Lefschetz decomposition and the cd-index of fans

TL;DR

This paper establishes a Lefschetz type decomposition for the cd-index of complete fans, providing a new proof of its non-negativity and extending concepts from toric geometry to nonsimplicial fans.

Contribution

It introduces an analogue of the Lefschetz operation for the cd-index of complete fans, generalizing known results from projective simplicial fans.

Findings

Lefschetz decomposition applies to the cd-index of complete fans.

Provides a new proof of non-negativity of the cd-index.

Extends Lefschetz theory to nonsimplicial fans.

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.