Autoduality of the Hitchin system and the geometric Langlands programme

This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.This thesis is not currently available in OR