Non-Abelian Hodge Theory

• Non-Abelian Hodge theories are frameworks that establish equivalences between flat connections, Higgs bundles, and Hodge structures on complex manifolds.
• They utilize moduli spaces such as Betti, de Rham, and Dolbeault, interconnected by harmonic metrics and twistor space constructions to interpolate between settings.
• Recent generalizations extend these results to positive characteristic, twisted, and analytic contexts, impacting hyperbolicity, arithmetic geometry, and the Langlands program.

Non-Abelian Hodge theories are a set of deep results and frameworks providing equivalences between categories of flat connections (i.e., local systems or representations of the fundamental group), Higgs bundles, and variations of Hodge structure on complex manifolds, as well as their moduli spaces. Originating from the work of Simpson, Corlette, Donaldson, Hitchin, and Mochizuki, non-abelian Hodge correspondences generalize classical Hodge theory to nonlinear settings, revealing intricate relationships between complex geometry, algebraic topology, differential equations, and arithmetic geometry.

1. Fundamental Correspondence and Moduli Spaces

On a compact Kähler manifold $X$, the central theorem asserts a category equivalence:

• Semisimple flat bundles with vanishing Chern classes,
• Polystable Higgs bundles with vanishing Chern classes,
• Polystable $\lambda$-connections with vanishing Chern classes.

Given a flat bundle $(E,\nabla)$, there exists a unique harmonic metric $h$ if and only if $(E,\nabla)$ is semisimple. The corresponding Higgs bundle $(E, \phi)$ is polystable if and only if it admits a harmonic metric. The moduli spaces involved are:

• Betti space: parametrizes local systems (representations of $\pi_1$) modulo conjugation ($M_B$),
• de Rham space: moduli of flat connections ($M_{dR}$),
• Dolbeault space: moduli of semistable Higgs bundles ($M_{Dol}$),
• Hodge space: moduli of $\lambda$-connections, interpolating between flat bundles and Higgs bundles ($M_{Hod}$) (Huang, 2019).

These moduli spaces are interconnected via real analytic diffeomorphisms; notably, the twistor space construction gives a deformation parameter $\lambda$, interpolating from purely Higgs ($\lambda=0$) to flat bundle ($\lambda=1$).

2. Simpson’s Non-Abelian Hodge Theory on Curves, Orbifolds, and Generalizations

For Riemann surfaces and orbifold curves $Y = \Gamma \backslash \mathbb{H}$ compactified to $X = \Gamma \backslash \mathbb{H}^*$, the nonabelian Hodge correspondence is refined to include local monodromy data at cusps (parabolic structures and filtrations) (Franc et al., 2018).

The correspondence relates:

• Filtered local systems (complex representations of $\Gamma$ with decreasing filtration at cusps),
• Regular, filtered flat connections with logarithmic poles (Deligne extensions),
• Filtered regular Higgs bundles with periodic filtration and nilpotent residue.

Moduli spaces $M_{Betti}$, $M_{dR}$, and $M_{Dol}$ share the same underlying complex structure and stratification by the action of $\mathbb{C}^*$ on the Higgs field, notably via the nilpotent cone of the Hitchin map.

Extensions to Fujiki class $\mathcal{C}$ manifolds rely on bimeromorphic Kähler modifications and descent of numerically flat bundles (Biswas et al., 2020).

3. Non-Abelian Hodge Theories in Positive Characteristic

Non-abelian Hodge theory admits analogues for algebraic curves in characteristic $p>0$. The fundamental equivalence relates:

• Flat $G$-connections (local systems) on $C$,
• $G$-Higgs bundles on the Frobenius twist $C'$.

A key result is the existence of an étale-local equivalence of moduli stacks over the Hitchin base (parametrized by Chevalley-invariants of the Higgs field or $p$-curvature). The correspondence uses spectral data, centralizer group schemes, and Azumaya algebras over spectral curves. The isomorphism is generally twisted by a Picard stack of splittings (Chen et al., 2013, Herrero et al., 2023). In positive characteristic, cohomological correspondences are established between Dolbeault and de Rham moduli spaces (filtered isomorphisms for perverse Leray filtrations) (Cataldo et al., 2021).

Logarithmic versions (admitting regular singularities) are constructed via stacky Frobenius twists and logarithmic de Rham stacks, extending Cartier descent and enabling a full stacky algebraic framework for non-abelian Hodge theory in characteristic $p$. The resulting correspondences encode boundary behaviors without extra parabolic data (Barz, 3 Dec 2025, Cataldo et al., 16 Jan 2025).

Twisted functoriality in the $p$-adic setting is also established: pullback of semistable Higgs bundles with vanishing Chern classes preserves semistability under twisted pullbacks reflecting obstruction classes in deformation/lifting theory (Sheng, 2021).

