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Non-abelian Hodge theory for algebraic curves in characteristic \(p\). (English) Zbl 1330.14015
Let \(C\) be a Riemann surface, and \(G_c\) be a compact Lie group with its complexification \(G\). In this case, one gets a bijection between stable \(G\)-Higgs bundles with vanishing Chern classes and irreducible \(G\)-local systems on \(C\). This is what C. T. Simpson called in [Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992; Zbl 0814.32003)] the non-abelian Hodge theory for \(C\). In the present paper the authors try to establish the non-abelian Hodge theory in characteristic \(p>0\) in the following way. Let \(G\) be a reductive group over an algebraically closed field \(k\) of characteristic \(p\), and \(C\) be a smooth projective curve over \(k\) with \(C'\) its Frobenius twist. They give a description of the moduli space of flat \(G\)-bundles in terms of that of \(G\)-Higgs bundles over \(C'\).

14D20 Algebraic moduli problems, moduli of vector bundles
14D23 Stacks and moduli problems
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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[1] A. Beauville and M.S. Narasimhan, S. Ramanan. Spectral curves and the generalised theta divisor. Journal für die Reine und Angewandte Mathematik, 398 (1989), 169-179 · Zbl 0666.14015
[2] A. Beilinson and V. Drinfeld. Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint. Available at http://www.math.uchicago.edu/ mitya/langlands/. · Zbl 0864.14007
[3] R. Bezrukavnikov and A. Braverman. Geometric Langlands conjecture in characteristic p: The GL_{\(n\)} case. Pure Applied Mathematics Quarterly, (1)3 (2007), 153-179. (Special Issue: In honor of Robert D. MacPherson, Part 3) · Zbl 1206.14030
[4] Bost, J.B., Algebraic leaves of algebraic foliations over number fields, Publications Mathmatiques Institut des Hautes Études Scientifiques, 93, 161-221, (2001) · Zbl 1034.14010
[5] B. Conrad, O. Gabber, and G. Prasad. Pseudo-Reductive Groups. New Mathematical Momographs, Vol. 17 (2010). · Zbl 1216.20038
[6] T.H. Chen and X. Zhu. Geometric Langlands in prime characteristic. (2014). arXiv:1403.3981. · Zbl 1390.14044
[7] Donagi, R.; Gaitsgory, D., The gerbe of Higgs bundles., Transformation Groups, 7, 109-153, (2002) · Zbl 1083.14519
[8] M. Groechenig. Moduli of flat connections in positive characteristic. (2012). arXiv:1201.0741. · Zbl 1368.14021
[9] L. Illusie. Complexe de deRham-Witt et cohomologie cristalline. Annales scientifiques de lÉcole Normale Supérieure, (4)12 (1979), 501-661 · Zbl 0436.14007
[10] Katz, N., Nilpotent connections and the monodromy theorem., Publications Mathmatiques Institut des Hautes Études Scientifiques, 39, 175-232, (1970) · Zbl 0221.14007
[11] Y. Laszlo and C. Pauly: On the Hitchin morphism in positive characteristic. International Mathematics Research Notices 3 (2001), 129-143 · Zbl 0983.14004
[12] B.C. Ngô. Hitchin fibration and endoscopy. Inventiones Mathematicae 164 (2006), 399-453 · Zbl 1098.14023
[13] B.C. Ngô. Le lemme fondamental pour les algèbres de Lie. Publications Mathmatiques Institut des Hautes Études Scientifiques 111 (2010), 1-169 · Zbl 0221.14007
[14] Ogus, A., Higgs cohomology, \(p\)-curvature, and the cartier isomorphism, Compositio Mathematica, 140, 145-164, (2004) · Zbl 1055.14021
[15] A. Ogus and V. Vologodsky: Nonabelian Hodge theory in characteristic p. Publications Mathmatiques Institut des Hautes Études Scientifiques 106 (2007), 1-138 · Zbl 1140.14007
[16] Simpson, C., Higgs bundles and local systems, Publications Mathmatiques Institut des Hautes Études Scientifiques, 75, 5-95, (1992) · Zbl 0814.32003
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