The equivariant tag has no wiki summary.
2
votes
1answer
166 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 ...
10
votes
2answers
339 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.
1
vote
0answers
62 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 ...
4
votes
1answer
201 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 ...
6
votes
1answer
251 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 ...
1
vote
1answer
163 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 ...
3
votes
0answers
77 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 ...
2
votes
1answer
135 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 ...
3
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0answers
118 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 ...
0
votes
1answer
190 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 ...
1
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0answers
70 views
Classifying Equivariant Maps Between Fin-Dim Irreducible Modules
Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known ...
2
votes
1answer
78 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
2answers
381 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 ...
4
votes
1answer
122 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 ...
2
votes
2answers
269 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 ...
2
votes
1answer
342 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 ...
4
votes
2answers
475 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$. ...
10
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0answers
428 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 $X$, $\text{Vect}_G(X)$, ...
5
votes
1answer
206 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 $G$ ...
0
votes
2answers
386 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.
5
votes
2answers
673 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 ...
4
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0answers
231 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 ...
13
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8answers
1k 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
1answer
263 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) ...
15
votes
2answers
654 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 ...
0
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0answers
209 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 ...
4
votes
0answers
299 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 ...
16
votes
0answers
1k 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 ...
4
votes
0answers
229 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 ...
6
votes
6answers
508 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 ...
3
votes
1answer
266 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 ...
6
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0answers
725 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 ...
16
votes
1answer
673 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 ...
3
votes
0answers
173 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 ...
4
votes
0answers
389 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 $X$?
10
votes
1answer
552 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 ...
1
vote
1answer
416 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 ...
1
vote
0answers
135 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 ...
5
votes
3answers
1k 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 ...
9
votes
1answer
617 views
Geometric interpretation of filtered rings and modules
Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...
4
votes
1answer
426 views
Are these notions of strongly equivariant D-modules equivalent?
It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related.
Let $\rho: G\times X \rightarrow X$ be an ...
