[Paper Review] On polytopes simple in edges

Vladlen Timorin|ArXiv.org|Oct 23, 2000

Advanced Combinatorial Mathematics14 references3 citations

TL;DR

This paper establishes a combinatorial analog of the Hard Lefschetz theorem for polytopes simple in edges with infrequent singularities—defined as polytopes where nonsimple vertices are sufficiently separated. By analyzing the cohomology of associated toric varieties and using a resolution of the dual fan, the authors prove unimodality and nonnegativity of h-vector components for k ≥ d/2, confirming Stanley’s conjecture for this class of polytopes.

ABSTRACT

We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's conjecture for such polytopes.

Motivation & Objective

• To establish a combinatorial analog of the Hard Lefschetz theorem for polytopes simple in edges.
• To verify Stanley’s conjecture on the unimodality and nonnegativity of h-vectors for polytopes with infrequent singularities.
• To provide a direct combinatorial proof of Khovanskii’s estimate on the nonexistence of finite-volume reflection groups in high-dimensional Lobachevskii spaces.
• To extend the framework of intersection cohomology to non-integral polytopes via a combinatorial construction.

Proposed method

• Constructing a standard resolution Σ of a d-polytope Δ simple in edges, leading to a simplicial subdivision Ψ of the dual fan Φ.
• Using the embedding of cohomology modules MΦ into OΨ to decompose the cohomology ring A(Σ) = OΨ into MΦ^k ⊕ N^k for k ≤ d/2.
• Proving that the restriction map R: OΨ → OΥ is injective on MΦ^k, which implies the direct sum decomposition.
• Identifying the kernel N^k of the restriction map with the ideal I^k generated by characteristic functions of inserted rays in Ψ − Φ.
• Applying the Hard Lefschetz operator multiplication by SΔ to show injectivity of MΦ^k → MΦ^{k+1} for k < (d−1)/2.
• Using the decomposition and injectivity to derive inequalities on the h-vector, including h_k ≤ h_{d−k} and unimodality for k ≥ d/2.

Experimental results

Research questions

• RQ1 Does a combinatorial analog of the Hard Lefschetz theorem hold for polytopes simple in edges with infrequent singularities?
• RQ2 Can Stanley’s conjecture on the unimodality and nonnegativity of the h-vector be proven for this class of polytopes?
• RQ3 Is there a direct combinatorial proof of Khovanskii’s estimate on the nonexistence of finite-volume reflection groups in high-dimensional Lobachevskii spaces?
• RQ4 How can intersection cohomology invariants be extended to non-integral polytopes using combinatorial constructions?

Key findings

• For polytopes with infrequent singularities, the h-vector satisfies h_k ≤ h_{d−k} for all k ≤ d/2.
• The h-vector components h_k for k ≥ d/2 are nonnegative and unimodal: h_{[d/2]} ≥ h_{[d/2]+1} ≥ ⋯ ≥ h_d.
• The multiplication by SΔ induces an injective map from MΦ^k to MΦ^{k+1} for k < (d−1)/2, establishing a combinatorial Hard Lefschetz property.
• The generalized h-vector Gh_k(Δ) equals the h-vector h_k(Δ) for k ≥ d/2 in polytopes with infrequent singularities.
• The kernel N^k of the restriction map OΨ → OΥ is generated by characteristic functions of rays in Ψ − Φ, enabling the decomposition OΨ = MΦ^k ⊕ N^k.
• The construction confirms Stanley’s conjecture for this class of polytopes, extending its validity beyond simple polytopes.