Homology, Homotopy and Applications

Volume 9 (2007)

Number 2

A cohomological interpretation of Brion’s formula

Pages: 321 – 336

DOI: http://dx.doi.org/10.4310/HHA.2007.v9.n2.a11

Author

Thomas Hüttermann (Department of Pure Mathematics, Queen’s University Belfast, Northern Ireland, United Kingdom)

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 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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