Klyachko's classification of toric vector bundles - an ongoing discussion

Question

I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra) I am reading Sam Payne's version of it as presented in the preliminaries of his article Moduli of Toric Vector Bundles and also consulting the relevant parts of Jose Luis Gonzalez's article Okounkov bodies on projectivizations of rank two vector bundles. I will use this post to write elaborations and ask questions that arise during my attempt. I already posted couple of preliminary questions here.

Klyachko's classification theorem (Payne 2.3):

The category of toric vector bundles on $X(\Delta)$ is naturally equivalent to the category of finite-dimensional k-vector spaces $E$ with collections of decreasing filtrations $\{E^\rho(i)\}$ indexed by the rays of $\Delta$, satisfying the following compatibility condition: For each cone $\sigma$, there exists a decomposition $$E=\bigoplus_{[u]\in M_\sigma} E_{[u]}$$ with $M_\sigma=M/(\sigma^\perp\cap M)$ such that $$E^\rho(i)=\sum_{[u](v_\rho)\geq i}E_{[u]} $$ for every $\rho\preceq\sigma$ and $i\in\mathbb{Z}.$

My attempt so far (this is elaborated in Payne in 2.1): Given a toric vector bundle $E$ on $X(\Delta)$ I was able to understand the construction of a finite dimensional vector space $E_{x_0}$ as the fiber at the identity element of the torus and its decomposition

$$E_{x_0}=\bigoplus_{[u]\in M_\sigma} E_{[u]}$$

via the embedding of the T-eigenspaces

$$ \Gamma(U_\sigma,E) = \oplus_{u\in M}\Gamma(U_\sigma,E)_u\hookrightarrow E^\sigma_u\subset E_x0$$

using the evaluation at the identity element as the injective map. I also understood the decreasing filtration $$\dots\supset E^\rho(i-1)\supset E^\rho(i)\supset E^\rho(i+1)\supset\dots$$

So the first part of the compatibility condition is alright. I am currently trying to understand several things regarding the second part, namely that $$E^\rho(i)=\sum_{[u](v_\rho)\geq i}E_{[u]} $$ for every $\rho\preceq\sigma$ and $i\in\mathbb{Z}.$ My questions:

1. Why do we use the sum here, I intuitively thought about union.
2. How does an element of $E^\rho(i)$ looks like?
3. Why does it hold for the above construction?

Answer 1

1. That sumations mean the generated by the subspace generated by all the $E_{[u]}$ satisfying that $\langle u, v_{\rho}\rangle\geq i$. In some sense, we can think that we have a switch to turn on/off different elements of the filtrations under the condition $\langle u, v_{\rho}\rangle\geq i$ for any $u$ and a given ray $\rho$ in $\sigma$. In other words (Klyachko's words), it is equivalent to say that the filtration generates a distributive lattice where the operations are intersection and $F_1+F_2=\langle F_1, F_2\rangle$.

2. Any $E^{\rho}(i)$ is a vector space. Warning! It depends on $i$. And the big goal constructing Klyachko filtrations is that for any $u$ holding $\langle u, v_{\rho}\rangle\geq i$ is possible decompose $E^{\rho}(i)$ as sumations of $E_{[u]}$.

3. I do not understand the question.

Why are you studying this topic?