# Questions tagged [equivariant]

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78
questions

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### Ample divisors on $T$-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample?
In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...

**2**

votes

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47 views

### Equivariant smooth approximation

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space ...

**6**

votes

**1**answer

265 views

### What is equivariant chains on a representation sphere?

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity.
What is the reduced chain complex $...

**3**

votes

**1**answer

100 views

### Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...

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74 views

### Equivariant Morse theory for non-compact Lie groups

Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...

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80 views

### Is there a smooth $W_{G_2}$-equivariant map from the flag manifold of $U(4)$ to that of $G_2$?

The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...

**5**

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144 views

### A group action on another group action quotient: how to best describe the resulting structure and does it have a name?

Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...

**23**

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**1**answer

622 views

### Vector bundles on $\mathbb{A}^n / G$

Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...

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53 views

### Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...

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177 views

### Definition of an equivariant connection and equivariant curvature

Can anyone give me a reference which precisely stated the definition of an equivariant connection and equivariant curvature?
Precisely, If G be a compact lie group acting linearly on a smooth ...

**8**

votes

**1**answer

348 views

### Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...

**5**

votes

**1**answer

174 views

### Graded commutativity of the $n$th Browder bracket

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...

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113 views

### Equivariant vector bundles whose quotient map preserves the stabilizer

Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question.
Assume that $E\to M$ is a vector bundle which has the potential of admitting ...

**2**

votes

**1**answer

177 views

### Resolution by locally free $G$-equivariant sheaves on varieties

I have been reading the section in the beginning of Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics by Bartocci et. al., and stumbled across the following sentence (page 26).
...

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71 views

### Does intermediate extension functor commutes with forgetful functor in equivariant derived category?

The forgetful functor from $D^b_G(X)$ to $D^b(X)$ carries $Perv_G(X)$ to $Perv(X)$ by definition $5.1$ in the book of Bernstein and Lunts. My question is do the following functors, intermediate ...

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65 views

### Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?

Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where
$$
[G\backslash X]_n=\underbrace{G\times \...

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116 views

### Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...

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98 views

### The space of Riemannian structures as an orbifold.

Consider a smooth closed manifold $M$. The space of Riemannian metrics is an open cone in the space of sections of some vector bundle. On this space the group of diffeomorphisms of $M$ acts by ...

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158 views

### Fundamental class in equivariant K-theory

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.
The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...

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127 views

### Spin equivariance of the Dirac operator-flat case

This question was posed on Math.SE but no one has answered it; it may be suitable for MathOverflow.
Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial ...

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157 views

### Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...

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votes

**1**answer

492 views

### 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 ...

**5**

votes

**2**answers

583 views

### Classification of (complex algebraic) vector bundles on punctured affine space

The Quillen-Suslin theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector ...

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67 views

### Equivariant Formula for High Dimensional Isolated set

The Atiyah-Bott-Berline-Vergne-Witten localization formula says
$S^1$ acting on compact manifold $M$ isolated fixing points. And for a closed equivariant form $\omega$, then
$$(2\pi)^{-\frac{\dim(M)}...

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63 views

### Equivariant corner straightening

Equivariant corner straightening is usually mentioned in the literature without further explanation. What would be a reference where this is done (more or less) carefully for compact Lie group actions ...

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152 views

### Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...

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95 views

### group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...

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200 views

### Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...

**4**

votes

**1**answer

347 views

### Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?

**3**

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498 views

### Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers?
Any references that state this explicitely?
Are there formulas ...

**6**

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106 views

### 2-functoriality of equivariant derived categories

I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)
For the more precise formulation, recall ...

**5**

votes

**2**answers

960 views

### Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory.
So ...

**3**

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311 views

### Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a $G$-...

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201 views

### Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...

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85 views

### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that $...

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**1**answer

202 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$...

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**1**answer

136 views

### What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...

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**1**answer

505 views

### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

**2**

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**1**answer

547 views

### Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...

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**2**answers

1k views

### Equivariant version of Morse theory

Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on StackExchange.

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124 views

### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...

**6**

votes

**1**answer

415 views

### Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$
: $X_n := G^n \times X$, $s_0:X_0 \...

**7**

votes

**1**answer

586 views

### Equivariant algebraic K-theory of affine space

Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...

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vote

**1**answer

317 views

### Do there exist equivariant sheafs that are not equivariant vector bundles?

For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...

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119 views

### Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...

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**1**answer

333 views

### Vector bundles on a weighted projective stack

Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack.
Each vector ...

**5**

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**1**answer

998 views

### Equivariant resolution of singularities

I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they ...

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**1**answer

350 views

### Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...

**3**

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**1**answer

198 views

### Classifying all Equivariant Bilinear Forms on a Finite-Dimensional Module

Given a finite dimensional (real) vector space $V$, and two non-degenerate bilinear forms $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$, one can use a basic linear algebra argument to show that there exists ...

**6**

votes

**2**answers

659 views

### Weights on equivariant cohomology?

Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$.
Is there a natural mixed Hodge structure on its equivariant cohomology?
Is ...