2025/12/07 by Ishii, Daisuke, Jahangir, Rizwan
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2512.06755
openalex publication_date 2025/12/07 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28
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 IH2kcomb(Σ, 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.