Bigraded Mixed Hodge Structures

• Bigraded mixed Hodge structures are defined by the interplay between the weight and Hodge filtrations, refining the classical Hodge decomposition.
• They employ spectral sequences, residue isomorphisms, and Gysin maps to assemble local geometric data into a comprehensive global invariant.
• These structures provide practical tools for computing L2-cohomology, linking analytic methods with algebraic invariants in complex geometry.

Bigraded structures in the theory of mixed Hodge structures encapsulate the fine interaction between two key filtrations—the weight and Hodge filtrations—on the cohomology of complex algebraic and analytic spaces. These bigradings extend the classical (pure) Hodge decomposition to the richer context of open, singular, or non-compact varieties, as well as to analytic invariants arising in settings such as coverings of Kähler or complex manifolds. The dual filtration framework enables refined decomposition theorems, functorial operations, and the assembly of local geometric data via spectral sequences and residue-Gysin maps. This article provides an exhaustive account of bigraded structures within mixed Hodge theory, drawing on intricate examples and analytic techniques established in the literature.

1. Filtrations and the Construction of Bigraded Mixed Hodge Complexes

A mixed Hodge structure on a complex or real vector space $V$ consists of an increasing weight filtration $W_\bullet$ and a decreasing Hodge filtration $F^\bullet$ on $V_\mathbb{C}$. For mixed Hodge complexes arising from geometric or analytic data, such as the $L^2$-cohomology of a covering manifold $p : \widetilde{X} \to X$ with a normal crossing divisor $D$, one considers the double-graded complex

$(p^*(2)A^\bullet(X, \log D), d)$

where forms are square-integrable ($L^2$) and may have logarithmic singularities along $D$. The bigrading manifests via:

• Weight filtration $W$: Defined by the order of logarithmic poles, locally in a chart where $D = \{z_1 z_2 \cdots z_k = 0\}$, forms in $W_m$ involve at most $k-m$ logarithmic differentials.
• Hodge filtration $F$: In the Kähler case, the $(p,q)$ decomposition gives $F^p$ as forms with at least $p$ holomorphic differentials.

The structure is rigorously constructed only in suitable quotient categories Mod($N(G)$)/$T$, where $N(G)$ is the von Neumann algebra associated to the covering group $G$, and $T$ is a torsion theory (such as $T_\mathrm{dim}$ for zero von Neumann dimension). This passage is required to ensure closed images for differentials and the strictness needed for spectral sequence arguments.

Filtered quasi-isomorphisms between analytic $(L^2)$ complexes and algebraic resolutions (e.g., Čech or triangulation) transfer the filtrations and allow the deployment of spectral sequences:

• Hodge spectral sequence: Typically degenerates at $E_1$.
• Weight spectral sequence: Structure and degeneration depends on the geometry (e.g., degenerates at $E_2$ or $E_1$ in Kähler hyperbolic contexts).

This setup yields a canonical bigrading in cohomology compatible with functorial and duality operations.

2. Graded Pieces, Residue Isomorphisms, and Gysin Maps

The bigrading is sharply realized in the isomorphism for graded pieces of the weight filtration: $$\operatorname{Res}_m: \operatorname{Gr}^W_m p^*(2)\Omega_X^\bullet(\log D) \xrightarrow{\sim} a_m^*p^*(2)\Omega_{D_m}^\bullet[-m]$$ where $D_m$ is the union of $m$-fold intersections of components of $D$, and $a_m$ is the inclusion. Cohomologically, the graded piece $\operatorname{Gr}^W_m$ corresponds to the $L^2$-cohomology of the covering of the stratum $D_m$, shifted by $-m$.

In the associated weight spectral sequence, the $E_1$-term is

$$E_1^{-m, p+q} \cong H^{p+q-2m}(p^{-1}(D_m), p^*(2)\mathbb{C})$$

The differentials at this stage are induced by Gysin maps: $$d_1([\alpha]) = \sum_{i} (-1)^{\varepsilon(i)} (\mathrm{Gysin}_i)_*([\alpha])$$ The sign $\varepsilon(i)$ is determined combinatorially from the Čech or dual CW complex boundary operator. These Gysin maps “glue” contributions from strata—all intersections of divisors—to yield the global mixed Hodge structure, realizing Deligne’s principle that local residue data and Gysin operations govern the assembly of the global (typically bigraded) cohomology.

