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 map $H^{\ast }(\mathrm{Bun}(G,C),\mathbb{Q})\to H^{\ast }({\mathcal{M}}_{\mathrm{Higgs}}^{\mathrm{ss}},\mathbb{Q})$ from the cohomology of the moduli stack of $G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the "variant cohomology" (and variant intersection cohomology) of the stack ${\mathcal{M}}_{\mathrm{Higgs}}^{\mathrm{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^{\ast }(M_{\mathrm{Higgs}}^{\mathrm{ss}},\mathbb{Q})\to H^{\ast }({\mathcal{M}}_{\mathrm{Higgs}}^{\mathrm{ss}},\mathbb{Q})$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, 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.

References in corpus (2)

• Kirwan surjectivity for quiver varieties
• Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure

Cited by in corpus (1)

• Aspects of the topology and combinatorics of Higgs bundle moduli spaces