A cohomological interpretation of Brion’s formula

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 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.

Keywords

polytope, cone, lattice point enumerator, toric variety, line bundle, Čech cohomology

2010 Mathematics Subject Classification

05A19, 14M25, 52B20

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Published 1 January 2007