3. Quotient Categories, Torsion Theory, and Analytic vs Algebraic Models

The analytic models (Hilbert complexes of $L^2$-forms, Sobolev complexes) often lack strictness in the module category over $N(G)$ due to analytic subtleties (failure of closed range, presence of zero-dimensional parts). Torsion theories such as $T_\mathrm{dim}$ or $T_u(G)$ are introduced, defining the category of “negligible” modules (e.g., those with zero von Neumann dimension).

By forming quotient categories (Mod($N(G)$)/$T$), the “bad” components are formally set to zero, and mixed Hodge structures acquire strictness: differentials become strict, extensions behave as in the pure case, and degeneration results hold. The cohomology in this quotient category then admits well-behaved bigrading, and spectral sequences degenerate as predicted by Deligne’s theory. Formally: $(H^n(X, p^*(2)\mathbb{C}), W, F) \quad \text{is a mixed Hodge structure in Mod(%%%%43%%%%)/%%%%44%%%%}$ This approach is essential in analyzing $L^2$-Betti numbers, twisted Euler characteristics, and in connecting analytic invariants on covering spaces to classical algebraic invariants via Atiyah-type equalities.

4. Applications to Topology, Moduli, and Analytic Invariants

The bigraded mixed Hodge structures on $L^2$-cohomology provide refined topological invariants for open manifolds, complements of divisors, and coverings in Kähler geometry:

• Open and non-compact varieties: The mixed Hodge structure on $L^2$-cohomology of a covering of $X \setminus D$ refines the classical topology of the complement and reflects the geometry of the covering.
• Analysis via von Neumann dimensions: The quotient category accounts for torsion and ensures that $L^2$-Betti numbers and the $L^2$-Euler characteristic relate directly to classical invariants.
• Degeneration results: In Kähler hyperbolic cases (Kähler forms d-bounded), the weight spectral sequence degenerates at $E_1$, and typically only the expected pure Hodge components remain; this yields vanishing results for higher-degree $L^2$-cohomology and constrains the Laplacian spectrum.

These methods intersect with singularity theory (via dual complexes of divisors), rigidity theorems, and asymptotic group invariants, linking global analysis, Hodge theory, and operator algebras.

5. Explicit Examples and Computational Schemes

Key examples illustrate bigraded structure and analytic-to-algebraic passage:

• Galois coverings: Given $p : \widetilde{X} \to X$, $X$ compact Kähler, $D$ normal crossing, the $L^2$-complex (with logarithmic forms) computes the $L^2$-cohomology, and its weight spectral sequence has $E_1$ described precisely in terms of $L^2$-cohomology of preimages of strata.
• Sobolev resolutions: Inclusion of high-order Sobolev spaces into smooth $L^2$-forms is a filtered quasi-isomorphism, providing a practical analytic computation method.
• Dual CW-complexes: Combinatorial objects encode the structure of divisors and their intersections, offering a combinatorial interpretation of the highest weight part of the mixed Hodge structure.
• Kähler hyperbolic manifolds: Spectral gap for the Laplacian in middle degree implies degeneration of the $L^2$-Hodge-to-de Rham spectral sequence and retention of only pure Hodge structures.

In each case, the bigraded structure is sharply expressed through residue isomorphisms, filtration formulas, and the functorial behavior of associated complexes.

6. Summary and Structural Significance

Bigraded structures in mixed Hodge theory provide a robust framework for decomposing cohomological invariants via two core filtrations. In the context of $L^2$-cohomology of coverings, analytic techniques (Sobolev spaces, operator algebra methods, sheaf-theoretic resolutions) converge with algebraic approaches (filtered spectral sequences, residue-Gysin correspondences) to establish strict mixed Hodge structures equipped with canonical bigradings. The graded pieces, brought together via Gysin maps, encode contributions from all strata of divisors and their interactions.

These bigraded structures not only illuminate analytic properties of open and non-compact varieties but also enable the transfer and computation of intricate invariants (Euler characteristics, Betti numbers) and underpin applications in moduli theory, dual complexes, and the paper of higher-dimensional geometry. In modern complex geometry, the synthesis of analytic, topological, and algebraic methods through bigraded mixed Hodge structures reveals deep links between local geometric features and global cohomological data.