The geometry of Higgs bundles

ended

Higgs bundles play a fundamental role in the current panorama of mathematics and theoretical physics through their many connections. Amongst the latter is the link with the geometric Langlands programme, a suitable generalization of the relation between a curve and its Picard variety, which moreover admits a natural quantum field theoretical interpretation. According to this, any G- local system on a curve yields a perverse sheaf on the moduli stack of G*-bundles (where G* is the Langlands dual to G). A simpler (abelianised) version of the geometric Langlands programme has been proven for Higgs bundles by Donagi and Pantev. A programme initiated by these two scientists aims at inducing the full Langlands correspondence from its abelianised version. Building on the work of the researcher and the hosts, we will fill in the gaps of this program and provide alternative tools broadening the current state of the art also beyond this action. In doing so, we will study central elements of the geometry of Higgs bundles from a new perspective. More precisely, we will give a way to understand the Bialynicki-Birula stratification via algebraic techniques, and, related to that, carefully study the irreducible components of the nilpotent cone, applying also the theory of SU(p,q)-Higgs bundles. Finally, we will explore the case of positive characteristic, with the aim to shed light on the Hecke eigenproperty in this setting.