Nonabelian Hodge Theory for Stacks

• Nonabelian Hodge Theory for Stacks is an advanced framework that extends the classical correspondence to incorporate stack structures and higher derived geometry.
• It establishes enriched correspondences among moduli stacks of flat connections, Higgs bundles, and local systems, detailing the roles of Hodge and weight filtrations.
• By bridging classical, positive, and p-adic settings, the theory offers new computational tools and insights for representation theory and arithmetic geometry.

Nonabelian Hodge theory for stacks extends the nonabelian Hodge correspondence from moduli spaces to moduli stacks, integrating stack-theoretic structures, higher derived geometry, and the interplay of nonabelian mixed Hodge structures (MHS) in both classical and nonclassical (e.g., positive or mixed characteristic, $p$-adic) settings. It establishes enriched correspondences between moduli stacks of flat connections, Higgs bundles, and local systems—incorporating actions of group-stacks, Postnikov towers, and derived mapping stacks—with deep implications for cohomology, representation theory, and arithmetic geometry.

1. Brill–Noether Stacks and Nonabelian Hodge Cohomology

Key objects are Brill–Noether stacks, denoted $B(S,V)$ for a reductive group $S$ over $\mathbb{C}$ and a finite-dimensional $S$-representation $V$ (with rational action $\rho: S\to\mathrm{GL}(V)$), of Postnikov type characterized by

$$\pi_1\bigl(B(S,V)\bigr) = S,\qquad \pi_n\bigl(B(S,V)\bigr) = V$$

and all other homotopy sheaves vanishing. The fibration sequence

$$(V, n)\xrightarrow{i} B(S, V) \xrightarrow{\tau} (S, 1)$$

realizes $(V,n)$ as an Eilenberg–MacLane stack and $(S,1)$ as the classifying stack of $S$-torsors, with $\rho$ encoding the monodromy of the $V$-local system over $BS$ (Boutchaktchiev, 2013).

Given a smooth complex algebraic variety $X$, nonabelian cohomology with coefficients in $B(S,V)$ is defined as the derived mapping stack

$\mathbb{H}^*(X,B(S,V)) := \Hom\bigl(X, B(S,V)\bigr)$

which parameterizes triples $(E, \theta, \eta)$: $(E, \theta)$ a flat principal $S$-bundle and $\eta \in H^n\bigl(X, E \times^S V\bigr)$ a cohomology class in the associated local system. This gives rise to a stratified fibration over Simpson's moduli of flat $S$-bundles.

2. Construction of Nonabelian Mixed Hodge Structures for Stacks

The Brill–Noether framework structures cohomology groups $H^i\bigl(X, B(S, V)\bigr)$ with two canonical filtrations:

• Hodge filtration $F^\bullet$: Derived from a natural $\mathbb{C}^*$-action (scaling Higgs fields), inducing a grading where $$F^pH^i = \bigoplus_{r\ge p} H^i\bigl(X, B(S, V)\bigr)^{(r)}$$, the $t$-eigenspace for $t$ acting by $t^r$.
• Weight filtration $W_\bullet$: Constructed from the lower central series of the pro-unipotent radical of the relative completion of $\pi_1(X)$ in $S$, combined with the Whitehead bracket induced by the stack fibration (Boutchaktchiev, 2013). For the associated Lie algebra $\mathfrak{g}$,

$W_m \mathfrak{g} = \Gamma_{-m+1} \mathfrak{g}$

where $\Gamma_k$ is the $k$th central series term.

The associated-graded groups for the weight filtration decompose as

$\mathrm{Gr}^W_k H^i \cong \begin{cases} H^i\bigl(X, \mathrm{gr}^W_\ast(\mathfrak{g})\bigr) & \text{(nonabelian part)} \ H^{i-n}\bigl(X, E^S V\bigr) & \text{(%%%%33%%%%-abelian part)} \end{cases}$

and each carries a pure Hodge structure of weight $k$.

In the vicinity of $\mathbb{C}^*$-fixed points in $\Hom(X, B(S, V))$, the pro-representing local algebra is the completed symmetric algebra on $H^1(X, \mathrm{Ad} E) \oplus H^n(X, E^S V)$, with the induced (local) MHS.

3. Cohomological Hall Algebras and Isomorphisms for Stacky Moduli

Nonabelian Hodge theory for stacks is fundamentally realized in the structure of Borel–Moore homology and cohomological Hall algebras (CoHA) of the Dolbeault (Higgs), de Rham (connection), and Betti (local system) stacks:

• $\mathcal{M}_{\text{Dol}}(X)$: stack of semistable Higgs bundles.
• $\mathcal{M}_{\text{dR}}(X)$: stack of flat connections.
• $\mathcal{M}_{\text{B}}(X)$: stack of local systems.

