All Questions
16
questions
10
votes
0
answers
263
views
Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...
- 1,111
6
votes
0
answers
173
views
Functions of polynomial growth on linear algebraic groups
$\DeclareMathOperator\GL{GL}$Let $G$ be a complex linear algebraic group, i.e. a subgroup in $\GL_n({\mathbb C})$, defined by a system of polynomial equations
$$
p_i(x)=0
$$
(here $p_i$ are ...
- 7,366
2
votes
1
answer
236
views
Compactifications of group varieties
Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$.
Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, ...
user335418
11
votes
1
answer
519
views
What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$?
Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence
$$1\to \mathbb{G}_a^n\times \mathbb{G}_m^m\to G\to A\to 1,$$
where $A$ ...
- 1,139
4
votes
1
answer
130
views
$\mathbb{C}^*$-equivariant smooth completion of a quasiprojective variety
A famous theorem by Sumihiro states that, given a normal quasi-projective variety $X$ with a regular $G$-action (where $G$ is a connected linear algebraic group), there is a G-equivariant
projective ...
- 1,647
3
votes
0
answers
113
views
Geometry of elements with prescribed multiplicity eigenvalues
Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...
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
6
votes
3
answers
1k
views
Fixed points under a finite group action on projective variety
Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits_{i=1}^N X_i$, whose irreducible components $X_i$ are all smooth and of the same dimension $d$, and ...
- 1,647
1
vote
1
answer
141
views
does $Aut^0$ act trivially on the Neron-Severi group?
Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0_{X/k}(k)$ act trivially on $NS(X)$?
- 181
7
votes
0
answers
417
views
Monodromy group from semisimple local system is reductive
Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $\rho: \pi_1(X,x)\rightarrow Gl(n,\mathbb{C})$ be a semisimple representation of fundamental group of $X$. The monodromy group $M(\rho, x)$...
- 1,081
7
votes
0
answers
186
views
Finite abelian groups acting on smooth varieties and "invariant" invertible functions
Let $G$ be a finite abelian group acting on a smooth quasi-projective variety $X$ over $\mathbb C$. Let $E(X) = \mathcal{O}(X)^*/\mathbb C^*$.
Under what conditions is $E(X)^G$ the trivial group?
...
- 71
0
votes
2
answers
401
views
Group actions on affine space which are almost good
Let $G$ be a finite group acting on $\mathbb A^n_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two.
Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, ...
- 11
8
votes
0
answers
175
views
Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset
This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety.
General question: Let $X$ be a smooth separated ...
- 81
4
votes
1
answer
250
views
Smooth algebraic stacks with precisely two $\mathbb C$-objects
In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...
- 193
1
vote
1
answer
214
views
Decomposing quasi-finite separated group schemes
Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
