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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 $$C'$$.

##### MSC:
 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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##### References:
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