As of May 31, 2023, we have updated our Code of Conduct.

# Questions tagged [torus-action]

The tag has no usage guidance.

32 questions
Filter by
Sorted by
Tagged with
302 views

### Faithful locally free circle actions on a torus must be free?

Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free? I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$. Another ...
1 vote
203 views

### Plus and minus Białynicki-Birula decomposition for normal variety

We work over $\mathbb{C}$. Let $X$ be a normal projective irreducible variety, and let $\mathbb{C}^*$ act nontrivially on $X$. The fixed point locus of $X$, namely $X^{\mathbb{C}^*}$, can be ...
161 views

### Understanding the proof of a theorem by Van Den Bergh

I'm trying to understand the proof of a theorem by Van Den Bergh, which is Proposition 6 in the paper Bessenrodt, Christine and Lieven Le Bruyn. “Stable rationality of certain PGLn-quotients.” ...
259 views

### Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus. Question: Is the following statement true? ...
131 views

375 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 ...
198 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 ...
388 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 ...
244 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
80 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 $\mathbb{F}_3$. ...
278 views

136 views

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

let $k$ be any field of char 0. $G$ is split reductive algebraic 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 ...
574 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 ...
472 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 ...
419 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 ...
689 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 ...
370 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 ...
349 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 ...
565 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 ...
556 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 ...
1k 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 ...
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 ...
### 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 \$...