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# 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.

## See also

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

## Authors

- René Birkner <rbirkner@math.fu-berlin.de>
- Nathan Ilten <nilten@cs.uchicago.edu>
- Lars Petersen <petersen@math.fu-berlin.de>

## Certification

Version 1.0 of this package (under the name "ToricVectorBundles") 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 (DOI: 10.2140/jsag.2010.2.11). That version can be obtained from the journal.

## Version

This documentation describes version 1.1 of OldToricVectorBundles, released August 21, 2014.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{OldToricVectorBundlesSource,
      title = {{OldToricVectorBundles: vector bundles on toric varieties. Version~1.1}},
      author = {Ren{\'e} Birkner and Nathan Ilten and Lars Petersen},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

    @article{OldToricVectorBundlesArticle,
      title = {{Computations with equivariant toric vector bundles}},
      author = {Ren{\'e} Birkner and Nathan Ilten and Lars Petersen},
      journal = {The Journal of Software for Algebra and Geometry: Macaulay2},
      volume = {2},
      year = {2010},
    }

## Exports

- Types
  - ToricVectorBundle -- the class of all toric vector bundles
  - ToricVectorBundleKaneyama -- the class of all toric vector bundles in Kaneyama's description
  - ToricVectorBundleKlyachko -- the class of all toric vector bundles in Klyachko's description
- 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
  - 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
  - `addBase(ToricVectorBundleKlyachko,List)` -- see addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description
  - `addBaseChange(ToricVectorBundleKaneyama,List)` -- see addBaseChange -- changing the transition matrices of a toric vector bundle
  - `addDegrees(ToricVectorBundleKaneyama,List)` -- see addDegrees -- changing the degrees of a toric vector bundle
  - `addFiltration(ToricVectorBundleKlyachko,List)` -- see addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
  - `areIsomorphic(ToricVectorBundleKlyachko,ToricVectorBundleKlyachko)` -- see areIsomorphic -- checks if two vector bundles are isomorphic
  - `base(ToricVectorBundleKlyachko)` -- see base -- the basis matrices for the rays
  - `cartierIndex(List,Fan)` -- see cartierIndex -- the Cartier index of a Weil divisor
  - `charts(ToricVectorBundle)` -- see charts -- the number of maximal affine charts
  - `cocycleCheck(ToricVectorBundleKaneyama)` -- see cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition
  - cokernel(ToricVectorBundleKlyachko,Matrix) -- the cokernel of a morphism to a vector bundle
  - `cotangentBundle(Fan)` -- see cotangentBundle -- the cotangent bundle on a toric variety
  - `deltaE(ToricVectorBundle)` -- see deltaE -- the polytope of possible degrees that give non zero cohomology
  - `details(ToricVectorBundle)` -- see details -- the details of a toric vector bundle
  - dual(ToricVectorBundle) -- the dual bundle of a toric vector bundle
  - `eulerChi(Matrix,ToricVectorBundle)` -- see eulerChi -- the Euler characteristic of a toric vector bundle
  - `eulerChi(ToricVectorBundle)` -- see eulerChi -- the Euler characteristic of a toric vector bundle
  - `existsDecomposition(ToricVectorBundleKlyachko,List)` -- see existsDecomposition -- checks if a list of matrices of weight vectors for each maximal cone admits a decomposition
  - exteriorPower(ZZ,ToricVectorBundle) -- the 'l'-th exterior power of a toric vector bundle
  - fan(ToricVectorBundle) -- the underlying fan of a toric vector bundle
  - `filtration(ToricVectorBundleKlyachko)` -- see filtration -- the filtration matrices of the vector bundle
  - `findWeights(ToricVectorBundleKlyachko)` -- see findWeights -- finds the possible weight vectors for the maximal cones
  - HH^ZZ ToricVectorBundle -- the i-th cohomology group of a toric vector bundle
  - HH^ZZ(ToricVectorBundle,List) -- the i-th cohomology of a toric vector bundle for a given list of degrees
  - HH^ZZ(ToricVectorBundle,Matrix) -- the i-th cohomology of a toric vector bundle in a given degree
  - `hirzebruchFan(ZZ)` -- see hirzebruchFan -- the fan of the n-th Hirzebruch surface
  - image(ToricVectorBundleKlyachko,Matrix) -- the image of a vector bundle under a morphism
  - `isGeneral(ToricVectorBundleKlyachko)` -- see isGeneral -- checks whether a toric vector bundle is general
  - isomorphism(ToricVectorBundleKlyachko,ToricVectorBundleKlyachko) -- the isomorphism if the two bundles are isomorphic
  - `isVectorBundle(ToricVectorBundle)` -- see isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle
  - kernel(ToricVectorBundleKlyachko,Matrix) -- the kernel of a morphism to a vector bundle
  - maxCones(ToricVectorBundle) -- the list of maximal cones of the underlying fan
  - net(ToricVectorBundleKaneyama) -- displays characteristics of a toric vector bundle
  - net(ToricVectorBundleKlyachko) -- displays characteristics of a toric vector bundle in Klyachko's description
  - `pp1ProductFan(ZZ)` -- see pp1ProductFan -- the fan of n products of PP^1
  - `projectiveSpaceFan(ZZ)` -- see projectiveSpaceFan -- the fan of projective n space
  - `randomDeformation(ToricVectorBundleKlyachko,ZZ)` -- see randomDeformation -- a random deformation of a given toric vector bundle
  - `randomDeformation(ToricVectorBundleKlyachko,ZZ,ZZ)` -- see randomDeformation -- a random deformation of a given toric vector bundle
  - rank(ToricVectorBundle) -- the rank of the vector bundle
  - rays(ToricVectorBundle) -- the rays of the underlying fan
  - `regCheck(ToricVectorBundleKaneyama)` -- see regCheck -- checking the regularity condition for a toric vector bundle
  - ring(ToricVectorBundle) -- the graded ring of the bundle
  - symmetricPower(ZZ,ToricVectorBundle) -- the 'l'-th symmetric power of a toric vector bundle
  - `tangentBundle(Fan)` -- see tangentBundle -- the tangent bundle on a toric variety
  - tensor(ToricVectorBundle,ToricVectorBundle) -- the tensor product of two toric vector bundles
  - `toricVectorBundle(ZZ,Fan)` -- see toricVectorBundle -- the trivial bundle of rank 'k' for a given fan
  - ToricVectorBundle ** ToricVectorBundle -- the tensor product of two toric vector bundles
  - ToricVectorBundle ++ ToricVectorBundle -- the direct sum of two toric vector bundles
  - ToricVectorBundle == ToricVectorBundle -- checks for equality
  - toricVectorBundle(ZZ,Fan,List,List) -- a toric vector bundle of rank 'k' with given filtrations or degrees
  - `twist(ToricVectorBundleKlyachko,List)` -- see twist -- twists a toric vector bundle with a line bundle
  - `weilToCartier(List,Fan)` -- see weilToCartier -- the line bundle given by a Cartier divisor

## For the programmer

The object OldToricVectorBundles is a package, defined in OldToricVectorBundles.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2016:0.