Combinatorial Hodge Classes

• Combinatorial Hodge classes are algebraically defined invariants derived from discrete geometric data that mirror classical Hodge theory via Poincaré duality, hard Lefschetz, and Hodge-Riemann relations.
• They underpin proofs of log-concavity and unimodality in matroid theory and provide algorithmic verifiability in the intersection cohomology of fans and tropical surfaces.
• These classes facilitate spectral methods and explicit calculations in topology and moduli theory, enabling discrete models to capture complex intersection and decomposition phenomena.

A combinatorial Hodge class is a class in a cohomology or homology theory constructed purely from combinatorial or discrete geometric data—such as matroids, fans, simplicial complexes, or modular graphs—that satisfies analogues of the classical Hodge-theoretic properties: Poincaré duality, hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. In combinatorial and tropical geometry, such classes serve as algebraically well-defined representatives of cycles or primitives, often mirroring the intersection-theoretic and decomposition behavior found in algebraic geometry, but achieved without recourse to complex or algebraic-analytic structure.

1. Combinatorial Hodge Classes in Matroid and Polyhedral Settings

The foundational realization of combinatorial Hodge classes appears in the study of matroids. For a loopless matroid $M$ of rank $d$ on a finite ground set $E$, the intersection cohomology module $IH(M)$ is constructed as an orthogonal complement in the Chow ring $CH(M)$, graded by cohomological degree, equipped with a cup product, and a nondegenerate pairing via the degree map on complete flags. The ring structure supports Poincaré duality: for each $k = 0, \dots, d$ the induced form on $IH^k(M) \times IH^{d-k}(M)$ is perfect. Hard Lefschetz holds: for any ample $\ell \in IH^1(M)$, the multiplication map $\ell^{d-2k}: IH^k(M) \to IH^{d-k}(M)$ is an isomorphism for $k \le d/2$. The Hodge-Riemann relations further require that, on the primitive subspace

$$P^k(M) = \ker\left(\ell^{d-2k+1}: IH^k(M) \rightarrow IH^{d-k+1}(M)\right),$$

the bilinear pairing

$$Q_k(\alpha, \beta) = (-1)^k \deg_M(\ell^{d-2k}\alpha\cdot\beta)$$

is positive definite. Elements of $P^k(M)$ are precisely the combinatorial Hodge (primitive) classes. The $sl_2$-action via Lefschetz and its adjoint decomposes $IH(M)$ into irreducibles with the primitive classes as highest-weight vectors. The intersection form's signature condition on $P^k(M)$ mirrors the sign condition from classical Hodge theory. This combinatorial Hodge package provides the algebraic infrastructure for results such as the top-heaviness conjecture, nonnegativity and unimodality for the Kazhdan-Lusztig and Z-polynomials, and representation-theoretic refinements in equivariant settings (Braden et al., 2020, Adiprasito et al., 2015, Baker, 2017).

Similarly, for the intersection cohomology of fans associated to projective toric varieties, one defines a combinatorial intersection cohomology algebra $comb^*(\Sigma,\mathbb Q)$ whose grading, perfect pairing, and Lefschetz operator structure are constructed from convex, piecewise-linear functions on the fan. Primitive combinatorial Hodge classes are

$$P^k(\Sigma) = \ker\left(L^{n-k+1}: comb^k(\Sigma) \to comb^{2n-k+2}(\Sigma)\right)$$

where $L$ is cup product with an ample piecewise-linear function, with the induced Hodge-Riemann bilinear form positive on $P^k(\Sigma)$. Cones of dimension $k$ provide combinatorial cycle classes in $comb^{2k}(\Sigma,\mathbb Q)$, whose span is conjectured to exhaust the group of rational Hodge classes in intersection cohomology (combinatorial Hodge conjecture) (Jahangir, 7 Dec 2025).

2. Simplicial and Algorithmic Realizations

In the context of finite simplicial complexes $X$, the language of discrete Hodge decomposition emerges naturally via the chain/cochain complexes and discrete Laplacian operators. The $k$-th combinatorial Hodge classes are defined as the harmonic $k$-cochains:

$\ker \Delta^k$

where $$\Delta^k = \delta^{k-1}\partial_k + \partial_{k+1}\delta^k$$ is the (simplicial) Laplacian. These spaces are canonical, admit orthonormal eigenbases, and correspond via a canonical isomorphism to $H^k(X)$. Computation reduces to the null-space problem for a sparse symmetric matrix, making combinatorial Hodge classes central to spectral algorithms in topology and data analysis (Essl, 2023).

On divisors of toric Calabi-Yau threefold hypersurfaces, Hodge numbers $h^i(\mathcal{O}_D)$ admit combinatorial realization as Betti numbers of a "puff" complex $P_D$ constructed from the stratification data of the dual polytope:

$h^i(\mathcal{O}_D) = h_i(P_D)$

for $i=0,1,2$, giving a purely combinatorial enumeration of sheaf cohomology and their algebraic representatives (Braun et al., 2017).

