Concentrated Mixed Hodge Spectrum Class

• The concentrated mixed Hodge spectrum class is an invariant capturing the interaction between Milnor fiber topology and mixed Hodge structures in hypersurface singularities.
• It is considered concentrated when its cohomological and Hodge contributions are confined to a single degree, implying purity of the underlying structure.
• Combinatorial and motivic formulae link its coefficients to Newton polyhedra and lattice counts, with significant applications in character varieties and moduli spaces.

A concentrated mixed Hodge spectrum class arises in the study of hypersurface singularities and character varieties, where the interaction between the topology of Milnor fibers and the Hodge-theoretic structure is encoded in a highly structured algebraic invariant. This class, often denoted as $\Sp(f)$ for a holomorphic function germ $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$, captures the monodromy and mixed Hodge theoretical information of the Milnor fiber near critical points, and is said to be "concentrated" when its cohomological and Hodge-theoretic contributions are supported in a single weight or cohomological degree. Such concentration often signals "purity" of the underlying mixed Hodge structure and has deep implications for the interpretation of singularity invariants, the structure of character varieties, and the formulation of motivic and Hodge-theoretic integration.

1. Formal Definition and Spectrum Polynomial

Given a non-isolated Newton non-degenerate hypersurface singularity $f$, the mixed Hodge spectrum class $\Sp(f)$ is defined as a formal Laurent series

$\Sp(f) = \sum_{\alpha \in \mathbb{Q}} n_{\alpha} t^{\alpha},$

where each coefficient

$$n_\alpha = \sum_{j \in \mathbb{Z}} \sum_{p \in \mathbb{Z}} (-1)^j \dim_\mathbb{C} \left(\operatorname{Gr}_F^p \operatorname{Gr}^W_{p+q} H^j(F_{f,0};\mathbb{C})_{\lambda}\right), \qquad q = j - 2p, \ \lambda = e^{-2\pi i \alpha},$$

encodes the mixed Hodge numbers of the eigencomponents of the monodromy action on the cohomology of the Milnor fiber $F_{f,0}$. The spectrum captures both the weight filtration $W_\bullet$ and the Hodge filtration $F^\bullet$ of the mixed Hodge structure on the cohomology (Saito, 2017).

2. Concentration and Purity

A spectrum class is said to be "concentrated" if, for all but finitely many eigenvalues $\lambda$ (excluding a finite set $R_f$ of "bad eigenvalues"), the reduced cohomology of the Milnor fiber is supported in a single cohomological degree, specifically,

$$\widetilde{H}^{j}(F_{f,0};\mathbb{C})_{\lambda} = 0,\quad j \neq n-1,$$

for $\lambda \notin R_f$. This property ensures that $\Sp(f)$ receives nonzero contributions only from the middle degree $j = n-1$. Purity in this context means that the mixed Hodge structure involved is in fact pure of a single weight—there are no nontrivial subquotients of different weights. This is equivalent to the spectrum class lying in the pure subring of the Grothendieck group of mixed Hodge structures (i.e., generated by structures $H$ with $W_w H = H$, $W_{w-1} H = 0$ for some $w$) (Komyo, 2014).

3. Combinatorial and Motivic Formulae

For Newton non-degenerate singularities, combinatorial formulae exist expressing the coefficients $n_\alpha$ of $\Sp(f)$ in terms of local weighted $h^*$-polynomials and polyhedral subdivisions closely related to the geometry of the Newton polyhedron $\Gamma_+(f)$. For $\lambda \notin R_f$,

$$n_\alpha = (-1)^{n-1} \sum_{F \prec \Gamma_f \text{ admissible}} l^*_\lambda(\Delta_F, \nu_F; 1) \, l_P(S_\nu, \Delta_F ;1),$$

with $l^*_\lambda(\Delta_F, \nu_F; u)$ and $l_P(S_\nu, \Delta_F; u)$ encoding signed lattice-point counts and the combinatorics of the Newton polytope subdivision, respectively (Saito, 2017). These invariants realize motivic and Hodge-theoretic information in terms of polyhedral and lattice data.

4. Jordan Structure and Monodromy Filtration

The interplay between the weight filtration $W_\bullet$ and the monodromy filtration $N_\bullet$ is sharply reflected in the concentrated mixed Hodge spectrum. For $\lambda \notin R_f$ (the "good" eigenvalues), the weight filtration on $H^{n-1}(F_{f,0};\mathbb{Q})_\lambda$ coincides with the monodromy (nilpotent) filtration centered at $n-1$. This is quantitatively realized by equating the Jordan block structure of the monodromy operator with the combinatorial spectrum data, making it possible to enumerate the size and multiplicity of Jordan blocks directly from the mixed Hodge-theoretic computations (Saito, 2017).

5. Character Varieties and Extension to Moduli Spaces

Purity and concentration results for mixed Hodge structures extend beyond singularity settings to the study of character varieties and moduli spaces of rank $n$ parabolic Higgs bundles or connections. For character varieties of indivisible type, the pure part of the mixed Hodge structure is isomorphic to the ordinary rational cohomology of associated quiver varieties (Komyo, 2014). In certain instances, all mixed Hodge structures in these moduli problems turn out to be pure or of a specific Hodge–Tate type, and the associated Hodge polynomials are independent of generic choices, reflecting far-reaching purity phenomena (Komyo, 2014).

6. Significance and Applications

Concentrated mixed Hodge spectrum classes provide powerful invariants that enable translation between topological, algebraic, and combinatorial properties of singularities and moduli spaces. In singularity theory, concentration results identify the precise locus of topological complexity and delimit the possible mixedness of Hodge structures. In representation theory and geometry, purity theorems for character varieties and moduli spaces encode deep relationships between flat bundles, quiver varieties, and rational cohomology. A plausible implication is that the combinatorial and motivic formulae for spectrum classes support motivic integration and the study of mirror symmetry phenomena in broader geometric frameworks.