Tony, et SIMPSON, Carlos. Twistor Hecke eigensheaves in genus 2. arXiv preprint arXiv:2403.17045, 2024. - https://doi.org/10.48550/arXiv.2403.17045
DONAGI, Ron, PANTEV, Tony. Langlands duality for Hitchin systems. Invent. math. 189, 653–735 (2012). - https://doi.org/10.1007/s00222-012-0373-8
DONAGI, Ron et PANTEV, Tony. Parabolic Hecke eigensheaves. Parabolic Hecke eigensheaves. Astérisque 435, 2022. - https://doi.org/10.24033/ast.1178
DONAGI, R., PANTEV, T., et SIMPSON, C. Direct images in non Abelian Hodge theory. arXiv preprint arXiv:1612.06388, 2016. - https://doi.org/10.48550/arXiv.1612.06388
NARASIMHAN, Mudumbai S. et RAMANAN, Sundararaman. Moduli of vector bundles on a compact Riemann surface. Annals of Mathematics, 1969, vol. 89, no 1, p. 14-51. - https://doi.org/10.2307/1970807
DRINFELD, V. G. Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL (2). American Journal of Mathematics, 1983, vol. 105, no 1, p. 85-114. - https://doi.org/10.2307/2374382
LAUMON, Gérard. Correspondance de Langlands géométrique pour les corps de fonctions. 1987. - https://doi.org/10.1215/S0012-7094-87-05418-4
HITCHIN, Nigel. Stable bundles and integrable systems. 1987. - https://doi.org/10.1215/S0012-7094-87-05408-1
HAUSEL, Tamás et THADDEUS, Michael. Mirror symmetry, Langlands duality, and the Hitchin system. Inventiones mathematicae, 2003, vol. 153, p. 197-229. - https://doi.org/10.1007/s00222-003-0286-7
KAPUSTIN, Anton et WITTEN, Edward. Electric-magnetic duality and the geometric Langlands program. Communications in Number Theory and Physics, 1(1):1–236, 2007. - https://doi.org/10.4310/CNTP.2007.v1.n1.a1
MOCHIZUKI, Takuro. Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $ D $-Modules, Part 1. American Mathematical Soc., 2007. - https://doi.org/10.1090/memo/0869
ARINKIN, D., BERALDO, D., CAMPBELL, J. , CHEN L, FAERGEMAN, J., GAITSGORY, D, LIN, K., RASKIN, S, ROZENBLYUM, N. .Proof of the geometric Langlands conjecture, 2024 (five papers combined) - https://people.mpim-bonn.mpg.de/gaitsgde/GLC/

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