3. Tropical and Moduli-Theoretic Combinatorial Hodge Classes

In tropical geometry, the analogues of classical Hodge classes are realized in the Néron-Severi group $NS(\Delta)\otimes\mathbb{R}$ for a tropical surface $\Delta$, defined as the group of Cartier divisors modulo algebraic equivalence. The intersection pairing on this space is nondegenerate, and the tropical Hodge index theorem establishes its signature as $(1,n-1)$ or $(0,n)$. The Lefschetz operator (intersection with a fixed divisor of positive self-intersection) picks out, as in the algebraic case, a primitive decomposition, and the primitive part consists of those classes orthogonal to the chosen ample divisor—these are the combinatorial Hodge (primitive) classes of the tropical surface. All intersection and signature phenomena are proved entirely in terms of the underlying combinatorial data of the $\Delta$-complex and its structure constants (Cartwright, 2015).

On moduli spaces of curves, explicit combinatorial expressions for Hodge classes are possible using Strebel/Jenkins differentials or via dual graphs and the combinatorics of admissible covers. For example, the first Chern class $\lambda_1$ of the Hodge bundle on moduli of cyclic covers admits an explicit expansion as a linear combination of boundary strata, with coefficients determined by combinatorial congruencies on the dual graphs (Owens et al., 2019). On the combinatorial model $\mathcal{M}_{g,n}[\vec p]$ parametrized by ribbon graphs and periods of abelian differentials, the monodromy of the Hodge tau-function around cycles corresponding to pentagon moves and Dehn twists provides combinatorial representatives for the (Poincaré dual of) Chern classes, establishing combinatorial analogues of the Mumford and Penner relations (Bertola et al., 2018).

4. Applications and Structural Consequences

The existence of combinatorial Hodge classes and the validity of the combinatorial Hodge package in the aforementioned settings has deep consequences:

• Log-concavity: In matroid theory, the Hodge-Riemann relations for the Chow ring or intersection cohomology module immediately yield log-concavity results for characteristic polynomials, independence complexes, and Kazhdan-Lusztig polynomials (Adiprasito et al., 2015, Braden et al., 2020).
• Algorithmic verifiability: In the fan setting, the conjecture that combinatorial cycle classes exhaust intersection cohomology can be algorithmically checked for any rational fan via finite computations with stalks, sheaf assembly, and matrix rank tests (Jahangir, 7 Dec 2025).
• Spectral methods: In applications to signal processing and data, combinatorial Hodge classes as harmonic representatives allow for efficient extraction of topological features from high-dimensional data via Laplacian eigenspace computations (Essl, 2023).
• Moduli theory: Combinatorial Hodge classes enable recursive and explicit computation of intersection numbers and Chern integrals in the tautological rings associated to spaces of covers or pseudostable curves, with all terms classified and indexed combinatorially (Owens et al., 2019, Cavalieri et al., 2024).

5. Open Problems and Future Directions

Several foundational avenues remain active:

• The search for purely combinatorial and explicit bases (e.g., explicit flag or graph-theoretic representatives) for the primitive subspaces $P^k(M)$ in the intersection cohomology of matroids is an open question. Conjecturally, these may admit indexing by local positivity conditions on flags of flats (Braden et al., 2020).
• In equivariant and representation-theoretic settings, extending the combinatorial Hodge theory to account for group actions and to relate to invariants valued in spaces of representations is under development (Braden et al., 2020).
• Generalizing combinatorial Hodge theory to oriented matroids or to more general Coxeter matroids—where the absence of algebro-geometric models means all definitions must be realized entirely with combinatorial constructs—remains an active challenge (Braden et al., 2020).
• In tropical and combinatorial moduli theory, a thorough development of a full $(p,q)$-Hodge theory compatible with the structures above is ongoing, with partial results in $2$-dimensional and curve cases (Cartwright, 2015, Bertola et al., 2018).

6. Canonical Examples: Explicit Descriptions

Context	Combinatorial Hodge classes	Primitive Condition / Structure
Matroid $M$	$P^k(M) = \ker(\ell^{d-2k+1})$ in $IH^k(M)$	Highest-weight vectors under $sl_2$, orthogonal under $Q_k$ (Braden et al., 2020)
Fan $\Sigma$	$P^k(\Sigma) = \ker(L^{n-k+1})$ in $comb^k(\Sigma)$	Orthogonal under the Hodge-Riemann form, represented by cycle classes (Jahangir, 7 Dec 2025)
Simplicial $X$	$\ker \Delta^k$ (harmonic $k$-cochains)	Nullspace of Laplacian, basis for $H^k(X)$ (Essl, 2023)
Tropical $\Delta$	$P = \ker(L_H)$ in $NS(\Delta)\otimes\mathbb{R}$	Classes orthogonal to ample divisor in Néron-Severi, signature $(1,n-1)$ form (Cartwright, 2015)

These explicit constructions, algorithms, and representations collectively ground the combinatorial Hodge class as a central algebraic object in modern combinatorics, discrete geometry, and applications to moduli, topology, and data science.