The Intersection Cohomology of a Fan and the Hodge Conjecture for Toric Varieties

TL;DR

This paper proposes a combinatorial approach to the Intersection Hodge Conjecture for projective toric varieties, linking algebraic cycles with combinatorial intersection cohomology and verifying the conjecture in low dimensions.

Contribution

It introduces a combinatorial version of the Hodge Conjecture for toric varieties, defines combinatorial Hodge classes, and verifies the conjecture for low-dimensional and simplicial cases.

Findings

Verified the conjecture for all projective toric varieties of dimension ≤ 3.

Confirmed the conjecture for simplicial projective toric varieties.

Developed an algorithmic framework for testing the conjecture on arbitrary fans.

Abstract

We formulate a combinatorial version of the Intersection Hodge Conjecture for projective toric varieties. The conjecture asserts that the subspace of rational Hodge classes in the intersection cohomology $IH^*(X_\Sigma)$ is generated by the classes of algebraic cycles. We define the space of combinatorial Hodge classes, $Hdg^k_{\mathrm{comb}}(\Sigma) \subset IH^{2k}_{\mathrm{comb}}(\Sigma, \mathbb{Q})$, using the combinatorial intersection cohomology theory for fans developed by Barthel, Brasselet, Fieseler, and Kaup. We conjecture that this space is spanned by the combinatorial cycle classes corresponding to torus-invariant subvarieties. We verify this conjecture for all projective toric varieties of dimension $n \le 3$ and for the class of simplicial projective toric varieties. Finally, we provide an algorithmic framework to verify the conjecture for arbitrary rational fans.