Mixed Hodge Components

• Mixed Hodge components are graded subquotients that encode the interaction of Hodge and weight filtrations in complex algebraic varieties.
• They underpin mixed Hodge structures in cohomology, offering insights into singularities, D-module behavior, and functorial properties.
• Applications include studies in mirror symmetry, birational geometry, and combinatorial models, aiding explicit computations and theoretical advances.

A mixed Hodge component refers to the particular graded subquotients within the structure imposed by mixed Hodge theory, encoding the interaction between the Hodge and weight filtrations on cohomological or (more generally) D-module objects arising in algebraic geometry. These components capture the algebraic and differential-geometric features of algebraic varieties and mixed Hodge modules, providing a refined tool for understanding their cohomology, singularities, and functorial behavior.

1. Mixed Hodge Structures: Filtrations and Components

A mixed Hodge structure (MHS) on a finite type module $H_A$ over $A = \mathbb{Z}, \mathbb{Q}$, or $\mathbb{R}$ consists of:

• An increasing weight filtration $W_\bullet$ on $H_A\otimes A\otimes \mathbb{Q}$,
• A decreasing Hodge filtration $F^\bullet$ on $H_\mathbb{C} := H_A \otimes A \otimes \mathbb{C}$,

such that for each $k$, the graded piece $$\operatorname{Gr}^W_k H_\mathbb{C} := W_kH_\mathbb{C} / W_{k-1}H_\mathbb{C}$$, together with the induced filtration $F^\bullet \operatorname{Gr}_k^W H_\mathbb{C}$, is a pure Hodge structure of weight $k$; that is, the filtrations $F^\bullet$ and its complex conjugate $\overline{F}^\bullet$ are $k$-opposed on $\operatorname{Gr}^W_k H_\mathbb{C}$ (Elzein et al., 2013).

The mixed Hodge components $H^{p,q}(\operatorname{Gr}^W_k H_\mathbb{C})$ are defined as

$$H^{p,q}(\operatorname{Gr}^W_k H_\mathbb{C}) := F^p \operatorname{Gr}_k^W H_\mathbb{C} \cap \overline{F^q \operatorname{Gr}_k^W H_\mathbb{C}},$$

and $$\operatorname{Gr}^W_k H_\mathbb{C} = \bigoplus_{p+q=k} H^{p,q}$$. For a general mixed Hodge structure $H$, there is a canonical linear-algebraic splitting $H_\mathbb{C} = \bigoplus_{p,q} I^{p,q}$ such that $I^{p,q}$ projects isomorphically onto $H^{p,q}(\operatorname{Gr}_{p+q}^W H_\mathbb{C})$ (Elzein et al., 2013).

2. Mixed Hodge Modules and Decomposition Theorems

The theory of mixed Hodge modules, due to Saito, encodes a compatible system comprising:

• a rational perverse sheaf,
• a filtered regular holonomic $\mathcal{D}$-module with a good $\mathcal{O}_X$-coherent Hodge filtration $F_\bullet$,
• weight filtrations on both modules,

satisfying strict compatibility conditions with respect to functors such as proper direct/inverse images, Verdier duality, and the nearby/vanishing cycle functors (Schnell, 2014).

Any mixed Hodge module $M$ admits a unique decomposition by strict support into submodules $M_Z$ supported on irreducible subvarieties $Z$. The associated graded objects for the weight and Hodge filtrations give mixed Hodge structures on the cohomology, and their double-graded pieces, $\operatorname{Gr}_k^W \operatorname{Gr}_p^F M$, are the mixed Hodge components (Schnell, 2014).

Key formulas include the spectral sequence

$$E_1^{p,q} = H^{p+q}(\operatorname{Gr}_{-p}^W \mathrm{DR}(M)) \Longrightarrow H^{p+q}(\mathrm{DR}(M)),$$

parallel to Deligne's sequence in classical mixed Hodge theory (Schnell, 2014).

3. Monodromic Decomposition and Mixed Hodge Components

In the context of monodromic mixed Hodge modules on $X \times \mathbb{C}_t$, the module decomposes as $M = \bigoplus_{\beta\in\mathbb{Q}} M^\beta$. The Hodge filtration splits accordingly:

$$F_p M = \bigoplus_{\beta} F_p M^\beta, \quad F_p M^\beta = F_p M \cap M^\beta.$$

Each $M^\beta$ is itself a (regular holonomic) $\mathcal{D}_X$-module on $X$ and carries induced Hodge and weight filtrations, endowing it with a (mixed) Hodge structure (Saito, 2020).

