A cohomological interpretation of Brion's formula

Abstract

A subset $P$ of $mathbb{R}^n$ gives rise to a formal Laurent series with monomials corresponding to lattice points in $P$ . Under suitable hypotheses, this series represents a rational function $R(P)$; this happens, for example, when $P$ is bounded in which case $R(P)$ is a Laurent polynomial. Michel Brion [2] has discovered a surprising formula relating the Laurent polynomial $R(P)$ of a lattice polytope $P$ to the sum of rational functions corresponding to the supporting cones subtended at the vertices of $P$ . The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample line bundle on a projective toric variety. The paper also contains some general remarks on the cohomology of torus-equivariant line bundles on complete toric varieties, valid over arbitrary commutative ground rings.