Intersection homology of toric varieties and a conjecture of Kalai

Abstract

We prove an inequality, conjectured by Kalai, relating the g-polynomials of a polytope P , a face F , and the quotient polytope P/F , in the case where P is rational. We introduce a new family of polynomials g(P, F ), which measures the complexity of the part of P "far away" from the face F ; Kalai's conjecture follows from the nonnegativity of these polynomials. This nonnegativity comes from showing that the restriction of the intersection cohomology sheaf on a toric variety to the closure of an orbit is a direct sum of intersection homology sheaves.