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OldToricVectorBundles :: ToricVectorBundleKaneyama

ToricVectorBundleKaneyama -- the class of all toric vector bundles in Kaneyama's description

Description

"Consider an equivariant vector bundle $E$ of rank $k$ on a toric variety $X$ corresponding to a fan $\Sigma$. Then $E$ is trivial on any invariant open affine subvariety of $X$ and moreover homogeneously generated by $k$ elements. Furthermore, the transition maps between these trivializations are homogeneous of degree zero. Thus, after fixing local homogeneous generators, we get a list of degrees of generators for each cone in $\Sigma$, along with a transition map for each pair of cones. Conversely, given a list of $k$ degrees for every cone of $\Sigma$ along with transition maps satisfying compatibility and regularity conditions for every pair of cones, one can construct an equivariant vector bundle of rank $k$ on $X$."

"This description of equivariant vector bundles, due to Kaneyama, is implemented for complete, pointed fans in the following way: It is only necessary to consider charts corresponding to maximal dimensional cones of $\Sigma$. Furthermore, each codimension-one cone of $\Sigma$ corresponds to a pair of maximal dimensional cones, and thus to a transition map. Due to the compatibility condition for transition maps, one can reconstruct the transition map corresponding to an arbitrary pair from the maps of this sort. If the dimension of $\Sigma$ is $n$ then for each maximal dimensional cone the degree list of the corresponding chart is saved as an $n$ times $k$ matrix over ",TO ZZ,", giving $k$ degree vectors in the dual lattice of the fan, one for each local generator of the bundle. Additionally, for every pair of maximal cones intersecting in a common codimension-one face, there is a matrix in GL($k$,",TO QQ,"), representing the transition map between these two affine charts. Indeed, suppose that cones $\sigma_1$ and $\sigma_1$ intersect in some codimension-one face, with corresponding affine charts $U_1$ and $U_2$. Then on the intersection, the $i$-th generator for $U_1$ has a unique representation as a linear combination in the generators for $U_2$ after being multiplied with characters to all have the required degree. The coefficients in this representation form the $i$-th column of the desired matrix."

"We briefly consider the example of $\mathbb{P}^2$, corresponding to the complete fan with rays through $(0,1)$, $(1,0)$, and $(-1,-1)$. Denote by $x$ the character of weight $[1,0]$ and by $y$ the character of weight $[0,1]$. Now the coordinate rings of the three standard affine charts of $\mathbb{P}^2$ are generated by respectively $(x^{ -1},x^{ -1}y)$, $(x,y)$, and $(xy^{ -1},y^{ -1})$. This means that the modules of differentials are generated by respectively $(d(x^{ -1}),d(x^{ -1}y))$, $(dx,dy)$, and $(d(xy^{ -1}),d(y^{ -1}))$. These modules give us local trivializations of the cotangent bundle on $\mathbb{P}^2$. The degrees of the generators for the first chart then are $[-1,0]$ and $[-1,1]$, for example. Now, since $d(x^{ -1})=-x^{ -2}dx$ and $d(x^{ -1}y) = -x^{ -2}ydx + x^{ -1}dy$, we get that the transition map between the generators of the first and second chart is given by the matrix with columns $(-1,0)$ and $(-1,1)$."

An instance of class ToricVectorBundleKlyachko, when displayed or printed, gives an overview of the characteristics of the bundle:

i1 : E = cotangentBundle(projectiveSpaceFan 2,"Type" => "Kaneyama")

o1 = {dimension of the variety => 2 }
      number of affine charts => 3
      rank of the vector bundle => 2

o1 : ToricVectorBundleKaneyama

To see all relevant details of a bundle use details. The data described above is all stored in a single hash table. In the example from above, the first chart has the key 0, and transition map described above has key (0,1):

i2 : details E

o2 = (HashTable{0 => (| -1 0 |, | -1 -1 |)}, HashTable{(0, 1) => | -1 -1 |})
                      | -1 1 |  | 0  1  |                        | 0  1  |
                1 => (| 1 0 |, | 1 0 |)                (0, 2) => | 1  0  |
                      | 0 1 |  | 0 1 |                           | -1 -1 |
                2 => (| 1 -1 |, | 0  1  |)             (1, 2) => | -1 -1 |
                      | 0 -1 |  | -1 -1 |                        | 1  0  |

o2 : Sequence

Caveat

This implementation only supports vector bundles where the corresponding transition maps have coefficients in QQ.

See also

Functions and methods returning a vector bundle on a toric variety (Kaneyama's description) :

Methods that use a vector bundle on a toric variety (Kaneyama's description) :

For the programmer

The object ToricVectorBundleKaneyama is a type, with ancestor classes ToricVectorBundle < HashTable < Thing.