Peng Zhou

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blog:2023-01-27 [2023/01/28 07:35] – pzhoublog:2023-01-27 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ===== Vector Bundles on P2\P^2 ===== ===== Vector Bundles on P2\P^2 =====
 It is always a good idea to share your thoughts, it might induce more sparks.  It is always a good idea to share your thoughts, it might induce more sparks. 
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 +We follow Knutson and Sharpe. 
  
 Consider the moduli space of rank nn vector bundles on P2\P^2. It is given by a disjoint union of components, labelled by $(\lambda, \mu, \nu) \in (X^*(T)/W)^3$, dominant weights Consider the moduli space of rank nn vector bundles on P2\P^2. It is given by a disjoint union of components, labelled by $(\lambda, \mu, \nu) \in (X^*(T)/W)^3$, dominant weights
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 First, we recall Klyacho's description of toric vector bundle on P2\P^2. Consider v1,v2,v3∈Nv_1, v_2, v_3 \in N the three ray generators of the toric fan. Let D1,D2,D3D_1, D_2, D_3 be the corresponding divisor. The subtorus TiT_i for viv_i fixes DiD_i. Consider the vector bundle EE on P2\P^2, we have TiT_i acting on E∣DiE|_{D_i}, with weights. They give me a collection of hyperplanes.  First, we recall Klyacho's description of toric vector bundle on P2\P^2. Consider v1,v2,v3∈Nv_1, v_2, v_3 \in N the three ray generators of the toric fan. Let D1,D2,D3D_1, D_2, D_3 be the corresponding divisor. The subtorus TiT_i for viv_i fixes DiD_i. Consider the vector bundle EE on P2\P^2, we have TiT_i acting on E∣DiE|_{D_i}, with weights. They give me a collection of hyperplanes. 
  
-Now, we can consider the restriction of EE over the torus fixed points. Then we have +Now, we can consider the restriction of EE over the torus fixed points. Here is {{ :blog:alexander_a_klyachko.pdf |Klyacho's}} 'filtration description' of a toric vector bundle.  
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 +So, why we have a filtration? We can say, if we do restriction to fixed, points, we get a weight decomposition of the fiber over there. More generally, we get a multi-polytope, as shown in [[https://web.ma.utexas.edu/users/sampayne/pdf/Moduli-toric-vector-bundles.pdf | Sam Payne's paper]] 
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 +How does this compare with the configuration space of decorated flags?  
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blog/2023-01-27.1674891329.txt.gz · Last modified: 2023/06/25 15:53 (external edit)