# OldToricVectorBundles -- cohomology computations of equivariant vector bundles on toric varieties

## 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.

OldToricVectorBundles 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.

• OldPolyhedra -- for computations with convex polyhedra, cones, and fans

## Certification

Version 1.0 of this package was accepted for publication in volume 2 of 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.

## Version

This documentation describes version 1.1 of OldToricVectorBundles.

## Source code

The source code from which this documentation is derived is in the file OldToricVectorBundles.m2.

## Exports

• Types
• Functions and commands
• addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description
• addBaseChange -- changing the transition matrices of a toric vector bundle
• addDegrees -- changing the degrees of a toric vector bundle
• addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
• areIsomorphic -- checks if two vector bundles are isomorphic
• base -- the basis matrices for the rays
• cartierIndex -- the Cartier index of a Weil divisor
• charts -- the number of maximal affine charts
• cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition
• cotangentBundle -- the cotangent bundle on a toric variety
• deltaE -- the polytope of possible degrees that give non zero cohomology
• details -- the details of a toric vector bundle
• eulerChi -- the Euler characteristic of a toric vector bundle
• existsDecomposition -- checks if a list of matrices of weight vectors for each maximal cone admits a decomposition
• filtration -- the filtration matrices of the vector bundle
• findWeights -- finds the possible weight vectors for the maximal cones
• hirzebruchFan -- the fan of the n-th Hirzebruch surface
• isGeneral -- checks whether a toric vector bundle is general
• isomorphism -- the isomorphism if the two bundles are isomorphic
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle
• pp1ProductFan -- the fan of n products of PP^1
• projectiveSpaceFan -- the fan of projective n space
• randomDeformation -- a random deformation of a given toric vector bundle
• regCheck -- checking the regularity condition for a toric vector bundle
• tangentBundle -- the tangent bundle on a toric variety
• toricVectorBundle -- the trivial bundle of rank 'k' for a given fan
• twist -- twists a toric vector bundle with a line bundle
• weilToCartier -- the line bundle given by a Cartier divisor
• Methods

## For the programmer

The object OldToricVectorBundles is .