2005/11/19 by Tom Braden, Braden, Tom
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0511488
openalex publication_date 2005/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g2. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that gk(P) = 0 implies gk(P^*) = 0 and gk+1(P) = 0.