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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1808.10311 |
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Table of Contents:
- 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 $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$ 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 $\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$, 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.