On each graded piece $\operatorname{gr}_k^W M$, this splitting persists, and the induced nilpotent monodromy acts as $$N: \operatorname{gr}_k^W M^\alpha \to \operatorname{gr}_{k-2}^W M^\alpha$$, so pure subobjects $M^\alpha$ carry pure Hodge structures of shifted weight. The direct-sum $\bigoplus_\beta M^\beta$ yields the mixed Hodge components of $M$ (Saito, 2020).

Notably, the Fourier–Laplace transform of a monodromic mixed Hodge module preserves monodromicity and the splitting of the Hodge and weight filtrations; the components transform via explicit shifts in indices (Saito, 2020).

4. Explicit Descriptions in Cohomological Contexts

In Deligne’s formulation for the cohomology of complex algebraic varieties, the weight and Hodge filtrations yield a Deligne splitting

$$H^k(X, \mathbb{C}) = \bigoplus_{p+q=k} H^{k,(p,q)}(X)$$

with

$$H^{k,(p,q)}(X) = \operatorname{Gr}^W_{p+q} H^k(X) \cap F^p H^k(X) \cap \overline{F^q H^k(X)}$$

(Zhang et al., 19 Aug 2025, Florentino et al., 2021). The dimensions $h^{p,q;k}(X) = \dim_\mathbb{C} H^{k;(p,q)}(X)$ are the mixed Hodge numbers; their vanishing or diagonalization (e.g., $h^{p,q;k}=0$ unless $p=q$) reveals the precise nature (e.g., mixed Tate) of the variety’s cohomology.

For structure algebras arising from combinatorics or arrangement complements, there is a concrete bigrading (e.g., $H^{p,q}(M) \simeq H^{-p,2q}(R_K)$ for a certain bigraded algebra $R_K$ in the context of moment-angle complexes or coordinate arrangement complements) (Eliyashev, 2014).

5. Mixed Hodge Components in D-Module and Hypersurface Settings

For holonomic $\mathcal{D}$-modules underlying mixed Hodge modules along a divisor $D = \{f=0\}$, the Kashiwara–Malgrange $V$-filtration interacts with the Hodge filtration via Beilinson-type formulas. The double-graded pieces

$$\operatorname{Gr}^W_k \operatorname{Gr}_p^F M = \frac{W_k M \cap F_p M}{W_k M \cap F_{p-1} M + W_{k-1} M \cap F_p M}$$

serve as the mixed Hodge components, carrying pure Hodge structures of weight $k$ and type $(p, k-p)$ (Davis et al., 20 Mar 2025).

The specialization functors (nearby/vanishing cycles) respect and often reflect these double gradings, and the corresponding components encode the geometry of singularities and the structure of multiplier and Hodge ideals. For example, the family of Hodge ideals $I_k(\alpha D)$ can be realized as the limit of mixed Hodge components as a parameter approaches infinity, connecting algebraic and Hodge-theoretic invariants (Davis et al., 20 Mar 2025).

6. Mixed Hodge Components in Explicit Models and Applications

In explicit computational contexts, such as character varieties of nilpotent groups, cluster varieties, or toric arrangements, the mixed Hodge numbers and components can often be calculated combinatorially or via recursive formulae. For varieties of Hodge–Tate or mixed Tate type, all off-diagonal mixed Hodge numbers vanish and the components are fully determined by the diagonal terms (Zhang et al., 19 Aug 2025, Florentino et al., 2021, Eliyashev, 2014).

For GKZ hypergeometric D-modules, the Hodge filtration coincides (up to shift) with the order filtration, and the only non-vanishing mixed Hodge components are determined by the principal grade (Reichelt et al., 2015).

7. Significance and Theoretical Implications

Mixed Hodge components encapsulate the core structure of cohomology and D-modules in algebraic geometry, mediating between topological, algebraic, and analytic perspectives. The ability to split, study, and compute these components underlies advances in:

• Mirror symmetry (e.g., non-commutative Hodge structures and quantum D-modules) (Reichelt et al., 2015),
• Birational and singularity theory (via Hodge and multiplier ideals) (Davis et al., 20 Mar 2025),
• Arithmetic and representation theory (via explicit formulae for mixed Hodge numbers of moduli spaces) (Florentino et al., 2021, Zhang et al., 19 Aug 2025),
• Arrangement complements and topological models (via bigraded algebraic models) (Eliyashev, 2014).

The mixed Hodge components thus provide a framework for both theoretical insights and explicit calculations across modern algebraic geometry and representation theory.