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The category of singularities as a crystal and global Springer fibers

Authors: D. Arinkin and D. Gaitsgory
Journal: J. Amer. Math. Soc. 31 (2018), 135-214
MSC (2010): Primary 14F05, 14H60
Published electronically: May 8, 2017
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Abstract: We prove the ``gluing conjecture'' on the spectral side of the categorical geometric Langlands conjecture. The key tool is the structure of crystal on the category of singularities, which allows one to reduce the conjecture to the question of homological triviality of certain homotopy types. These homotopy types are obtained by gluing from a global version of Springer fibers.

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  • [AG] D. Arinkin and D. Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. (N.S.) 21 (2015), no. 1, 1–199. MR 3300415
  • [Be] D. Beraldo, On the extended Whittaker category, arXiv:1411.7982.
  • [BIK] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 573–619 (English, with English and French summaries). MR 2489634
  • [DrGa] V. Drinfeld and D. Gaitsgory, Compact generation of the category of D-modules on the stack of 𝐺-bundles on a curve, Camb. J. Math. 3 (2015), no. 1-2, 19–125. MR 3356356
  • [Ga1] Dennis Gaitsgory, ind-coherent sheaves, Mosc. Math. J. 13 (2013), no. 3, 399–528, 553 (English, with English and Russian summaries). MR 3136100
  • [Ga2] Dennis Gaitsgory, Sheaves of categories and the notion of 1-affineness, Stacks and categories in geometry, topology, and algebra, Contemp. Math., vol. 643, Amer. Math. Soc., Providence, RI, 2015, pp. 127–225. MR 3381473
  • [Ga3] Dennis Gaitsgory, Outline of the proof of the geometric Langlands conjecture for 𝐺𝐿₂, Astérisque 370 (2015), 1–112 (English, with English and French summaries). MR 3364744
  • [Ga4] D. Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve, available at gaitsgde/GL/.
  • [GR1] Dennis Gaitsgory and Nick Rozenblyum, Crystals and D-modules, Pure Appl. Math. Q. 10 (2014), no. 1, 57–154. MR 3264953
  • [GR2] D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, available at gaitsgde/GL/.
  • [Ste] Greg Stevenson, Subcategories of singularity categories via tensor actions, Compos. Math. 150 (2014), no. 2, 229–272. MR 3177268

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Additional Information

D. Arinkin
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

D. Gaitsgory
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): May 13, 2015
Received by editor(s) in revised form: January 18, 2017
Published electronically: May 8, 2017
Additional Notes: The research of the first author is partially supported by NSF grant DMS-1101558.
The research of the second author is partially supported by NSF grant DMS-1063470.
Article copyright: © Copyright 2017 American Mathematical Society