# Tagged Questions

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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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T$-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $... 0answers 433 views ### What is the right notion of equivariant Cech cohomology? What is the right definition of equivariant Cech cohomology is so that given a$G$-space$X$,$H^1_G(X;H)$classifies$G$-equivariant principal$H$-bundles on$X$? 1answer 610 views ### What does this naive attempt at$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$. This means we have a map$\...
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 ...