Intersection Hodge Conjecture for Toric Varieties

Updated 14 December 2025

Intersection Hodge Conjecture for projective toric varieties is the study of surjectivity of cycle class maps in intersection cohomology, using combinatorial and residue-theoretic techniques.
The methodology employs BBFK combinatorial intersection cohomology and algorithmic verification to compute and confirm the algebraicity of rational Hodge classes in low-dimensional and simplicial cases.
Key results include explicit verifications for simplicial fans and quasi-smooth intersections, while open questions focus on extending these methods to non-simplicial and mixed Hodge scenarios.

The intersection Hodge conjecture for projective toric varieties concerns the surjectivity of the cycle class map from algebraic cycles to rational Hodge classes in intersection cohomology, framed through the combinatorics of fans and the geometric properties of toric varieties. Developments in the combinatorial intersection cohomology theory, especially the Barthel–Brasselet–Fieseler–Kaup (BBFK) approach, enable explicit combinatorial formulations and algorithmic verifications of the conjecture. Significant progress has been made for projective toric varieties of low dimension, simplicial fans, and quasi-smooth complete intersections, with major advances on the algebraicity of middle Hodge classes using cohomological and residue-theoretic techniques.

1. Classical Intersection Hodge Conjecture for Toric Varieties

For a complete rational fan $\Sigma \subset N_{\mathbb{R}}$ defining a projective toric variety $X_\Sigma$ of complex dimension $n$ , the intersection cohomology groups $IH^*(X_\Sigma, \mathbb{Q})$ possess a pure Hodge structure of weight $*$ . The space of codimension- $p$ rational Hodge classes is given by

$Hdg^p(X_\Sigma, \mathbb{Q}) := IH^{2p}(X_\Sigma, \mathbb{Q}) \cap IH^{p,p}(X_\Sigma).$

The intersection cycle-class map

$cl_{IH}: Z^p(X_\Sigma)_{\mathbb{Q}} \to IH^{2p}(X_\Sigma, \mathbb{Q})$

assigns each algebraic cycle its fundamental class. The intersection Hodge conjecture (IHC) posits:

$\operatorname{Im}(cl_{IH}) = Hdg^p(X_\Sigma, \mathbb{Q}) \quad \forall p.$

For projective toric varieties, Fieseler's theorem asserts that $IH^{2p}(X_\Sigma, \mathbb{Q}) = Hdg^p(X_\Sigma, \mathbb{Q})$ , reducing the IHC to surjectivity of $cl_{IH}$ (Jahangir, 7 Dec 2025).

2. Combinatorial Intersection Cohomology and BBFK Theory

BBFK combinatorial intersection cohomology treats $\Sigma$ as a poset of cones, constructing a minimal-extension sheaf $\mathcal{E}$ and its intersection cohomology complex $IC_\Sigma$ . The combinatorial intersection cohomology is defined as

$IH^k_{\mathrm{comb}}(\Sigma, \mathbb{Q}) := \mathbb{H}^k(\Sigma, IC_\Sigma).$

Properties include direct grading, Poincaré duality $IH^k_{\mathrm{comb}} \cong IH^{2n-k}_{\mathrm{comb}}$ , and, for projective fans, the Hard Lefschetz theorem (Karu). In this setting, cycle classes associated to torus-invariant subvarieties $V(\tau)$ , for cones $\tau \in \Sigma$ , are constructed via proper pushforwards in combinatorial cohomology. The space of combinatorial Hodge classes is

$Hdg^k_{\mathrm{comb}}(\Sigma) := \operatorname{Span}_{\mathbb{Q}}\{ [V(\tau)]_{\mathrm{comb}} : \dim \tau = k \} \subset IH^{2k}_{\mathrm{comb}}(\Sigma, \mathbb{Q})$

(Jahangir, 7 Dec 2025).

3. The Combinatorial Intersection Hodge Conjecture

The combinatorial conjecture asserts that $Hdg^k_{\mathrm{comb}}(\Sigma)$ spans all of $IH^{2k}_{\mathrm{comb}}(\Sigma, \mathbb{Q})$ . Via the canonical isomorphism $\varphi: IH^*_{\mathrm{comb}}(\Sigma, \mathbb{Q}) \simeq IH^*(X_\Sigma, \mathbb{Q})$ , this becomes

$\varphi(Hdg^k_{\mathrm{comb}}(\Sigma)) = Hdg^k(X_\Sigma, \mathbb{Q}),$

guaranteeing the algebraicity of all rational Hodge classes in intersection cohomology for toric varieties (Jahangir, 7 Dec 2025).

