On the Kirwan map for moduli of Higgs bundles

Abstract

Let C be a smooth complex projective curve and G a connected complex reductive group. We prove that if the center Z(G) of G is disconnected, then the Kirwan mapHiggs (G, C), Q from the cohomology of the moduli stack of Gbundles to the moduli stack of semistable G-Higgs bundles, fails to be surjective: more precisely, the variant cohomology (and variant intersection cohomology) of M ss Higgs (G, C) is always nontrivial. We also show that the image of the pullback map H * M ss Higgs (G, C), Q → H * M ss Higgs (G, C), Q , from the cohomology of the moduli space of semistable G-Higgs bundles to the stack, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen-style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.

Intersection Cohomology of Rank 2 Character Varieties of Surface Groups

“…The discussion above shows that the sequences (8) are exact for C(X ,L ) and it is Aequivariant by the commutativity of (9). Taking invariants, we show then that (8) holds for C(X,L) unconditionally.…”

Intersection cohomology of rank two character varieties of surface groups

“…The discussion above shows that the sequences (8) are exact for C(X ′ , L ′ ), and it is A-equivariant by the commutativity of (9). Taking invariants, we show then that (8) holds for C(X, L) unconditionally.…”

“…As a final remark, note that the proof of Theorem 5.7 shows the more general statement that Kirwan surjectivity implies the tautological generation of the low-degree intersection cohomology for M (C, SL n ). However, this surjectivity is an open problem for n > 2; cf [9].…”