Questions tagged [geometric-invariant-theory]
for questions on geometric invariant theory (or GIT), including stability criteria and symplectic quotients.
242 questions
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How big is the complement of stable locus $\operatorname{Bun}G$
Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type).
What is the ...
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Descent of projective bundles
A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients.
It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
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Quotients of toric varieties
This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...
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Fiber product arising from reductive group action on varieties
Let $G$ be a reductive group acting on a smooth projective $X$. Let $P$ be a parabolic subgroup of $G$, and $Y$ a locally closed subvariety invariant under $P$. Assume in addition $Y$ is smooth. ( ...
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1
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(Co)tangent sheaves to good quotients
Suppose given a variety $X$ over an algebraically closed field $K$, $\mathrm{char}K = 0$, equipped with an action of a reductive group $G$. Suppose also that $X$ admits a good quotient $p\colon X\to Y:...
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Lifting of curves in characteristic zero
Let $K$ be an algebraically closed field of characteristic zero. Let $G$ be an affine reductive group over $K$, and let $H$ be a closed reductive subgroup of $G$.
Since $H$ is reductive the GIT ...
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Fibre of GIT morphism
Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$. I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$
Let $ v \...
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2
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Motivating the Quotient of an Algebraic Variety
Let $X$ be a variety with a $G$-action by an algebraic group.
My question refers to a motivating example from:
https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf
Here is the relevant ...
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1
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1
answer
250
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Secondary fan and KN strata
Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...
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1
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Preparation for GIT (Geometric Invariant Theory)
I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford ...
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Quotient of a Fano variety by a torus
We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$.
I think we can canonically linearize the ...
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4
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239
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Invariants of linear endomorphisms of tensor products
Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero.
Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
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3
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235
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Moduli space of nilpotent Lie algebras
Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration.
I'm interested in some ...
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4
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117
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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 $−2πi\Omega$ ...
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104
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Alternatives to the ring of invariants depicting the orbit closures?
Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
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2
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1
answer
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Reference on reductive group acting on quotient algebra
In unpublished notes by Yi Hu (which appear to be no longer online), I found the following:
Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally ...
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2
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1
answer
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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})^{...
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347
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Good quotients and coarse moduli spaces
I started study the moduli space theory, by Newstead's book ''Lectures on moduli problems and orbit spaces". Theorem 3.21 says that given a variety $X$ and a line bundle $L$ over $X$, then for any L-...
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140
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Topological criterion for GIT semistability
Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. ...
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119
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Invariants and subgroups
Let $G$ be an affine algebraic group over some algebraically closed field $K$, and let $H$ be a closed subgroup.
Assume that $G$ acts algebraically on an affine variety $X$.
Assume that $X'\subseteq ...
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9
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1
answer
294
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A duality result for Coxeter groups
Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
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GIT quotients of open subsets
Let X be a projective variety on with a action of reductive group G. Let L be a G Linearised ample line bundle on X. Let U be a G stable open subset of X. Let $U^{ss}:=X^{ss}\cap U$. Is it true that $...
user100841
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1
answer
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GIT quotient vs. largest Hausdorff quotient
Let a group $G$ act on a (not necessarily irreducible) algebraic variety over ${\bf C}$.
It seems to be well-known that the quotient in the sense of geometric invariant theory (i.e., the categorical ...
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Progress since Luna's theorem on smooth invariants
In 1976, Luna proved the following important theorem of smooth invariant theory:
Let $G$ be a real reductive Lie group and a representation of $G$ on a real finite dimensional vector space $V$. ...
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6
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141
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Binary forms and equivariant derived category
One of the classical questions in invariant theory is the classification of binary forms, i.e., the description of polynomial invariants of the ${\rm SL}_2(\mathbb{C})$-action on ${\rm Sym}^d \mathbb{...
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1
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1
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un-ordered distinct $n$-tuples of points on $\mathbb P^1$
I want to study the space $Y$ of all un-ordered $d$-tuples of points on $\mathbb P^1$. By considering the space $V_d$ of homogeneous polynomials of degree $d$ in two variables, one may identify $Y$ ...