Canonical isomorphisms of Borel–Moore homology exist: $$H^{\mathrm{BM}}_*(\mathcal{M}_{\text{Dol}}(X)) \cong H^{\mathrm{BM}}_*(\mathcal{M}_{\text{dR}}(X)) \cong H^{\mathrm{BM}}_*(\mathcal{M}_{\text{B}}(X))$$ with the CoHA structure (constructed via a correspondence diagram involving short exact sequence stacks and virtual pullbacks) being preserved under these isomorphisms (Hennecart, 2023).

These Hall algebras realize PBW-type theorems, relate to BPS Lie algebras (free Lie algebras generated by intersection complexes of simple loci), and exhibit integrality and positivity phenomena in the cohomology of stacky moduli (Davison et al., 2022, Davison, 2021).

4. Positive and Mixed Characteristic Theory and Stacks

In positive characteristic, nonabelian Hodge correspondences are constructed for stacks of principal bundles and parahoric torsors, building on the Ogus–Vologodsky and Lan–Sheng–Zuo framework. For a $W_2(k)$-liftable variety $X/k$ (perfect field, $p>0$), and a split reductive $G$: $$\operatorname{Higgs}_{G,p-1}(X') \simeq \operatorname{Flat}_{G,p-1}(X)$$ where the equivalence is constructed stack-theoretically (allowing nilpotent $p$-curvature/Higgs field exponent $\leq p-1$). This correspondence extends to root stacks and parahoric torsors and preserves Ramanathan $R$-stability (Sheng et al., 16 May 2024).

The geometric realization in positive characteristic also enables reduction-to-$p$ techniques for algebraic hyperbolicity of moduli stacks (e.g., of Calabi–Yau varieties), using semistability of the Higgs bundle associated via the Cartier transform (Brunebarbe, 2022).

5. $p$-adic Nonabelian Hodge Theory and Stacks

For $X$ a smooth projective curve over $\mathbb{C}_p$ and $G$ a reductive group, $p$-adic nonabelian Hodge correspondences at the stack level are established via twisted isomorphisms: $\psi: H \times^P M_H \to M_v$ where $M_H$ is the $G$-Higgs moduli stack, $M_v$ the v-stack of $G$-torsors in the v-topology, and $H$ a torsor for the Picard stack $P$ of regular centralizer $J$-torsors over the Hitchin base. The twist is canonical and accounts for the "mysterious exponential" of classical $p$-adic Simpson theory. This construction defines a geometric homeomorphism of moduli spaces, embeds $p$-adic representation varieties as open substacks, and globalizes the $p$-adic Simpson correspondence to stacks (Heuer et al., 2 Feb 2024).

Related constructions with the Hodge–Tate stack and derived Simpson functors refine the correspondence from vector bundles to the level of perfect complexes and incorporate Higgs–Sen enhancements in mixed characteristic (Anschütz et al., 2023).

6. Nonabelian Hodge Theory for Stacks: Consequences and Applications

This stacky theory sharpens classical phenomena and conjectures:

• Filtration phenomena: The stacky P=W conjecture asserts that under the isomorphism of Borel–Moore homologies, the mixed Hodge weight filtration on the Betti stack matches the perverse filtration (with respect to the Hitchin map) on the Dolbeault stack, recovering the classical results for moduli spaces and generalizing to stack-theoretic settings (Davison, 2021).
• CoHA and BPS structures: The symmetric algebra structure on the Borel–Moore homology reflects the decomposition theorem for 2-Calabi–Yau categories, with primitives corresponding to intersection cohomology classes; positivity and integrality are encoded in the structure of the associated Lie and Hall algebras (Davison et al., 2022).
• Explicit formulas: Iterated integral descriptions for nonabelian cohomology and local MHS at stack points make explicit the Hodge decomposition and connect to schematic homotopy types (Boutchaktchiev, 2013, Boutchaktchiev, 2013).
• Behavior in positive characteristic: The ring-level isomorphisms of $\ell$-adic cohomology for moduli stacks of Higgs bundles and connections are established, compatible with cup product and localization on nilpotent cones, with explicit exploitations of $\mathbb{G}_m$-actions (Herrero et al., 6 Aug 2024).

7. Generalizations and Future Directions

Natural directions extend Brill–Noether and related stacks to more general Postnikov towers, allowing multiple nontrivial homotopy groups and higher Whitehead products. Conjecturally, each stage supports a pro-unipotent group with mixed Hodge structures whose interactions are governed by higher analogues of the Whitehead bracket, and the associated nonabelian period maps and $p$-adic variations. Stacky methods make concrete the structure of nonabelian Hodge theory for richer moduli problems—higher connections, arithmetic moduli, and higher algebraic stacks—while providing explicit computational tools based on iterated integrals (Boutchaktchiev, 2013).