Combinatorial Cycle Classes in the Intersection Cohomology of Projective Toric Varieties

Abstract

We investigate cycle-class realizations inside the combinatorial intersection cohomology for fans developed by Barthel, Brasselet, Fieseler, and Kaup (BBFK). For projective toric varieties, the intersection cohomology is Hodge-Tate, and thus the space of rational Hodge classes coincides with the full rational even-degree intersection cohomology. We formulate a compatibility statement between combinatorial and geometric cycle classes and explore it in the torus-invariant setting under standard functoriality assumptions. The central question we address is whether these invariant combinatorial cycle classes span the even-degree combinatorial intersection cohomology $IH^{2k}_{\mathrm{comb}}(\Sigma, \mathbb{Q})$. Assuming the stated BBFK--BL compatibility, we verify this linear-generation statement for projective toric varieties of dimension at most $3$; the simplicial case follows unconditionally from standard rational cohomology descriptions. We illustrate the framework with a non-simplicial example in dimension $3$ for which the Betti numbers and spanning property are derived directly from Stanley's toric $h$-vector formula and Fieseler's surjectivity theorem.