4. Categorical, Twisted, and Simplicial Generalizations

Recent advances extend non-abelian Hodge correspondences to moduli stacks parametrizing quiver-shaped diagrams of bundles, connections, and Higgs fields. Algebraic Artin stacks of diagrams over Dolbeault, de Rham, and Hodge geometries are shown to be well-behaved, supporting categorified versions of the NAH correspondence (Simpson’s correspondence on objects, morphisms, and higher diagrams) (Azam et al., 13 Dec 2025).

Twisted nonabelian Hodge theory (with torsors and gerbes) is formalized, establishing equivalences between categories of twisted flat connections (modules over sheaves of twisted differential operators, e.g., Beilinson–Bernstein $D_{X,\alpha}$) and twisted semistable Higgs bundles via principal $\infty$-bundles and and mapping stacks. These constructions directly serve geometric Langlands, twisted D-modules, and Fourier–Mukai dualities (Garcia-Raboso, 2015).

5. Hodge Filtrations, Cohomology, and Perverse Structures

A central insight is the identification of filtrations on cohomology of character varieties and moduli spaces:

• The weight filtration on the Betti side (via mixed Hodge structure; Deligne),
• The perverse Leray filtration induced by the Hitchin map on the Dolbeault moduli.

For $G = GL_2, PGL_2, SL_2$, the perverse and weight filtrations are shown to coincide under the non-Abelian Hodge correspondence (Cataldo, 2010). The decomposition theorem and Ngô’s support theorem establish the perverse–weight equivalence, a phenomenon anticipated in higher rank.

Direct images in non-Abelian Hodge theory (from families of curves with tame harmonic bundles or tame polystable parabolic Higgs bundles) can be described algebraically. The $L^2$ Dolbeault direct image is computed via modifications accounting for monodromy weight filtrations, using $V$-filtration machinery on D-modules and twistor modules, reconciling cohomological invariants across the correspondence (Donagi et al., 2016).

Vanishing theorems (Kodaira, Kawamata–Viehweg, Saito) admit generalizations in the non-abelian context using mixed twistor D-modules and their functorial properties. These results unify classical statements, analytic and algebraic, within the framework of non-abelian Hodge theoretic formalism (Wei, 2022).

6. Extensions to Real, Analytic, and Dynamic Structures

Analytic non-abelian Hodge theory considers completions of the fundamental group in the context of Banach and C*-algebras. The category of pluriharmonic local systems in Hilbert spaces corresponds to -representations of a universal pro-C-algebra, which carries a continuous $S^1$-action giving pure non-abelian Hodge structures (weight 0) as dynamical systems. Associated cohomological theories exhibit "Hodge and twistor splittings" and a principle of two types (Pridham, 2012).

7. Impact on Hyperbolicity, Arithmetic, and Langlands Program

Non-abelian Hodge theoretic methods yield major results on hyperbolicity. For quasi-projective varieties $X$, the existence of "big" or "large" reductive representations of $\pi_1(X)$ (via the Simpson–NAH correspondence) ensures that outside a proper exceptional locus $Z\subset X$, all subvarieties are of log general type, and entire curves are contained in $Z$ (Cadorel et al., 17 Dec 2025). The spectral data of Higgs bundles, via the construction of symmetric log differentials, are central for the proof of the Green–Griffiths–Lang conjectures in these non-abelian settings.

Geometric Eisenstein series methods extend the Langlands philosophy to non-abelian Hodge moduli; decompositions of nilpotent sheaves on Dolbeault, Hodge, and twistor moduli spaces (and their hyperholomorphic branes) echo spectral–cuspidal decompositions, fitting with Kapustin–Witten duality, and providing the categorical infrastructure for Dolbeault Langlands and hyperkähler S-duality conjectures (Hanson, 30 Dec 2025).

8. Fixed Part Theorem and Rigidity Loci

The non-abelian analogue of Deligne’s Fixed Part theorem establishes that a finite monodromy orbit of a polarized variation of Hodge structure over the fibers of a locally trivial fibration implies the existence of an extension of the local system to the total space, preserving Mumford–Tate group and Hodge filtration—characterized via isomonodromic extension of local systems and Hodge-theoretic rigidity (Esnault et al., 2024).

Select foundational and recent contributions:

• (Franc et al., 2018) Nonabelian Hodge theory and vector valued modular forms.
• (Huang, 2019) Non-Abelian Hodge Theory and Related Topics.
• (Biswas et al., 2020) Nonabelian Hodge theory for Fujiki class $\mathcal C$ manifolds.
• (Cataldo et al., 2021) A Cohomological Non Abelian Hodge Theorem in Positive Characteristic.
• (Herrero et al., 2023) Semistable Non Abelian Hodge theorem in positive characteristic.
• (Esnault et al., 2024) A non-abelian version of Deligne's Fixed Part Theorem.
• (Cadorel et al., 17 Dec 2025) Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories.
• (Hanson, 30 Dec 2025) Geometric Eisenstein series in non-abelian Hodge theory and hyperholomorphic branes from supersymmetry.
• (Azam et al., 13 Dec 2025) Moduli stacks of quiver connections and non-Abelian Hodge theory.