All Questions
Tagged with geometric-invariant-theory complex-geometry
16 questions
11
votes
1
answer
918
views
When Are Quotients Complete Intersections?
Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a ...
- 2,097
7
votes
1
answer
718
views
GIT and singularities
Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.
Is there a relationship between the singular ...
- 1,383
7
votes
0
answers
466
views
Kähler quotients of affine varieties and GIT
Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
- 779
5
votes
2
answers
269
views
Group actions on affine varieties with closed orbits
The following is motivated by a (now-deleted) MSE-question by @aglearner.
Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the ...
- 12.3k
5
votes
1
answer
450
views
Invariant ideal generated by invariant elements
Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions?
...
- 1,383
5
votes
2
answers
1k
views
A question about Marsden-Weinstein reduction theory
Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...
user21574
5
votes
0
answers
278
views
Smooth quotients and separation of orbits
Consider a unipotent algebraic group $G$ over $\mathbb{C}$ acting polynomially on $\mathbb{C}^n$. Suppose that the quotient exists as an analytical geometric quotient, i.e., $\mathbb{C}^n/G$ is a ...
- 189
4
votes
0
answers
227
views
Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
- 7,320
4
votes
0
answers
113
views
Cover by $K$-invariant affine open sets
Let $X$ be a non-singular complex algebraic variety (quasi-projective if necessary) and $K$ a connected compact Lie group acting on $X$ real algebraically, i.e. the action map $K \times X \to X$ is ...
- 1,383
4
votes
0
answers
117
views
Complexifed Gauge action on determinant line bundle and change of metric
Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
- 789
3
votes
1
answer
760
views
Quotient of complex manifold by a free and locally proper action (difficulty with reading German)
Let $X$ be a complex manifold with an action of $G=GL(n,\mathbb{C})$ which is free and locally proper (each point of $X$ has a $G$-invariant neighborhood on which $G$ acts properly.)
Satz 24 of the ...
- 297
3
votes
1
answer
670
views
How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?
I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
2
votes
1
answer
179
views
Analytic sections of a GIT quotient lying in the Kempf-Ness set
I would like to understand whether the following is true. Given a complex reductive group $G$ (for me $G = \operatorname{PGL}_n(\mathbb{C})$) acting on a vector space $V$ (for me $V = M_n(\mathbb{C})^{...
- 431
1
vote
1
answer
189
views
Is the conjugation action linearizable?
Let $G$ be a reductive algebraic group over some algebraically closed field $k$. Recall that, given an algebraic variety $X$ with an action of $G$, it is said that this action is linearizable if there ...
- 507
1
vote
0
answers
76
views
Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
- 41
1
vote
0
answers
93
views
Sufficient condition for moduli space of slope-stable bundles to be non-empty
I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...
- 11