Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

Question

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant.

It turns out that, given this setup, essentially all the algebraic-geometric information of $X$ can be encoded in a finite combinatorial structure (the fan of $X$). This makes toric varieties very appealing to algebraic geometers, since they're much easier to calculate with than arbitrary varieties. And indeed, toric varieties have been intensively studied since they were introduced in the early 1970s.

It seems natural to generalise this by replacing $T$ by some other linear algebraic group $G$, and studying the resulting class of varieties. To reiterate, that means we are looking at $G$-varieties $X$ with a dense equivariant embedding $G \hookrightarrow X$.

But I have never seen anyone mention this more general setup. So my questions are:

Q 1. Is there a clear conceptual reason that, for general $G$, this setup does not lead to a nice theory?

Q 2. If so, can we put some restrictions on $G$ so that we do get a nice theory?

Q 3. Finally, are there any references that discuss this general setup?

Answer 1

It turn out that torus are the only compact abelian groups.

So both the algebra and the topology are especially simple and well mastered.

Answer 2

For $G =\mathbb{C}^{*}$ there is a birational theory. Certain elements of Mori's minimal model program can be performed equivariantly in this category. One reason for this is that every extremal ray of the Mori cone contains a curve which is $\mathbb{C}^{*}$-invariant.

Maybe have a look at the following paper.

https://arxiv.org/abs/1911.12129

Answer 3

Actually there is a concept of Spherical varieties. It is a variety that admits an action of a reductive algebraic group G wherein a Borel subgroup has an open dense orbit. So some kind of generalization from torus to connected solvable group is there. The names of people who have contributed to this study that I remember are Michel Brion, Luna, Vust.

Lakshmibai also has done some work on Schubert Varieties X in G/P that are spherical varieties for some Levi subgroup of G.