Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

Details Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

1993-02-09

arxiv.org/abs/alg-geom/9302003v1

Mathematics

Algebraic Geometry

29 pages, latex 2.09

Scientific paper

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}.

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