A cohomological interpretation of Brion's formula
Abstract
A subset K of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in K. Under suitable hypotheses, this series represents a rational function R(K). Michel Brion has discovered a surprising formula relating the rational function 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 noetherian ground rings.
Cite
@article{arxiv.math/0607464,
title = {A cohomological interpretation of Brion's formula},
author = {Thomas Huettemann},
journal= {arXiv preprint arXiv:math/0607464},
year = {2014}
}
Comments
15 pages; uses Paul Taylor's "diagrams" and "QED" macro packages; v2: "noetherian" hypothesis removed, minor typos corrected