All Questions
Tagged with equivariant vector-bundles
11 questions
25
votes
1
answer
839
views
Vector bundles on $\mathbb{A}^n / G$
Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
- 2,310
5
votes
2
answers
1k
views
Classification of (complex algebraic) vector bundles on punctured affine space
The Quillen-Suslin theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector ...
- 23.4k
5
votes
2
answers
727
views
Jets of Equivariant Vector Bundles
Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
- 3,409
3
votes
1
answer
452
views
Vector bundles on a weighted projective stack
Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack.
Each vector ...
- 811
3
votes
0
answers
126
views
group actions of $S^3$ on the configuration space of projective plane
Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
- 2,223
2
votes
1
answer
401
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,974
2
votes
1
answer
372
views
When is restriction an equivalence of categories of equivariant vector bundles?
Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor
$i^*: Vect^G(X) \...
- 853
1
vote
1
answer
426
views
Do there exist equivariant sheafs that are not equivariant vector bundles?
For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...
1
vote
1
answer
322
views
Short exact sequence of equivariant line bundles on $\mathbb P^1$
I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^...
1
vote
0
answers
237
views
Equivariant vector bundles whose quotient map preserves the stabilizer
Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question.
Assume that $E\to M$ is a vector bundle which has the potential of admitting ...
- 366
0
votes
1
answer
204
views
Equivariant sheaves on $\mathbb P^1$
Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
- 2,974