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Geometric Langlands in prime characteristic. (English) Zbl 1390.14044
The geometric Langlands conjecture proposed by Beilinson and Drinfeld, asserts that the derived category $$D(Bund_G^0)$$ of D-modules on moduli stack $$Bun_G$$ on a smooth projective curve, is equivalent to the derived cateogry $$D(QCoh(LocSys^0_{\check{G}}))$$ of quasi-coherent sheaves on the moduli stack $$LocSys_{\check{G}}$$ of $$\check{G}$$-local systems on $$C$$, where $$\check{G}$$ is the Langlands dual group of $$G$$. The original conjecture is in the setting of geometry over $$\mathbb{C}$$. This paper estabished a similar duality “generically” when the field is of Characteristic $$p$$, where $$G$$ is assumed to be semisimple and $$C$$ is at least of genus $$2$$. This work is a generalization of the duality in characteristic $$p$$ established by Bezrukavnikov and Braverman [4] when $$G=\mathrm{GL}_n$$.
The classical limit of geometric Langlands is a Fourier-Mukai type duality between Hitchin fibrations. It was established “generically” by Donagi and Pantev [14] when the field is $$\mathbb{C}$$ by using transcendental methods. A feature of characteristic $$p$$ D-module, is that there is a bridge between D-module and certain twisted sheaf on Hitchin fibration. This is exactly the idea of this paper (and also the work by Bezrukavnikov and Braverman [loc. cit.]), to deduce the geometric Langlands duality in characteristic p from Hitchin fibration duality. In the case of $$\mathrm{GL}_n$$, the Hitchin fibration can be reduced to a Fourier-Mukai duality on Jacobian of curves, since generailly the Hitchin fiber is the Jacobian of the asccoiated spectral curve. For general $$G$$, the story is more complicated. The authors established a Fourier-Mukai type duality on Beilinson 1-motive, which is a slightly generalization of abelian variety.
The equivalence between $$D(Bun_G^0)$$ and $$D(QCoh(LocSys^0_{\check{G}}))$$ established by the authors, is still not completely confirmed to be the geometric Langlands transform (i.e. satisfy appropriate Hecke operators conditions) conjectured by Beilinson and Drinfeld [3].

##### MSC:
 14D24 Geometric Langlands program (algebro-geometric aspects) 22E57 Geometric Langlands program: representation-theoretic aspects
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##### References:
 [1] Arinkin, D., Appendix to []. · Zbl 1408.14048 [2] Arinkin, D. and Gaitsgory, D., Singular support of coherent sheaves, and the geometric Langlands conjecture, Preprint (2012), arXiv:1201.6343. · Zbl 1423.14085 [3] Beilinson, A. and Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint (1991), http://www.math.uchicago.edu/earinkin/langlands/. [4] Bezrukavnikov, R. and Braverman, A., Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case, Pure Appl. Math. Q.3 (2007), 153-179. doi:10.4310/PAMQ.2007.v3.n1.a5 · Zbl 1206.14030 [5] Bezrukavnikov, R., Mirković, I. and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2)167 (2008), 945-991. doi:10.4007/annals.2008.167.945 · Zbl 1220.17009 [6] Bezrukavnikov, R. and Travkin, R., Quantization of Hitchin integrable system via positive characteristic, Preprint (2016), arXiv:1603.01327. [7] Breen, L., Un théoreme d’annulation pour certains Ext de faisceaux abéliens, Ann. Sci. Éc. Norm. Supér. (4), 5, 339-352, (1975) · Zbl 0313.14001 [8] Bosch, S., Lutkebohmert, W. and Raynaud, M., Néron models, (Springer, Berlin, 1990). doi:10.1007/978-3-642-51438-8 · Zbl 0705.14001 [9] Brochard, S., Foncteur de Picard d’un champ algébrique, Math. Ann., 343, 541-602, (2009) · Zbl 1165.14023 [10] Chen, T.-H. and Zhu, X., Non-abelian Hodge theory for curves in characteristic p, Geom. Funct. Anal.25 (2015), 1706-1733. doi:10.1007/s00039-015-0343-6 · Zbl 1330.14015 [11] Deligne, P., La formule de dualité globale, in SGA 4, tome 3, Expose XVIII, (Springer, Berlin, 1973), 481-587. · Zbl 0259.14006 [12] Donagi, R. and Gaitsgory, D., The gerbe of Higgs bundles, Transform. Groups7 (2002), 109-153. doi:10.1007/s00031-002-0008-z · Zbl 1083.14519 [13] Donagi, R. and Pantev, T., Torus fibrations, gerbes, and duality (with an appendix by Dmitry Arinkin), Mem. Amer. Math. Soc.193 (2008), no. 901. · Zbl 1140.14001 [14] Donagi, R. and Pantev, T., Langlands duality for Hitchin systems, Invent. Math.189 (2012), 653-735. doi:10.1007/s00222-012-0373-8 · Zbl 1263.53078 [15] Frenkel, E., Gaitsgory, D. and Vilonen, K., Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math. (2)153 (2001), 699-748. doi:10.2307/2661366 · Zbl 1070.11050 [16] Frenkel, E. and Witten, E., Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys.2 (2008), 113-283. doi:10.4310/CNTP.2008.v2.n1.a3 · Zbl 1223.14014 [17] Gaitsgory, D., Outline of the proof of the geometric Langlands conjecture for$$\operatorname{GL}_{2}$$, Preprint (2013), arXiv:1302.2506. [18] Groechenig, M., Moduli of flat connections in positive characteristic, Preprint (2013), arXiv:1201.0741. [19] Hausel, T. and Thaddeus, M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math.153 (2003), 197-229. doi:10.1007/s00222-003-0286-7 · Zbl 1043.14011 [20] Hitchin, N., Stable bundles and integrable systems, Duke Math. J., 54, 91-114, (1987) · Zbl 0627.14024 [21] Laumon, G., Transformation de Fourier generalisee, Preprint (1996), arXiv:alg-geom/9603004. [22] Laumon, G. and Moret-Bailly, L., Champs algébriques, (Springer, Berlin, 2000). · Zbl 0945.14005 [23] Mukai, S., Duality between D (X) and D (X̂) with its application to Picard sheaves, Nagoya Math. J., 153-175, (1981) · Zbl 0417.14036 [24] Mukai, S., Fourier functor and its application to the moduli of bundles on an abelian variety, in Algebraic geometry, Sendai, 1985, (North-Holland, Amsterdam, 1987), 515-550. [25] Ngô, B. C., Hitchin fibration and endoscopy, Invent. Math., 164, 399-453, (2006) · Zbl 1098.14023 [26] Ngô, B. C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci., 111, 1-169, (2010) · Zbl 1200.22011 [27] Ogus, A. and Vologodsky, V., Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci.106 (2007), 1-138. doi:10.1007/s10240-007-0010-z · Zbl 1140.14007 [28] Osipov, D. and Zhu, X., A categorical proof of the Parshin reciprocity laws on algebraic surfaces, Algebra Number Theory5 (2011), 289-337. doi:10.2140/ant.2011.5.289 · Zbl 1237.19007 [29] Toen, B., Derived Azumaya algebras and generators for twisted derived categories, Invent. Math., 189, 581-652, (2012) · Zbl 1275.14017 [30] Travkin, R., Quantum geometric Langlands in positive characteristic, Preprint (2011), arXiv:1110.5707. [31] Yun, Z., Global Springer theory, Adv. Math., 228, 266-328, (2011) · Zbl 1230.14048
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