Using the descriptions of Kaneyama and Klyachko this package implements the construction of equivariant vector bundles on toric varieties.
Note that this package implements vector bundles in Kaneyama's description only over pure and full dimensional fans.
uses the OldPolyhedra
package by René Birkner
. At least version 1.1 of OldPolyhedra
must be installed via installPackage
to use OldToricVectorBundles
Each vector bundle is saved either in the description of Kaneyama or the one of Klyachko. The first description gives the multidegrees (in the dual lattice of the fan) of the generators of the bundle over each full dimensional cone, and for each codimension-one cone a transition matrix (See ToricVectorBundleKaneyama
). The description of an equivariant vector bundle given by Klyachko consists of filtrations of a fixed vector space for each ray in the fan of the base variety. Furthermore, these filtrations have to satisfy a certain compatibility condition (See ToricVectorBundleKlyachko
For the mathematical background see
Tamafumi Kaneyama,On equivariant vector bundles on an almost homogeneous variety, Nagoya Math. J. 57, 1975.
Alexander A. Klyachko,Equivariant bundles over toral varieties, Izv. Akad. Nauk SSSR Ser. Mat., 53, 1989.
Markus Perling,Resolution and moduli for equivariant sheaves over toric varieties, PhD Thesis, 2003.