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Questions tagged [torus-action]

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10
votes
0answers
204 views

Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
10
votes
1answer
250 views

Automorphisms of $GL_n(\mathbb{Z})$

I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...
6
votes
2answers
103 views

Linearization of hamiltonian torus action

Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such ...
8
votes
2answers
310 views

Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite? Consider the ...
4
votes
1answer
147 views

Index formula with nonisolated fixed points

Consider a compact Riemannian manifold of even dimension $n$ admitting a $U(1)$ action. If the fixed points of the action are isolated, then Witten [1; eq. 35] gives the character-valued index of the ...
1
vote
0answers
66 views

How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?

Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. ...
5
votes
1answer
158 views

Explicit local normal form symplectomorphism at torus fixed point of a coadjoint orbit

Let $K$ be a compact, connected (probably also simple) Lie group and with a maximal torus $T$. Regular coadjoint orbits $\mathcal{O}_{\lambda} \cong K/T$, parameterized by a regular element $\lambda \...
3
votes
0answers
77 views

Decompositions from torus actions and compactness of (sub-)level sets

Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$. From the induced $U(1)$-action and its (...
3
votes
1answer
260 views

Torus actions with more than one fixed point

I am looking for a reference for the following result: Let $X$ be a projective variety and $\mathcal{L}$ be an ample line bundle over $X$. Suppose that there is a torus $T=(\mathbb{G}_m)^n$ which ...
1
vote
2answers
192 views

Action of $\mathbb C^*$ on a closed orbit of a torus

Let $\varphi\colon (\mathbb C^*)^m\times\mathbb A^n\to\mathbb A^n$ be an algebraic effective action of a torus on affine space and $X$ be a Zariski closure of an orbit of this action. Suppose we also ...
0
votes
1answer
109 views

Do tori in a symplectic group always have invariant maximal isotropic subspaces?

$\newcommand{\mbf}{\mathbf}$ Hi all, I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help. For ...
4
votes
0answers
107 views

Twisting stable maps to C* equivariant space by a line bundle

Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus B$...
7
votes
4answers
1k views

Classification of Tori of GL2, up to conjugation

Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,...
0
votes
0answers
116 views

Is $k$-diagonalizable element in split maximal torus of $G(k)$?

let $k$ be any field of char 0. $G$ is split reductive algebriac group over k. Let p in G(k) be k-diagonalizable. Does there exist a split maximal torus of G(k) containing p? I know that is ture for ...
3
votes
2answers
463 views

Projective line as a quotient by a torus

Let $k$ be a field, and let $T$ an $n$-dimensional split torus over $k$. Let $X$ be a $k$-scheme with algebraic $T$-action. Solve for X: $$X / T \cong \mathbf{P}^1_k$$ (The quotient should be a ...
3
votes
2answers
358 views

a question about the isotropy subgroup of circle action on manifolds with isolated fixed point

Given a circle action on a closed, oriented smooth manifold $M^{2n}$ with isolated fixed points. My question is, does there always exist a point $p\in M$ such that the isotropy subgroup of $p$ is ...
0
votes
0answers
315 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 ...
13
votes
2answers
618 views

Does there exist smooth circle action on manifolds M^{4n} with exactly three fixed points such that n\neq 1

I have a question in mind for some time. That is, does there exist a smooth circle action on a closed manifold $M^{4n}$ ($n\geq 2$) with exactly three fixed points? Remarks:(1) For n=1, the examples ...
7
votes
0answers
359 views

The scheme-theoretic flow-in locus

Let $R$ be a ring with an $\mathbb{N}\times\mathbb{Z}$-grading. The $\mathbb{N}$-grading allows you to construct the scheme $X = \operatorname{Proj} R$, and the $\mathbb{Z}$-grading defines an action ...
2
votes
0answers
308 views

Are schematic fixed points of a torus action on an affinized twistor deformation flat?

This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...
2
votes
2answers
480 views

Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay?

I'm not sure how long this iterative questions can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the ...
5
votes
1answer
467 views

Under what hypotheses are schematic fixed points of a flat deformation themselves flat?

This is something of a follow-up question to this one; I hope people won't think this is a duplicate. At least in my head, it seems like a distinct enough question to merit a fresh start. All my ...
8
votes
1answer
909 views

Understanding the unreducedness of a subscheme supported on fixed points

EDIT: Based on comments, I've decided to essentially rewrite this question. My apologies to those who commented below, whose comments seem a bit off topic when you try to match them with the current ...
4
votes
1answer
721 views

Line Bundles on Torus Quotient

Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can ...
9
votes
2answers
699 views

Hamiltonian S^1 actions with isolated fixed points

I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian S^1 action with isolated fixed points, that does not admit a compatible S^1 ...