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One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of $ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z $. The exponential map defines a morphism of Lie groups $ \exp: \mathbb{G}_{a}^{n} \to \mathbb{G}_{m}^{n} $. The exponential map is not an isomorphism, but if $ H $ is the additive, Lie sub-group defined by the lattice of points $ \mathbb{Z}^{n} 2\pi \sqrt{-1} $, then $ \mathbb{G}_{a}^{n}/H \cong \mathbb{G}_{m}^{n} $

This suggests that there should be an additive geometry and that if one "mods out" by certain conditions, then they should obtain toric geometry. Namely there should be some similar data to the fan $ \Sigma $, cones $ \sigma \in \Sigma $ and all other elements of the geometry of toric varieties.

One reason that this should be worth pursuing is not necessarily for expanding on the theory of toric varieties, but to add to the theory of invariant theory of vector groups. The invariant theory of vector groups is not well understood. Many deep problems are related to the invariant theory of vector groups, but have been impenetrable due to lack of knowledge regarding invariant theory of vector groups. Some example are the Jacobian conjecture, Mori's conjecture regarding uniruledness and Mori's conjecture that there exist no rational hypersurfaces of degree at least four.

If we were able to improve the theory of toric varieties so that it included a dictionary relating toric geometry to some sort of "additive geometry", then we might enable future mathematicians to better understand the theory of vector groups and shine light on these problems.

Has anyone tried to do this? Is anyone interested?

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    $\begingroup$ The equivariant compactifications of the additive group are not "discrete." Think of the standard action of $\mathbb{G}_a^2$ (as a unipotent subgroup of the Borel of $\textbf{PGL}_3$) acting on $\mathbb{P}^2$. The hyperplane at infinity is fixed, so you can blow up any finite set of points on that hyperplane to get another equivariant compactification. $\endgroup$ Commented Feb 23, 2023 at 10:46
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    $\begingroup$ To go one step further, for every closed subscheme $X$ of $\mathbb{P}^{n-1}$, identify $\mathbb{P}^{n-1}$ as the hyperplane at infinite of $\mathbb{P}^n$ with its standard action of $\mathbb{G}_a^n$. The the blowing up of $\mathbb{P}^n$ along $X$ is an equivariant compactification of $\mathbb{G}_a^n$. Therefore, the "classification" of equivariant compactifications of $\mathbb{G}_a^n$ is at least as "wild" as the classification of closed subschemes of $\mathbb{P}^{n-1}$, i.e., the Hilbert scheme of $\mathbb{P}^{n-1}$. $\endgroup$ Commented Feb 23, 2023 at 11:41
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    $\begingroup$ @M.G. When it comes to the Jacobian conjecture, the Kernel conjecture in $ n+1 $ variables implies the Jacobian conjecture in $ n $-variables. This is in Polynomial Automorphism and the Jacobian Conjecture by Arno van den Essen Chapter 2.2. $\endgroup$
    – Schemer1
    Commented Feb 24, 2023 at 6:16
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    $\begingroup$ @M.G. with regards to Mori's conjecture on uniruledness consider the following. An $ n $-dimensional variety $ Z $ is ruled if and only if it is birational to $ \mathbb{P}^{1}_{k} \times Y $. If $ \phi: \mathbb{P}^{1}_{k} \times Y \dashrightarrow Z $ is the birational map, then we may shrink the domain so that $ \phi: \mathbb{A}^{1}_{k} \times U \to V $ is an isomorphism. The linear algebraic group acts on $ \mathbb{A}^{1}_{k} \times U $ in the obvious way and via the isomorphism this extends to and action on $ V $. So a variety is ruled if and only if (cont) $\endgroup$
    – Schemer1
    Commented Feb 24, 2023 at 6:25
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    $\begingroup$ $\mathbb G_a$ and $\mathbb G_m$ are not isomorphic, neither as group schemes (as pointed out in Daniel Loughran's answer) nor as analytic spaces in any sense - the exponential map is not injective, so it certainly doesn't give an isomorphism. $\endgroup$
    – Wojowu
    Commented Feb 24, 2023 at 6:26

3 Answers 3

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The theory of Luna and Vust (Plongements d'espaces homogènes. Comment. Math. Helv. 58 (1983), 186–245.) on equivariant compactifications of homogeneous varieties works actually for any connected linear algebraic group $G$, so also for vector groups. In that theory, the cocharacters of a torus are to be replaced by invariant valuations of the function field of $G$. These determine roughly the irreducible components of the boundary.

Even for reductive groups and more so for vector groups the theory is extremely intractable, though. In fact, probaly the most important insight of Luna-Vust was that a good compactification theory exists precisely for spherical varieties. Pretty much everything else is just a mess.

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Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map.

But there is a foundational paper on the topic you seek:

Brendan Hassett, Yuri Tschinkel, Geometry of equivariant compactifications of ${\bf G}_a^n$. Internat. Math. Res. Notices 1999, no. 22, 1211-1230.

There is a big difference between the two theories: For any smooth projective variety, if it has a structure as a toric variety then this structure is unique (up to a suitable equivalence). Essentially this is because any two maximal tori in a reductive algebraic group over an algebraically closed field are conjugate.

But for equivariant compactifications of $\mathbb{G}_a^n$ there can be many such structures. Hassett and Tschinkel show that $\mathbb{P}^6$ has infinitely many such structures; in fact a continuum's worth, so the set of possible structures cannot be described by any discrete combinatorial data.

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There is a family of equivariant compactifications of $\mathbb{G}_a^n$ called Schubert varieties of hyperplane arrangements, which behave just like toric varieties. This is studied in the recent paper https://arxiv.org/abs/2209.00052

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