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A toric variety $X$ over $k$ is a variety which contains an algebraic torus ($T= \mathbb{G}_k^s$) as a dense open subset such that the action of the torus on itself extends to the whole of $X$. Slogan: Essentially toric varieties are just fattened tori with an action.

Let $X$ be an algebraic variety which contains a dense open subvariety $U$ and there is an algebraic group $G$ acting on $X$. Are there sufficient conditions when the natural action of $G$ on $U$ extends to the whole $X$?

The main cases in the scope of my interest are projective varieties (or weakened to 'proper'). Which role plays the base field $k$ in this extension problem.

Let me also remind that an action by an algebraic group or more general a group scheme $G$ on a algebraic variety $S$ is a morphism $f: G \times S \to S$ which respects group multiplication morphism $m: G \times G \to G$. Formally this assumption is equivalent to these two laws.

Note that's a copy of identical question I asked in MSE a week ago without getting an answer.

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    $\begingroup$ Think about any G-variety $X$. Take an affine G-invariant open subset $U\subset X$ (which you can do if $G$ is reductive, for instance). Blow-up a smooth point in the complement of $U$ which is not G-invariant, then $G$ won't lift to the blow-up. This says that the answer to your question is almost always no. I doubt that the condition must be imposed on the action of $G$ on $U$, but rather on the ample divisor defining the embedding of $X$ and its restriction to $U$. $\endgroup$ Jan 25 at 3:57
  • $\begingroup$ The main case I can think of is when $X$ is determined by applying some kind of functor to $U$. If $X$ happens to be the Proj of the section ring of some ample $G$-invariant line bundle on $U$, for example. $\endgroup$
    – Will Sawin
    Jan 26 at 23:45
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The theory of toric embeddings has been generalized to arbitrary homogeneous spaces in the celebrated paper

Luna, D.; Vust, Th. Plongements d'espaces homogènes. (French) [Embeddings of homogeneous spaces] Comment. Math. Helv. 58 (1983), no. 2, 186–245.

Its scope are connected algebraic groups over algebraically closed fields of characteristic zero. Moreover, $U$ is assumed to be a homogeneous variety but that's not really essential.

Of their many reults, one might be of interest for you. Since $U$ is a $G$-variety the Lie algebra $\mathfrak g$ will act by means of vector fields on $U$. If I remember correctly, the following should be true:

Let $G$ be a connected algebraic group (over $\mathbb C$) acting on an irreducible $G$-variety $U$ und let $U\hookrightarrow X$ an open embedding into an irreducible variety $X$. Assume that the $\mathfrak g$-action on $U$ extends to one oon $X$. Then there is open embedding $X\hookrightarrow \overline X$ such that the $G$-action on $U$ extends to a $G$-action on $\overline X$.

Note that if $X$ is complete then $\overline X=X$.

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  • $\begingroup$ So just to be clear in your last paragraph, if we take $X = U$, then the action on $U$ extends to some $\overline U$ where $\overline U$ is complete -- is that correct? $\endgroup$ Jan 25 at 17:14
  • $\begingroup$ The last paragraph says that if the $\mathfrak g$-action extends to a completion $X$ of $U$ then the $G$-action extends, as well. So no, one cannot immediately construct equivariant completions. That's a separate theorem of Sumihiro. $\endgroup$ Jan 26 at 19:22

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