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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