Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone

Abstract

Let G be a real reductive Lie group, and H C the complexification of its maximal compact subgroup H ⊂ G. We consider classes of semistable G-Higgs bundles over a Riemann surface X of genus g 2 whose underlying H C -principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of G-Higgs bundles over X to the moduli space of H C -bundles over X, in contrast with the situation when g = 1, and to show reducibility of the nilpotent cone of the moduli space of G-Higgs bundles, for G complex.1991 Mathematics Subject Classification. 14H60, 32L05.

Flawed groups and the topology of character varieties

“…A hyperbolic surface group Γ (the fundamental group of a closed orientable surface Σ of genus g ≥ 2) is flawless. This follows from Theorems 3.15 and 3.12 in [FGN19], together with the fact that X Γ (G) is homeomorphic to the moduli space of G-Higgs bundles of trivial topological type over a Riemann surface with underlying topological surface Σ (see also Remark 2.8 below).…”

“…Generalizing the abelian case, in part by combining methods used in [FL09] and [FL14], the result is also shown to be true for all finitely generated nilpotent groups Γ in [Ber15]. On the other hand, whenever Γ is the fundamental group of a closed orientable surface of genus g ≥ 2 (called a hyperbolic surface group henceforth) such a deformation retraction (indeed, even a homotopy equivalence) is impossible, as follows from [BF11,FGN19].…”

“…(2) We know that closed hyperbolic surface groups are flawless and that was shown using Higgs bundle theory. Given the work in [FGN19], we conjecture that all non-abelian Kähler groups are flawless. (3) A group is supersolvable if it admits an invariant normal series where all the factors are cyclic groups.…”