# Questions tagged [equivariant]

The equivariant tag has no usage guidance.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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 \...

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### 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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### 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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### 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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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### Equivariant versus retractive spaces: a reference request

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric)...

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### Equivariant K-theory of $S^1$-action on $S^2$

Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of $K_{...

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### How to calculate the equivariant cohomology ring of $P^2$?

It is well known that Kirwan's injection theorem gives an ring injection from $H^{\ast}_T(M)$ to $H^{\ast}_T(M^T)$ which is induced by the inclusion $M^T \to M$, where $T$ is a torus acting on ...

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### Jets of Equivariant Vector Bundles

Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...

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### Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...

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### Equivariant handle decompositions

Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...

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### Reference for equivariant Riemann-Roch formula?

Is there any reference for equivariant Riemann-Roch formula: book, paper, notes or something? I want to compute the weight of the action of C^* on the top wedge of cohomology group.

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### Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?

A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...

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### What is the equivariant derived category good for?

Given a topological group acting on a topological space, Bernstein and Lunts construct the equivariant derived category, which looks like the derived category of the quotient would, if action was free ...

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### Reference request: Equivariant Topology

I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students ...

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### When is restriction an equivalence of categories of equivariant vector bundles?

Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor
$i^*: Vect^G(X) \...

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### What is the role of equivariance in the Atiyah-Singer index theorem?

I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof:
Define (using purely K-theoretic means) a homomorphism $K_G(TX) \to R(G)$...

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### Lifting of torus action to line bundle

Let $\mathbb{P}(V) = \mathbb{P}(\mathbb C \oplus \mathbb C)$ be with a $\mathbb C^*$ action : $\lambda (u,v) = (u,\lambda v)$.
There are two fixed points of this action, say $0$ and $\infty$. What ...