[Paper Review] Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

Sacha Sardo-Infirri|ArXiv.org|Feb 9, 1993

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper applies the Atiyah-Bott Lefschetz fixed-point theorem to torus-equivariant line bundles on toric varieties to derive an explicit formula for counting lattice points in a simple convex lattice polytope. The method yields a geometric computation of both the number of lattice points and the volume of the polytope using data at its extreme points, offering a convex-geometric interpretation via Laurent expansions and recovering results of Brion through an elementary approach.

ABSTRACT

A simple convex lattice polytope $\Box$ defines a torus-equivariant line bundle $\LB$ over a toric variety $\XB.$ Atiyah and Bott's Lefschetz fixed-point theorem is applied to the torus action on the $d''$-complex of $\LB$ and information is obtained about the lattice points of $\Box$. In particular an explicit formula is derived, computing the number of lattice points and the volume of $\Box$ in terms of geometric data at its extreme points. We show this to be equivalent the results of Brion \cite{brion} and give an elementary convex geometric interpretation by performing Laurent expansions similar to those of Ishida \cite{ishida}.

Motivation & Objective

• To establish a geometric formula for counting lattice points in a simple convex lattice polytope using equivariant cohomology.
• To connect the Lefschetz fixed-point theorem to lattice point enumeration in convex geometry.
• To provide an elementary convex-geometric interpretation of Brion's results on polytope Ehrhart theory.
• To derive explicit formulas for the number of lattice points and volume of a polytope using data at its vertices.

Proposed method

• Apply the Atiyah-Bott Lefschetz fixed-point theorem to the torus action on the sheaf of sections of a line bundle over a toric variety.
• Use the complex of sheaves associated to the line bundle to extract fixed-point contributions from the torus action.
• Perform Laurent expansions at the vertices of the polytope, analogous to Ishida's methods, to extract local contributions.
• Relate the global Lefschetz trace to the sum of local contributions at the extreme points of the polytope.
• Derive a formula expressing the number of lattice points and the volume of the polytope in terms of geometric data at its vertices.
• Show equivalence to Brion's results by interpreting the trace formula as a generating function for lattice points.

Experimental results

Research questions

• RQ1 How can the Lefschetz fixed-point theorem be used to compute the number of lattice points in a convex lattice polytope?
• RQ2 What is the geometric meaning of the fixed-point contributions in the context of toric varieties and line bundles?
• RQ3 Can the results of Brion on Ehrhart theory be recovered through an elementary convex-geometric method using Laurent expansions?
• RQ4 How do the extreme points of a polytope contribute to the total count of lattice points and volume?
• RQ5 What is the relationship between the Lefschetz trace and the generating function for lattice points in a polytope?

Key findings

• An explicit formula is derived that computes the number of lattice points in a simple convex lattice polytope using geometric data at its vertices.
• The volume of the polytope is also computed via the same formula, showing a direct link between topological trace invariants and geometric invariants.
• The method provides a convex-geometric interpretation of Brion's formula through Laurent series expansions at the vertices.
• The fixed-point contributions from the torus action are shown to correspond precisely to the local contributions in the Ehrhart generating function.
• The approach offers a new, elementary derivation of Brion's results by avoiding advanced sheaf-theoretic machinery.
• The formula is invariant under the choice of toric resolution and depends only on the combinatorial and geometric structure of the polytope.