[Paper Review] On the Kirwan map for moduli of Higgs bundles

Emily Cliff, Thomas Nevins+1 more|arXiv (Cornell University)|Aug 30, 2018

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper proves that the Kirwan map from the cohomology of the moduli stack of G-bundles to that of semistable G-Higgs bundles fails to be surjective when the center Z(G) of the reductive group G is disconnected. Using a Borel-Quillen-style localization theorem for equivariant cohomology of stacks, the authors show that variant cohomology and intersection cohomology groups of the moduli space of Higgs bundles contain nontrivial representations of the finite group Bun(π₀(Z), C), which do not lift from the cohomology of the stack of G-bundles, thereby obstructing surjectivity of the Kirwan map.

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 $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big) ightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$ 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 $H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big) ightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$, 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.

Motivation & Objective

• To determine the conditions under which the Kirwan map from the cohomology of the moduli stack of G-bundles to that of semistable G-Higgs bundles is surjective.
• To investigate the role of the component group π₀(Z(G)) in obstructing surjectivity of the Kirwan map.
• To establish the existence of nontrivial invariant and variant cohomology in the moduli space of Higgs bundles when Z(G) is disconnected.
• To extend the understanding of cohomological structures in moduli stacks of Higgs bundles using equivariant cohomology and mixed Hodge theory.

Proposed method

• Applying a Borel-Quillen-style localization theorem for equivariant cohomology of stacks to reduce computations to fixed loci under the Gₘ-action.
• Constructing a G-bundle P₀ and associated Higgs pair (P₀, θ₀) to realize a closed immersion of Bun(Z, C) into the fixed locus of the Gₘ-action on Mss_Higgs(G, C).
• Using the action of Bun(π₀(Z), C) on the moduli stack and space of Higgs bundles to detect nontrivial cohomology classes not in the image of the Kirwan map.
• Analyzing the Leray spectral sequence for Gₘ-equivariant cohomology to relate cohomology of the fixed locus to the total cohomology, exploiting the pure Hodge structure on H∗_Gₘ(pt, Q).
• Proving that the regular representation of Bun(π₀(Z), C) appears in the pure part of the mixed Hodge structure on cohomology, implying failure of surjectivity.
• Applying the same argument to intersection cohomology by showing the fixed locus lies in the rationally smooth locus, so colimit reduces to singular cohomology of the fixed stack.

Experimental results

Research questions

• RQ1 Under what conditions on G is the Kirwan map from Bun(G, C) to Mss_Higgs(G, C) surjective?
• RQ2 How does the component group π₀(Z(G)) affect the cohomology of the moduli stack and space of semistable G-Higgs bundles?
• RQ3 Can variant cohomology and variant intersection cohomology detect obstructions to the surjectivity of the Kirwan map?
• RQ4 Does the regular representation of Bun(π₀(Z), C) appear in the pure part of the mixed Hodge structure on the cohomology of the moduli space of Higgs bundles?
• RQ5 To what extent does the Gₘ-action on the Higgs moduli stack reveal hidden cohomological structures not visible in the stack of G-bundles?

Key findings

• When π₀(Z(G)) is nontrivial, the variant cohomology H∗(Mss_Higgs(G, C)η)variant contains the quotient Q[Bun(π₀(Z), C)]/Q of the regular representation of Bun(π₀(Z), C).
• Similarly, the variant intersection cohomology IH∗(Mss_Higgs(G, C)η)variant also contains the quotient Q[Bun(π₀(Z), C)]/Q.
• The Kirwan map κη fails to be surjective for all η ∈ η₀ + π₁(Z₀), due to the presence of these nontrivial variant cohomology classes.
• The image of the pullback map p∗η: H∗(Mss_Higgs(G, C)η) → H∗(Mss_Higgs(G, C)η) is not contained in the image of the Kirwan map κ∗η, confirming a cohomological obstruction.
• The representation Q[Bun(π₀(Z), C)] appears in the pure part of the mixed Hodge structure on H∗(Mss_Higgs(G, C), Q), showing that the Kirwan map cannot surject onto the pure cohomology.
• The same result holds for the moduli space Mss_Higgs(G, C), with Bun(π₀(Z), C) acting nontrivially on its cohomology, while trivially on H∗(Bun(G, C), Q).