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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^...
QSR's user avatar
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2 votes
0 answers
241 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 ...
Meer Ashwinkumar's user avatar
5 votes
1 answer
564 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?
user89937's user avatar
3 votes
0 answers
662 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 ...
Qfwfq's user avatar
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6 votes
0 answers
114 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 ...
Matthias Wendt's user avatar
5 votes
2 answers
1k 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 ...
user56980's user avatar
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0 answers
443 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 $G$-...
Dominik's user avatar
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4 votes
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216 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 ...
Sven Cattell's user avatar
3 votes
0 answers
102 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 $...
Megan's user avatar
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2 votes
1 answer
257 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 $$...
Megan's user avatar
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3 votes
1 answer
143 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{...
Megan's user avatar
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1 vote
1 answer
690 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 ...
Nilay Kumar's user avatar
2 votes
1 answer
699 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 ...
Satoshi  Nawata's user avatar
16 votes
2 answers
2k 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.
Daniel Robert-Nicoud's user avatar
1 vote
0 answers
153 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 ...
jj_p's user avatar
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6 votes
1 answer
512 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 \...
Sasha Pavlov's user avatar
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8 votes
1 answer
757 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 ...
Carl's user avatar
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1 vote
1 answer
426 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 ...
Christian Fischmann's user avatar
3 votes
0 answers
160 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 ...
Reladenine Vakalwe's user avatar
3 votes
1 answer
452 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 ...
H. Shindoh's user avatar
10 votes
1 answer
2k 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 ...
Libli's user avatar
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0 votes
1 answer
451 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 ...
Noel Brown's user avatar
3 votes
1 answer
259 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 ...
Sasha Alexander's user avatar
6 votes
2 answers
812 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 ...
Jan Weidner's user avatar
  • 13.2k
4 votes
1 answer
182 views

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)...
John Klein's user avatar
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4 votes
2 answers
522 views

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_{...
Zhaoting Wei's user avatar
  • 9,019
2 votes
1 answer
612 views

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 ...
Xuxu's user avatar
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5 votes
2 answers
727 views

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$. ...
Jean Delinez's user avatar
  • 3,409
21 votes
0 answers
1k views

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 $X$, $\text{Vect}_G(X)$, ...
Zhaoting Wei's user avatar
  • 9,019
6 votes
1 answer
489 views

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 $G$ ...
John Pardon's user avatar
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2 votes
2 answers
1k views

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.
yee yao's user avatar
  • 23
6 votes
2 answers
917 views

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 ...
Paul Johnson's user avatar
  • 2,372
4 votes
0 answers
292 views

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 ...
Jan Weidner's user avatar
  • 13.2k
18 votes
8 answers
3k views

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 ...
2 votes
1 answer
372 views

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) \...
Alex Collins's user avatar
18 votes
2 answers
1k views

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 $K_G(TX) \to R(G)$...
Akhil Mathew's user avatar
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0 votes
0 answers
460 views

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 ...
tonywang's user avatar
4 votes
0 answers
402 views

Equivariant sheaves and simplicial varieties

I would like to proof the following theorem: Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on ...
Jan Weidner's user avatar
  • 13.2k
22 votes
0 answers
2k views

Is the equivariant cohomology an equivariant cohomology?

Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology). $\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...
Sergey Melikhov's user avatar
5 votes
0 answers
334 views

T-Equivariant trivialization of a principal G-bundle

Let $k$ be a field, let $G$ be an algebraic group scheme over $k$ and let $T = \textrm{Spec } k[x, x^{-1}]$ be a one-dimensional torus. Does there exist a scheme $X$ over $k$, an algebraic $T$-action ...
expz's user avatar
  • 562
6 votes
6 answers
599 views

Equivariant homology of $\Omega X$\/-space (references needed)?

Let $(X, *)$ be pointed a (1-connected) space, and let $\Omega X$ denote its based loops space. Then, as one knows very well, $\Omega X$ is a group up to homotopy (this includes all the necessary ...
gshar's user avatar
  • 291
3 votes
1 answer
284 views

For a G-variety, what could one say about the motif of the corresponding simplicial variety

Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex ...
Mikhail Bondarko's user avatar
10 votes
0 answers
2k views

Why is the Nil-Hecke Algebra appearing?

The Nil-Hecke algebra is defined to be the subalgebra of the endomorphism ring of $\mathbb{C}[x_1,\ldots,x_n]$ generated by the operators of multiplication by $x_i$ and the divided difference ...
Peter McNamara's user avatar
21 votes
3 answers
2k views

Formality of classifying spaces

Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...
Geordie Williamson's user avatar
4 votes
0 answers
243 views

Equivariant sheaves basics reference

I am looking for a reference for basic facts about actions of linear algebraic groups and their Lie-algebras on $\mathcal O_X$-modules. For example I could not find a reference the following: Let $...
Jan Weidner's user avatar
  • 13.2k
5 votes
0 answers
624 views

What is the right notion of equivariant Cech cohomology?

What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?
Jesse Wolfson's user avatar
11 votes
1 answer
964 views

What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...
skupers's user avatar
  • 8,167
1 vote
1 answer
629 views

The fiber of the sheaf of invariants

Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet ...
Rurik's user avatar
  • 669
1 vote
0 answers
182 views

G-Modules on X=G/H modules on X/H ?

I think it is true that $G$-equivariant sheaves on $X$ are equal to $G/H$ equivariant sheaves on $X/H$. More precisely I'm interested in the following statement: Given an algebraic group $G$ with ...
Jan Weidner's user avatar
  • 13.2k
9 votes
3 answers
2k views

Is this a definition of equivariant derived category?

Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We ...
Jan Weidner's user avatar
  • 13.2k