When an action on open dense subvariety by an algebraic group extends to variety

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.

• 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 $U$. Jan 25 at 3:57
• 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. Jan 26 at 23:45

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 $$\overline X=X$$.

• 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? Jan 25 at 17:14
• 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. Jan 26 at 19:22