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 have an algebraic action $\psi\colon\mathbb C^*\times X\to X$ of one-dimensional torus on $X$. Does always exist a morphism $f\colon\mathbb C^*\to (\mathbb C^*)^m$ such that $\psi=\varphi\circ f$?
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1$\begingroup$ No. Consider the "standard" action of $\mathbb{C}^*$ on $\mathbb{A}^1$ by $\lambda\cdot x= \lambda x$, and also the second action of $\mathbb{C}^*$ on $\mathbb{A}^1$ by $\lambda \ast x = \lambda(x-1)+1$. $\endgroup$– Jason StarrCommented Mar 1, 2016 at 14:29
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$\begingroup$ @JasonStarr Is your second action really an action? It seems to me that $(\lambda \cdot \mu) \ast x \neq \lambda \ast (\mu\ast x)$ $\endgroup$– Ariyan JavanpeykarCommented Mar 1, 2016 at 15:12
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$\begingroup$ @AriyanJavanpeykar. Define $y=\lambda(x-1)+1$. Then $\mu\ast y$ equals $\mu(y-1)+1$. But $y-1$ equals $\lambda(x-1)$. Thus $\mu\ast y$ equals $\mu\lambda(x-1)+1$. So $\mu\ast(\lambda\ast x)$ equals $(\mu\lambda)\ast x$. smiley face $\endgroup$– Jason StarrCommented Mar 1, 2016 at 16:11
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$\begingroup$ @JasonStarr Thank you for the explanation. Shame on me. smiley face $\endgroup$– Ariyan JavanpeykarCommented Mar 1, 2016 at 19:00
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2 Answers
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Comment above posted as an answer. No, that is not true. One counterexample is when $X=\mathbb{A}^1 = \text{Spec} \ \mathbb{C}[x]$, $m$ equals $1$, $\phi(\lambda,x)$ equals $\lambda x$, and $\psi(\lambda,x)$ equals $\lambda(x-1)+1$.
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In fact $X$ is a toric variety. "In general" the automorphism group of this variety is the torus himself. But in some cases it is not. For instance if $\varphi$ is the diagonal action of $\bf {C^*} ^n$ on $\bf C^n$ the closure of the orbit is the affine space itself and the automorphims group of this space is much bigger ; it contains the homothety group which fixes some point as remarked by Jason Starr, or in higher dimension the entire group $GL(n,\bf C)$ hence any conjugate subrgoup of the diagonal torus, and even more complicated stuff. The complete classification of toric varietes and their automorphism groups is in Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4) 3 1970 507–588.
