[Paper Review] Lefschetz decomposition and the cd-index of fans

Kalle Karu|ArXiv.org|Sep 9, 2005

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper establishes a Lefschetz-type decomposition for the cd-index of complete fans by introducing a generalized Lefschetz operation on the cohomology of the first barycentric subdivision of a fan. Using a novel inductive construction based on Poincaré sheaves and pairing structures, it proves the non-negativity of the cd-index for complete fans, extending the classical Lefschetz decomposition from simplicial to nonsimplicial complete 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.

Motivation & Objective

• To extend the Lefschetz decomposition from simplicial to nonsimplicial complete fans by generalizing the notion of a Lefschetz operator.
• To provide a cohomological explanation for the non-negativity of the cd-index in complete fans, analogous to the classical Lefschetz theorem in algebraic geometry.
• To define a Lefschetz-type operation on the multi-graded cohomology of the first barycentric subdivision of a fan, ensuring isomorphisms that reflect the structure of the cd-index.
• To establish a sheaf-theoretic framework using Poincaré sheaves and duality to construct the required linear maps on cohomology.

Proposed method

• Introduces a multi-graded cohomology theory on the first barycentric subdivision $B\Delta$ of a complete fan $\Delta$, with grading by $\mathbb{N}^n$.
• Uses the space $\mathcal{A}(B\Delta)$ of conewise polynomial functions and quotients by the maximal homogeneous ideal to define the cohomology $H^S(B\Delta)$.
• Applies a change of variables to express the Poincaré polynomial $P_{B\Delta}(t_1,\ldots,t_n)$ as a homogeneous $cd$-polynomial $\Psi_\Delta(c,d)$.
• Constructs a generalized Lefschetz operation via an inductive 'main construction' on Poincaré sheaves, using pairing structures and duality.
• Defines linear maps $L_i: H^S(B\Delta) \to H^S(B\Delta)$ of degree $e_i$ that mimic multiplication by $x_i$, ensuring compatibility with the $cd$-monomial decomposition.
• Employs a torus action and genericity arguments to show the existence of maps satisfying $K^\perp \cap L(K) = 0$, ensuring injectivity and isomorphism properties.

Experimental results

Research questions

• RQ1 Can a Lefschetz-type decomposition be constructed for the cd-index of a complete nonsimplicial fan, analogous to the classical case in toric geometry?
• RQ2 How can the non-negativity of the cd-index be proven using cohomological methods rather than combinatorial or positivity arguments?
• RQ3 What is the appropriate generalization of the Lefschetz operator in the multi-graded setting of the barycentric subdivision cohomology?
• RQ4 Can the existence of such a Lefschetz operation be established via sheaf-theoretic constructions and duality?

Key findings

• The cd-index $\Psi_\Delta(c,d)$ of any complete fan $\Delta$ is non-negative, providing a new cohomological proof of this known result.
• A generalized Lefschetz operation is constructed on the multi-graded cohomology $H^S(B\Delta)$, with maps $L_i$ of degree $e_i$ that induce isomorphisms on components corresponding to $cd$-monomials containing $t_i+1$.
• The construction relies on an inductive 'main construction' on Poincaré sheaves, using duality and pairing structures to ensure compatibility and non-degeneracy.
• The existence of such a Lefschetz operation implies the non-negativity of the cd-index, as the dimensions of the cohomology components correspond to the coefficients of $\Psi_\Delta(c,d)$.
• The method extends the classical Lefschetz decomposition from simplicial fans to general complete fans by replacing the toric variety cohomology with sheaf-theoretic constructions on the barycentric subdivision.