Questions tagged [geometric-invariant-theory]
for questions on geometric invariant theory (or GIT), including stability criteria and symplectic quotients.
233
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GIT semi-stability on graded Artinian local $\Bbbk$-algebras
Let $\Bbbk$ be a algebraically closed field of characteristic zero. A graded Artinian local $\Bbbk$-algebra is $(A,\mathfrak{m},\bigoplus A_i)$ such that $(A,\mathfrak{m})$ is an Artianian local $\...
- 910
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1
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$G$- Fixed Point Scheme explicitly
Let $G$ be an abstract finite group acting on a separated $k$-scheme $X$. ($k$ a field; note we can canonically promote $G$ to a $k$- scheme). Then a result by Demazure and Grothendieck (in "...
- 5,780
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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
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GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,711
4
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1
answer
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Symplectic structure of Higgs branch
I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
- 313
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270
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GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,711
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69
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"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
- 767
4
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Are the two notions of free $\mathbb{G}_a$-actions equivalent?
Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation
$$\...
- 910
1
vote
0
answers
141
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There exists noncommutative geometric invariant theory?
In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...
- 3,104
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1
answer
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Are there geometric $\mathbb{G}_a$-quotients with trivial stabilizers, not being principal bundles?
Consider algebraic $\mathbb{C}$-schemes. The group scheme $\mathbb{G}_a$ is the scheme $\mathbb{A}^1$ with the addition. This is not a reductive group. Here I want to know some examples of $\mathbb{G}...
- 910
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234
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Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...
- 767
4
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answers
205
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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 = \...
- 7,250
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Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof).
Suppose $G$ is a ...
- 509
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105
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Iterated quotients in GIT
Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.
Suppose also that $G$ is equipped with a normal abelian subgroup $N$ such ...
- 18.7k
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Does the orbit in geometric invariant theory have natural scheme structure
Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as ...
- 5,780