2014/07/13 by Mark Andrea de Cataldo, de Cataldo, M. A., Luca Migliorini +3 · 5 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1407.3497
openalex publication_date 2014/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the topology of toric maps. We show that if f\colon X→ 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.