Homology, Homotopy and Applications

A cohomological interpretation of Brion's formula

Thomas Hüttermann

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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.

Article information

Homology Homotopy Appl., Volume 9, Number 2 (2007), 321-336.

First available in Project Euclid: 23 January 2008

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Mathematical Reviews number (MathSciNet)

Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 05A19: Combinatorial identities, bijective combinatorics 14M25: Toric varieties, Newton polyhedra [See also 52B20]

polytope cone lattice point enumerator toric variety line bundle Čech cohomology


Hüttermann, Thomas. A cohomological interpretation of Brion's formula. Homology Homotopy Appl. 9 (2007), no. 2, 321--336. https://projecteuclid.org/euclid.hha/1201127340

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