--- title: Nonabelian Hodge Filtration url: https://www.emergentmind.com/topics/nonabelian-hodge-filtration type: topic --- # Nonabelian Hodge Filtration Nonabelian Hodge filtration designates a family of filtration-theoretic structures that arise when the de Rham, Dolbeault, and Betti realizations of nonabelian Hodge theory are compared. In the most literal moduli-theoretic sense, it is the Griffiths-transverse filtration encoded by Simpson’s Hodge deformation, whose associated graded is the Dolbeault moduli; in adjacent settings, the same phrase refers to weighted filtrations at punctures, irregular Hodge filtrations on transformed \(D\)-modules, mixed Hodge filtrations on nonabelian homotopy data, and positive-characteristic graded or twisted constructions that serve as substitutes for a classical Hodge filtration [2509.06050] [2205.15475] [2204.13381] [2105.01385]. ## 1. Hodge moduli and the canonical filtration picture The classical nonabelian Hodge template is the identification of de Rham and Dolbeault moduli. In stack-theoretic form, for a smooth complex projective variety \(X\) and a linear algebraic group \(G\), one has \[ \Map\!\big(X_{\mathrm{dR}},BG\big)\simeq \Map\!\big(X_{\mathrm{Dol}},BG\big)^{\mathrm{ss},0}, \] so flat \(G\)-torsors correspond to semistable Higgs \(G\)-torsors with vanishing Chern data. This equivalence is the untwisted baseline from which more refined filtration statements are extracted [1501.05872]. A more precise filtration formalism appears on the relative Hodge moduli. For a smooth projective morphism \(\alpha:X\to S\) and a connected reductive group \(G\), the Hodge moduli space \(M^{sm}_{Hod}(X/S,G)\) carries a smooth morphism \[ \tilde f:M^{sm}_{Hod}(X/S,G)\to S\times \mathbb A^1 \] with a \(\mathbb G_m\)-action compatible with the standard action on \(S\times\mathbb A^1\); the fiber over \(S\times\{1\}\) is the relative de Rham moduli and the fiber over \(S\times\{0\}\) is the relative Dolbeault moduli. In this framework, \(\mathbb G_m\)-equivariant quasi-coherent sheaves on the Hodge side are equivalent to objects with a Griffiths transverse filtration, and the associated graded object is the Dolbeault moduli. The nonabelian Gauss–Manin connection extends to a \(\mathbb G_m\)-equivariant morphism \[ \tilde \nabla:\tilde f^*T_{S\times \mathbb A^1/\mathbb A^1}(-S\times\{0\})\to T_{M^{sm}_{Hod}(X/S,G)/\mathbb A^1}, \] and its associated graded map is the nonabelian Kodaira–Spencer map \[ \theta_{KS}=\tau\circ \rho_{KS}:T_S\to g_*T_{M^{sm}_{Dol}(X/S,G)/S}, \] which satisfies \([\theta_{KS},\theta_{KS}]=0\) and is \(\mathbb G_m\)-graded. This is the clearest instance in which “nonabelian Hodge filtration” is an actual filtration attached to the Hodge deformation itself [2509.06050]. ## 2. Weighted filtrations on curves and parahoric structures On noncompact curves, filtration data occur directly on the objects of the correspondence. For a Fuchsian or modular group representation, a filtered representation \((\rho,F)\) is equipped, at each cusp \(s\), with a decreasing exhaustive left-continuous filtration \(F_{\beta,s}V\), with associated graded pieces \[ \Gr_{\beta,s}(\rho,F)=\frac{F_{\beta,s}V}{\bigcup_{\alpha>\beta}F_{\alpha,s}V}, \qquad \Gr_s(\rho,F)=\bigoplus_{\beta\in\mathbb R}\Gr_{\beta,s}(\rho,F). \] The filtered degree and slope are \[ \deg(\rho,F)=\sum_{s\in S}\sum_{\beta\in\mathbb R}\beta\,\dim_{\mathbb C}\Gr_{\beta,s}(\rho,F), \qquad \mu(\rho,F)=\frac{\deg(\rho,F)}{\dim_{\mathbb C}V}. \] The same formalism appears for filtered bundles \(E_{\beta,s}\), filtered regular connections, and filtered regular Higgs bundles; a tame harmonic metric then mediates between the connection and Higgs descriptions. In this context the “nonabelian Hodge filtration” is not a single global filtration on cohomology, but the package of cusp filtrations that controls residues, growth, degrees, and stability [1812.06180]. For a general connected complex reductive group \(G\), the