Intersection homology of toric varieties and a conjecture of Kalai

Abstract

We prove an inequality, conjectured by Kalai, relating the g-polynomials of a poly- tope P ,af ace 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. Mathematics Subject Classication (1991). Primary 14F32; Secondary 14M25, 52B05.