Description
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.
ToricVectorBundles uses the
Polyhedra package by
René Birkner. At least version 1.1 of
Polyhedra must be installed via
installPackage to use
ToricVectorBundles.
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.
Certification 
Version 1.0 of this package was accepted for publication in volume 2 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-06-15, in the article Computations with equivariant toric vector bundles. That version can be obtained from the journal or from the Macaulay2 source code repository, after installing subversion, with the following shell command:
svn export -r 11367 svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/ToricVectorBundles.m2
The following command will display the log messages accompanying any changes to the file in the repository since publication.
svn log -r 11368:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/ToricVectorBundles.m2
The following command will summarize the changes to the file in the repository since publication, in the format the program diff uses: lines starting with + have been added, and lines starting with - have been removed. (Changes to white space or end of line style will not be reported.)
svn diff -x "-b --ignore-eol-style" -r 11367:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/ToricVectorBundles.m2
The differences between two releases in the repository mentioned in the log can be displayed by replacing 11367:HEAD by the pair of release numbers separated by a colon.