All Questions
13
questions
1
vote
0
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218
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Reference request: Weil's uniformization theorem
The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...
- 1,837
2
votes
0
answers
542
views
Finite groups principal bundles
I am studying principal bundles from the point of view of algebraic geometry and I have come up with the following question. For the sake of clarity, a principal $G$-bundle over a scheme $X$ is just a ...
- 439
6
votes
1
answer
318
views
Extending $G$-torsors on open subsets of affine space
Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). ...
- 889
2
votes
0
answers
181
views
Diagonal action on external product of trivial principal bundles
(Title, tags and phrasing of this problem might not very good, so please feel free to edit it.)
In the course of writing a long and technical proof, I recently came across the following problem:
Let ...
- 21
0
votes
1
answer
346
views
basic question on quotient stacks
Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
4
votes
0
answers
256
views
Group scheme with isomorphic fibers
Let $X$ be a smooth irreducible algebraic curve over $\mathbb C$. Let $\mathcal G\rightarrow X$ be a smooth affine group scheme over $X$ such that for any closed points $p\in X$, we have $\mathcal G_p\...
- 1,871
0
votes
2
answers
422
views
Existence of $B$-reduction of a $G$-torsor on a curve
Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.
Given a $G$-torsor $E$ on $X$ in the ...
- 5,522
5
votes
2
answers
288
views
Lindel's theorem for semisimple simply connected G
Let $k$ be a field.
$G/k$ be a simply connected semisimple algebraic group.
Let $X/k$ be a smooth affine $k$-scheme.
Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...
- 525
8
votes
1
answer
1k
views
Are principal bundles isotrivial?
Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such ...
- 1,550
3
votes
2
answers
715
views
Higgs bundle and stable bundle
Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...
- 3,452
3
votes
1
answer
167
views
homogenous bundles
Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...
- 543
7
votes
5
answers
847
views
Principal bundles over groups
If we have an extension of groups (say algebraic groups or group schemes) $1\to F\to P\to G\to 1$, then $P$ is a principal $F$-bundle over $G$ (is it locally trivial?). How about going in the opposite ...
- 15.5k
7
votes
1
answer
2k
views
Does local triviality in the fppf topology imply local triviality in the etale topology?
Given an algebraically closed field $k$, a smooth group scheme $G$ over $k$
and a principal $G$-bundle $X \rightarrow Y$, which is locally trivial in the fppf topology.
Is this bundle also locally ...
- 1,391