On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

Abstract

We explore the cohomological structure for the (possibly singular) moduli of SLn-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree > 2g − 2. We prove a support theorem for the SLn-Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree.Our main tool is an Ngô-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes. Contents 0. Introduction 1 1. Support theorems for Hitchin fibrations 5 2. Proper approximations and support theorems 8 3. The Hausel-Thaddeus conjecture 13 4. Vector bundles and Higgs bundles 18 References 21

Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients

“…If the bundles have rank two, then the corresponding Brill-Noether loci are of rank one, the theory of which is well-known [1]. This makes it feasible to obtain an explicit formula for the Poincaré series of moduli spaces of unstable Higgs bundles of rank two, a first step towards determining whether the cohomology of these moduli spaces is as rich as that of the moduli space of semistable Higgs bundles, which represents an active area of research (see for example [30,29,52,49]). For higher rank bundles, the centres of the blow-ups at each stage represent partial desingularisations of higher rank Brill-Noether loci, which are far from fully understood (see for example [10,11,50]); Theorem B may therefore help shed new light on these loci.…”

Hodge-to-singular correspondence for reduced curves

“…The cohomology of M (n, d) has been extensively studied in the literature, especially under the assumption that n and d are coprime, namely when M (n, d) is smooth; see for instance [40,33,36,35,38,54,31,16,54,19]. Recently studies about the intersection cohomology IH * (M (n, d), Q) of singular Dolbeault moduli spaces have started to emerge; see for instance [26,27,48,50,51,49,14,13,43,64]. 1 From this viewpoint, the decomposition theorem for the Hitchin map is a key tool to investigate the intersection cohomology of M (n, d): it allows to decompose IH * (M (n, d), Q) into building blocks, which are cohomology of some perverse sheaves on A n .…”

“…For convenience, we do not introduce hypertoric quiver varieties here, and we postpone their definition to Section 4 3. We recall the definition of perverse filtration in(49).…”