3
votes
0answers
225 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 ...
3
votes
1answer
275 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
votes
1answer
159 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 ...
7
votes
5answers
578 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 ...
6
votes
1answer
596 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 ...