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Orbit of a transverse manifold under the action of an algebraic group
Consider an algebraic group acting on a affine manifold. Suppose that S is an affine submanifold transverse to the action. Are there some conditions on the group, the action or S such that the orbit ...
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Generators of the same degree in a graded ring and GIT quotient
Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
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4
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2
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The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts
Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL_n$ or $G=U_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is ...
- 2,789
2
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1
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G-sweep of irreducible sub variety
Let $G$ be a connected reductive algebraic group and $X$ be a $G$-variety. Let $Y$ be a $G$-invariant irreducible sub variety of $X$ which has non-trivial intersection with the semi stable locus $X^{...
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lines in projective spaces [closed]
Let $\{v_1,v_2, \cdots , v_n, w_1,w_2, \cdots ,w_n\}$ be a basis of $\mathbb C^{2n}$. For a $n$-dimensional subspace $V \in Gr(n,\mathbb C^{2n})$ define another $n$ dimensional subspace $\bar{V} \in ...
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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 ...
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Ring of invariants and Borel subgroup
Let $G$ be a connected algebraic group (can assume $G$ to be reductive) acting on a $k$-algebra $A$. Let $B$ be a Borel subgroup of $G$.
Q. Is it generally true that the the ring of invariants $A^...
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0
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146
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Injectivity of a standard map in quiver representation
Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
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Standard Monomial basis for other types
For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for ...
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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 ...
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1
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Interactions (functors) between equivariant sheaves for different groups?
Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity).
To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
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319
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Is $\widetilde M_{0, n}$ a Mori Dream space?
I'm reading on $\overline M_{0, n}$ and $\widetilde M_{0, n}$. I know that $\overline M_{0, n}$ is a Mori Dream space for $n \leq 6$ and not a Mori Dream space for $n \geq 13$. Is there a similar ...
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Cohen-Macaulay rings in GIT
I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and ...
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172
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Projective and Quasiprojective quotients
Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-...
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406
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Categorical quotients for quasi-affine varieties
Let $X$ be an affine variety and let $G$ be a reductive algebraic group acting on $X$. Let $U \subset X$ be a $G$-invariant open set.
Under what hypothesis there exists a categorical quotient of $U$ ...
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What are the scalar conformal invariants of weight -3/2 in 3 dimensions?
I am looking for all the scalar conformal invariants (diffeomorphism-invariant polynomials $P[g]$ in the metric $g_{ij}$, its inverse $g^{ij}$ and its derivatives $g_{ij,klm\dots}$ such that $P[\...
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1
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proj of an Algebra [closed]
Let $\mathbb Z_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,...
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Semistability of tensor products under automorphisms of tensored vector spaces
Let $A,B,C,D,E,F$ be vector spaces over a field.
Let $x\in A \otimes B \otimes C$ and $y \in D \otimes E \otimes F$ be tensors that are semistable with respect to the natural actions of $\text{SL}(A) ...
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1
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Why is Mumford's GIT-quotient so effective?
According to remark 6.14 in Shigeru Mukai's An introduction to invariants and moduli (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine ...
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Question about GIT: when is the map $\pi:X//_\theta G\rightarrow X//G$ birational?
I want to ask somebody who is more familiar with the theory of GIT quotients than I am, if there is a nice list of conditions on the action of a reductive group $G$ on an affine variety $X$ over ...
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0
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213
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Categorical quotient of open subsets of affine varieties
Let $X$ be a complex affine variety and $G$ be a complex reductive group acting on $X$. Let $X//G=\operatorname{Spec}\mathbb{C}[X]^G$ and
$$\pi:X\to X//G$$
be the GIT quotient of $X$ by $G$. Suppose ...
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1
answer
322
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Supposed generalization of $X/(G \times H)\simeq (X/G)/H$ for GIT-quotients
I wonder whether it is true that the composition of two GIT-quotients is another GIT-quotient. It should be an analogue of a set-theoretic formula $X/(G \times H)\simeq (X/G)/H$ but with GIT-quotients ...
- 1,980
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0
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278
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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 ...
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2
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371
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Is this quotient of a threefold known? What are its singularities?
Assume $G$ is the Klein four group $G=\{1,\sigma_1,\sigma_2,\sigma_3\}$.
Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via:
$$\sigma_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } ...
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