[Paper Review] Ehrhart polynomials and stringy Betti numbers

Mircea Mustaţă, Sam Payne|ArXiv.org|Apr 23, 2005

Advanced Combinatorial Mathematics9 references4 citations

TL;DR

This paper establishes a connection between stringy Betti numbers of Gorenstein toric varieties and the Ehrhart polynomials of polyhedral regions, using orbifold cohomology and motivic integration to derive a combinatorial formula for the $δ$-vector of reflexive polytopes. The key contribution is a counterexample to Hibi's conjecture on $δ$-vector unimodality in dimension 6, demonstrating non-unimodal $δ$-vectors via explicit computation and a general construction for even dimensions.

ABSTRACT

We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of delta-vectors of reflexive polytopes.

Motivation & Objective

• To establish a precise link between stringy Betti numbers of Gorenstein toric varieties and Ehrhart polynomials of polyhedral regions.
• To provide a combinatorial formula for the $δ$-vector of reflexive polytopes using orbifold cohomology and motivic integration.
• To disprove Hibi's conjecture on the unimodality of $δ$-vectors of reflexive polytopes by constructing explicit counterexamples.
• To generalize the connection between $δ$-vectors and $h$-vectors of triangulations to non-crepant resolutions via stringy invariants.

Proposed method

• Use toric geometry to interpret $δ$-vectors as stringy Betti numbers of Gorenstein toric varieties via motivic integration.
• Define a polyhedral complex $Q \subset N_{\mathbb{R}}$ as $\{v \in N_{\mathbb{R}} \mid \Psi_K(v) \leq 1\}$, where $\Psi_K$ is a piecewise linear function associated to the canonical divisor.
• Apply the orbifold cohomology formula from Borisov–Chiang–Huang to express the generating function of $\delta_P$ as a sum over faces $F$ of the triangulation, weighted by $h$-vectors of associated fans $\Delta_F$.
• Use a regular triangulation of the boundary of a reflexive polytope $P$ to compute contributions from lattice points in $\operatorname{Box}(F)$, where $\Psi_K(v)$ determines the weight.
• Prove the main formula $(1-t)^d \sum_{v \in N} t^{\Psi_K(v)} = \sum_{F \in \mathcal{T}, v \in \operatorname{Box}(F)} t^{\Psi_K(v)} h_{\Delta_F}(t)$ via lattice point decomposition in cones.
• Construct explicit counterexamples in dimension 6 and higher by generalizing a 6-dimensional reflexive polytope with $\delta_P = (1,6,8,6,8,6,1)$, showing non-unimodal $\delta$-vectors.

Experimental results

Research questions

• RQ1 Can stringy Betti numbers of Gorenstein toric varieties be expressed as generating functions of Ehrhart polynomials of polyhedral regions?
• RQ2 Does the $\delta$-vector of a reflexive polytope arise as a positive linear combination of $h$-vectors of simplicial polytopes via orbifold cohomology?
• RQ3 Is Hibi's conjecture on the unimodality of $\delta$-vectors of reflexive polytopes true for all dimensions?
• RQ4 Can non-unimodal $\delta$-vectors be systematically constructed for reflexive polytopes in dimension $d \geq 6$?
• RQ5 What is the precise combinatorial structure of the $\delta$-vector in terms of triangulations and lattice point contributions in the box of faces?

Key findings

• The $\delta$-vector of a reflexive polytope is expressed as a positive linear combination of $h$-vectors of simplicial polytopes via orbifold cohomology, providing a new combinatorial formula.
• A counterexample to Hibi's conjecture is constructed in dimension 6 with $\delta_P = (1,6,8,6,8,6,1)$, which is not unimodal due to the descent at $\delta_2 = 8 > \delta_1 = 6$ and $\delta_3 = 6 < \delta_4 = 8$.
• The paper generalizes this counterexample to $2m$-dimensional reflexive polytopes with $\delta_P = (1,2m,2m+2,2m,2m+2,\ldots,2m,1)$, showing $[\frac{m-1}{2}]$ descents in the first half of the vector.
• The generating function of the $\delta$-vector is shown to equal the sum over faces $F$ of the triangulation of $t^{\Psi_K(v)} h_{\Delta_F}(t)$, where $v \in \operatorname{Box}(F)$, establishing a precise link between geometry and combinatorics.
• The formula is proven both via algebraic geometry (using orbifold cohomology and motivic integration) and via a purely combinatorial lattice point decomposition argument.
• The construction confirms that non-unimodal $\delta$-vectors exist for reflexive polytopes in dimension $d \geq 6$, resolving an open case of Hibi's conjecture.