Remarks on the combinatorial intersection cohomology of fans

Abstract

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 $g_2$. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that $g_k(P)=0$ implies $g_k(P^{\ast })=0$ and $g_{k+1}(P)=0$.

arXiv comment34 pages. Typos fixed; final version, to appear in Pure and Applied Math Quarterly

References in corpus (2)

• The cd-index of fans and lattices
• Lefschetz decomposition and the cd-index of fans