4. Verification in Low Dimensions and Simplicial Fan Cases

Explicit evidence supports the conjecture for $n \leq 3$ and all simplicial projective toric varieties:

For $n=1$ , $X_\Sigma \cong \mathbb{P}^1$ is smooth; $IH^* = H^*$ is generated by point and line classes.
For $n=2$ , $IH^2$ is spanned by torus-invariant divisors, while $IH^4$ is generated by any smooth fixed point.
For $n=3$ , $IH^2$ corresponds to invariant divisors, $IH^4$ to curve classes via Hard Lefschetz, and $IH^6$ to maximal cones.
For simplicial fans, the Danilov–Jurkiewicz presentation gives $H^*(X_\Sigma, \mathbb{Q}) \cong \mathbb{Q}[x_\rho \mid \rho \in \Sigma(1)] / ($ linear + Stanley–Reisner relations $)$ , establishing generation by degree-$2$ cycle classes (Jahangir, 7 Dec 2025).

Results for quasi-smooth intersections in projective simplicial toric varieties further extend the Hodge Conjecture for these ambient spaces, including hypersurfaces and higher-codimension intersections (Bruzzo et al., 2021, Montoya, 2023).

5. Algebraicity of Hodge Classes via Noether–Lefschetz and Cayley Constructions

Recent arithmetic, geometric, and combinatorial techniques have settled the conjecture for large classes:

For very general quasi-smooth intersections $X \subset X_\Delta$ where degree conditions are satisfied, every $H^{k,k}(X, \mathbb{Q})$ class is algebraic. This holds when the sum of the divisor degrees minus the anticanonical is nef and $n + s = 2(k+1)$ (Bruzzo et al., 2021).
Under the Cayley trick, codimension- $s$ intersections correspond to hypersurfaces in higher-dimensional toric varieties. Primitive Hodge pieces for these hypersurfaces are related to those of the intersection: $H^{k,k}_{\mathrm{prim}}(Y) \cong H^{1,1}_{\mathrm{prim}}(X)$ at $d+s=2(k+1)$ (Montoya, 2023).
Asymptotic results show that for high-degree hypersurfaces in odd-dimensional Oda projective simplicial toric varieties, all primitive rational middle Hodge classes are algebraic, realized by quasi-smooth subvarieties for sufficiently large degree (Noether–Lefschetz locus arguments) (Bruzzo et al., 2021).

These results broaden the known cases, encompassing toric Fano varieties and weighted projective spaces under suitable conditions.

6. Algorithmic Verification for Arbitrary Rational Fans

The BBFK theory enables systematic, combinatorial verification of the conjecture for arbitrary rational fans:

Compute local intersection cohomology $IH^*_{\mathrm{comb}}$ for links of cones via $g$ -polynomial recursion.
Assemble the minimal extension sheaf $IC_\Sigma$ from local data and restriction maps.
Resolve $IC_\Sigma$ to compute global groups $IH^k_{\mathrm{comb}}(\Sigma)$ .
Compute pushforwards for each cone of dimension $k$ , providing representations of $[V(\tau)]_{\mathrm{comb}}$ .
Matrix rank computation determines if the cycle classes span $IH^{2k}_{\mathrm{comb}}$ ; a full rank confirms the conjecture for each $k$ .

Pseudocode outline:

for sigma in Σ:
    IH_loc[sigma] = local_IH(link(sigma))
build_sheaf(IC_Σ, IH_loc)
IH_comb = hypercohomology(IC_Σ)   # yields dims d[k]
for k in range(n+1):
    Vectors = []
    for tau in Σ with dim tau == k:
        v = pushforward_class(tau, IC_Σ)
        Vectors.append(v)
    if rank(matrix(Vectors)) < d[k]:
        report "Conjecture fails in codim k"
        exit
report "Conjecture holds for Σ"

(Jahangir, 7 Dec 2025)

7. Limitations, Open Questions, and Extensions

Current results hold for projective toric varieties that are simplicial (toric orbifolds) and for quasi-smooth intersections. The full conjecture in the non-simplicial case remains open, as do cases where intersections do not satisfy the necessary nefness. The Cayley trick and associated residue maps provide powerful tools for reducing complete intersections to hypersurface settings, but extension to mixed Hodge components or arbitrary singular ambient varieties remains unresolved (Bruzzo et al., 2021, Montoya, 2023).

Key open problems include:

Extending the combinatorial conjecture to non-simplicial toric varieties.
Covering mixed Hodge degrees or non-complete intersections.
Algorithmic and computational refinement for non-orbifold ambient spaces.

These advances establish a combinatorial and cohomological framework for verifying the intersection Hodge conjecture in toric settings and motivate further exploration in broader classes of algebraic varieties.