Lefschetz decomposition and the cd-index of fans

2005/09/09 by Kalle Karu, Karu, Kalle
Mathematics · #06A07 #14M25 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO #msc:06A07 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/0509220

14 pages, 1 figure

arxiv created 2005/09/09 · arxiv updated 2009/12/01

Abstract

The goal of this article is to give a Lefschetz type decomposition for the cd-index of a complete fan. To a complete simplicial fan one can associate a toric variety X, the even Betti numbers h_i of X and the numbers g_i = h_i-h_i-1. If the fan is projective, then non-negativity of g_i follows from the Lefschetz decomposition of the cohomology. In the case of a nonsiplicial complete fan one can analogously compute the flag h-numbers h_S and, by a change of variable formula, the cd-index. We give an analogue of the Lefschetz operation for the cd-index. This gives another proof of the non-negativity of the cd-index for complete fans.

Related

• The cd-index of fans and lattices
• Relative Hard Lefschetz Theorem for Fans
• Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes
• Decomposition theorem for the cd-index of Gorenstein* posets
• The Lefschetz defect of Fano varieties