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.
- 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, DOI 10.1007/s00029-014-0167-5
- D. Beraldo, On the extended Whittaker category, arXiv:1411.7982.
- 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, DOI 10.24033/asens.2076
- 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, DOI 10.4310/CJM.2015.v3.n1.a2
- Dennis Gaitsgory, ind-coherent sheaves, Mosc. Math. J. 13 (2013), no. 3, 399–528, 553 (English, with English and Russian summaries). MR 3136100, DOI 10.17323/1609-4514-2013-13-3-399-528
- 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, DOI 10.1090/conm/643/12899
- Dennis Gaitsgory, Outline of the proof of the geometric Langlands conjecture for $GL_2$, Astérisque 370 (2015), 1–112 (English, with English and French summaries). MR 3364744
- D. Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve, available at http://www.math.harvard.edu/ gaitsgde/GL/.
- Dennis Gaitsgory and Nick Rozenblyum, Crystals and D-modules, Pure Appl. Math. Q. 10 (2014), no. 1, 57–154. MR 3264953, DOI 10.4310/PAMQ.2014.v10.n1.a2
- D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, available at http://www.math.harvard.edu/ gaitsgde/GL/.
- Greg Stevenson, Subcategories of singularity categories via tensor actions, Compos. Math. 150 (2014), no. 2, 229–272. MR 3177268, DOI 10.1112/S0010437X1300746X
- D. Arinkin
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: email@example.com
- D. Gaitsgory
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Email: firstname.lastname@example.org
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc. 31 (2018), 135-214
- MSC (2010): Primary 14F05, 14H60
- DOI: https://doi.org/10.1090/jams/882
- MathSciNet review: 3718453