Geodesic
A cohomological interpretation of Brion’s formula
Homology, homotopy, and applications, Tome 9 (2007) no. 2, pp. 321-336.

Voir la notice de l'article provenant de la source International Press of Boston

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.
DOI : 10.4310/HHA.2007.v9.n2.a11
Classification : 05A19, 14M25, 52B20
Keywords: polytope, cone, lattice point enumerator, toric variety, line bundle, Čech cohomology 
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     title = {A cohomological interpretation of {Brion{\textquoteright}s} formula},
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     pages = {321--336},
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Thomas Hüttermann. A cohomological interpretation of Brion’s formula. Homology, homotopy, and applications, Tome 9 (2007) no. 2, pp. 321-336. doi : 10.4310/HHA.2007.v9.n2.a11. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2007.v9.n2.a11/

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