Lefschetz decomposition and the cd-index of 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 hi of X and the numbers gi = hi-hi-1. If the fan is projective, then non-negativity of gi follows from the Lefschetz decomposition of the cohomology. In the case of a nonsiplicial complete fan one can analogously compute the flag h-numbers hS 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.
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