With the notation and conventions introduced above it is now possible to state the fundamental theorem of Klyachko which completely describes toric vector bundles in linear algebraic terms:
The category of toric vector bundles on the toric variety $X$ is equivalent to the category of finite dimensional $k$-vector spaces $E_0$ with collections of increasing filtrations $\{E^{\rho}(i)| i \in{} \mathbb{Z}\}$, indexed by the rays of $\Sigma$, satisfying the following compatibility condition: For each cone $\sigma \in{} \Sigma$ there is a decomposition $E_0 = \oplus_{u \in{} M_\sigma} E_u$ such that $E^{\rho}(i) = \sum_{(u,v_\rho) \leq i} E_u$ for every ray $\rho \in{} \sigma$ and every $i \in{} \mathbb{Z}$.
"In contrast to the implementation of Kaneyama's description this one works for every toric variety $X$ i.e., there are no restrictions on the fan $\Sigma$. For each ray $\rho$ of the fan $\Sigma$ there are two matrices comprising the necessary filtration data. The first one is an invertible matrix $A(\rho) \in{} $ GL("k,QQ") whose columns contain a basis of the vector space $E_0$ which is associated to the filtration corresponding to the ray $\rho$. The second one is a ",TT "1 x k"," integer matrix, the so called filtration matrix. It determines at which step an element of the basis given in the first matrix actually contributes to a certain subspace in the filtration, i.e., if the j-th entry of the filtration matrix is i then the j-th basis vector appears at the i-th step in the filtration. Hence $E^{\rho}(i)$ is generated by all basis vectors listed in $A(\rho)$ whose corresponding entry in the filtration matrix is less or equal to $E_0$."
"To link up to the description of Kaneyama we will also discuss the example of the cotangent bundle $\mathbf{\Omega}_X$ of $X = \mathbb{P}^2$. Recall that $X$ can be given by the complete fan with rays $\rho_1 = (1,0)$, $\rho_2 = (0,1)$, and $\rho_3 = (-1,-1)$. There are three maximal cones, namely $\sigma_1$ spanned by $\rho_1,\rho_2$, $\sigma_2$ spanned by $\rho_2,\rho_3$, and $\sigma_3$ spanned by $\rho_3,\rho_1$. Each of them corresponds to a torus invariant affine chart $U_{\sigma_i}$. It follows that the $k[\sigma_1^v \cap M]$-module $\Gamma(U_{\sigma_1},\Omega_X)$ is generated by $dx := d(x^{[1,0]})$, and $dy := d(x^{[0,1]})$, and analogously for the remaining charts. We now fix a basis of $\Omega_0$ by evaluating the sections $dx,dy$ at the unit $t_0$. This gives rise to filtrations $\Omega^\rho(i)$. We only consider the example $\rho = \rho_3$. The filtrations for the two other rays can be found by analogous calculations. Now, $k[U_{\rho_3}] = k[x^{-1},x^{-1}y,xy^{-1}]$. Then, $\Gamma(U_{\rho_3},\Omega_X)$ is generated as a $k[U_{\rho_3}]$-module by $-x^{-2}dx, -x^{-2}ydx + x^{-1}dy$. Thus, $\Gamma(U_{\rho_3},\Omega_X)_{[1,0]} = 0$, $\Gamma(U_{\rho_3},\Omega_X)_{[0,0]}$ is generated by $xy^{-1}(-x^{-2}ydx + x^{-1}dy)$, and $\Gamma(U_{\rho_3},\Omega_X)_{[-1,0]}$ is two-dimensional. Since $[1,0], [0,0]$, and $[-1,0]$ pair with $v_{\rho_3}=(-1,-1)$ to respectively $-1, 0$, and $1$, the filtration $\Omega^{\rho_3}(i)$ jumps at $1$ and $0$ with corresponding basis vectors $(0,-1)$ and $(-1,1)$. Since $\Omega_X$ already is a vector bundle we do not have to check the compatibility conditions."
An instance of class ToricVectorBundleKlyachko, when displayed or printed, gives an overview of the characteristics of the bundle:
i1 : E = cotangentBundle(projectiveSpaceFan 2) o1 = {dimension of the variety => 2 } number of affine charts => 3 number of rays => 3 rank of the vector bundle => 2 o1 : ToricVectorBundleKlyachko |
To see all relevant details of a bundle use details. The data described above are stored in a single hash table. In the example from above, the keys are the rays of the fan, and each of them comes with a base matrix and a filtration matrix:
i2 : details E o2 = HashTable{| -1 | => (| 0 -1 |, | 1 0 |)} | -1 | | -1 1 | | 0 | => (| 0 1 |, | 1 0 |) | 1 | | 1 0 | | 1 | => (| 1 0 |, | 1 0 |) | 0 | | 0 1 | o2 : HashTable |
The object ToricVectorBundleKlyachko is a type, with ancestor classes ToricVectorBundle < HashTable < Thing.