Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

TL;DR

This paper applies the Lefschetz fixed point theorem to toric varieties to derive explicit formulas for counting lattice points and computing the volume of convex polytopes, connecting algebraic geometry with convex geometry.

Contribution

It introduces a novel application of the Lefschetz fixed point theorem to lattice point enumeration in convex polytopes, providing explicit formulas and geometric interpretations.

Findings

Derived explicit formulas for lattice point counts and volume.

Established equivalence with Brion's results.

Provided elementary convex geometric interpretations.

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}.