# Questions tagged [torus-action]

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### Singular schemes with a torus action and embedded points

I've got a couple rather geometric questions about the following setup. Let $X$ be a scheme over an algebraically closed field ($\mathbb{C}$, say) with the action of a torus $T$, such that the action ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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 $P_L=L\setminus B$...