Combinatorics and topology of proper toric maps

Abstract

Abstract We study the topology of toric maps. We show that if f : X → Y {f\colon X\to Y} is a proper toric morphism, with X simplicial, then the cohomology of every fiber of f is pure and of Hodge–Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the f-vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of Y, which is positive precisely when the corresponding closed subset of Y appears as a support in the Decomposition Theorem. The description of this invariant involves the stalks of the intersection cohomology complexes on X and Y, but in the case when both X and Y are simplicial, there is a simple formula in terms of the relative f-vector.