appropriate replacement of parabolic structure is parahoric. One fixes weights \[ \theta=\{\theta_x\}_{x\in D},\qquad \theta_x\in Y(T)\otimes_{\mathbb Z}\mathbb R, \] and defines the local parahoric subgroup \[ G_\theta(K)=\{g(z)\in G(K)\mid z^\theta g(z)z^{-\theta}\ \text{has a limit as }z\to 0\}. \] These local groups glue to a Bruhat–Tits group scheme \(G_\theta\) on the compact curve, and the algebraic Dolbeault and de Rham objects become logahoric \(G_\theta\)-Higgs torsors and logahoric \(G_\theta\)-connections. Stability is measured by parahoric degree, and the categorical correspondence identifies Dolbeault, de Rham, and Betti sides after matching local weight, residue, and monodromy data. The local transformation laws \[ \beta=\alpha-(S_\alpha+S_\alpha),\qquad \gamma=-(S_\alpha+S_\alpha) \] show that the weight system itself is transported through the harmonic metric. Here the filtration is the mechanism that makes the noncompact correspondence work beyond the \(GL_n(\mathbb C)\)-case [2205.15475]. ## 3. Irregular and hypergeometric Hodge filtrations In the \(D\)-module setting, nonabelian Hodge filtration frequently appears as an irregular Hodge filtration. For a monodromic \(D_E\)-module \(M\) on a vector bundle \(E\to X\), the Euler vector field \(E_E\) yields a decomposition \[ M=\bigoplus_{\beta\in\mathbb R}M^\beta, \qquad M^\beta=\bigcup_{l\ge 0}\ker\!\left((E_E-\beta)^l:\pi_*M\to\pi_*M\right), \] and if \(M\) underlies a monodromic mixed Hodge module then its Hodge filtration splits compatibly: \[ F_pM=\bigoplus_{\beta\in\mathbb R}F_pM^\beta, \qquad F_pM^\beta=F_pM\cap M^\beta. \] The Fourier–Laplace transform \(M^\wedge\) again carries a mixed Hodge module structure, and the irregular Hodge filtration is computed by \[ {}_{\alpha+p}M^\wedge = \bigoplus_{\beta\in\mathbf R}F_{p+\lfloor \alpha-\beta\rfloor}M^\beta, \qquad \alpha\in[0,1),\ p\in\mathbf Z. \] At integer indices one has the comparison theorem \[ {}_pM^\wedge = F_pM^\wedge, \] so the irregular Hodge filtration recovers the Hodge filtration on the Fourier–Laplace transform at all integer levels [2204.13381]. Hypergeometric systems provide explicit computational models for these filtrations. For GKZ systems \(M_A^\beta\), the Hodge filtration is identified with the order filtration up to shift; under the normality and homogeneity assumptions on \(\widetilde A\), \[ F^H_{p+(n-d)}M_{\widetilde A}^{\beta}=F_p^{\mathrm{ord}}M_{\widetilde A}^{\beta}, \] and this explicit control is used to produce variations of pure polarized non-commutative Hodge structures on reduced quantum \(D\)-modules of nef complete intersections in toric varieties [1503.01004]. For non-resonant hypergeometric connections with rational parameters, the irregular Hodge filtration is computed geometrically from a Laurent polynomial model and a Newton polytope filtration; its jumps occur at the values \[ \theta(k)=(n-m)\alpha_k+\#\{i\mid \beta_i<\alpha_k\}+(n-k)-\sum_{i=1}^n\alpha_i+\sum_{j=1}^m\beta_j, \] and the graded ranks satisfy \[ \operatorname{rk}\,\mathrm{gr}^p_{F_{\irr}}H(\alpha;\beta)=\#\{k\mid \theta(k)=p\}. \] The same analysis implies that certain hypergeometric sums are everywhere ordinary on \(|\mathbb G_{m,\mathbb F_p}|\), in the sense that the Frobenius Newton polygon equals the irregular Hodge polygon [2308.05138]. ## 4. Positive-characteristic substitutes for Hodge filtration In characteristic \(p>0\), the inverse Cartier transform of Ogus–Vologodsky replaces the classical de Rham–Dolbeault bridge, but the filtration picture becomes subtler. For a morphism of log smooth varieties \[ f:Y^{\log}\to X^{\log} \] with chosen \(W_2(k)\)-liftings, twisted functoriality is expressed by a twisted pullback \(f^\circ(E,0)\) built from the obstruction class \(\mathrm{ob}(f)\in \mathrm{Ext}^1(f^*\Omega^1_{X/k},\mathcal O_Y)\), and the inverse Cartier transform satisfies \[ C^{-1}_{Y^{\log}}\circ f^\circ \cong f^*\circ C^{-1}_{X^{\log}} \quad \text{on } \mathrm{HIG}^{