Let $ Z $ be an $ n $-dimensional, projective, variety, over a field of arbitrary characteristic and let $ \iota: \mathbb G_{m}^{n} \to Z $ be a morphism such that for any $ z \in Z $, the fibre $ \iota^{-1}(z) $ is either empty or a single point. Note that I am not requiring that $ \iota(\mathbb G_{m}^{n}) $ is an open, sub-variety of $ Z $. Does anyone know of some object which measures whether $ Z $ is toric? Namely, does someone know of something that measures when the action of $ \mathbb G_{m}^{n} $ extends to an action on $ Z $?
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1$\begingroup$ $\mathbb G_m$ worked for me, I edited your post $\endgroup$– Nicolas HemelsoetMar 10, 2023 at 7:54
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$\begingroup$ Thank you @NicolasHemelsoet. It may be a computer issue. $\endgroup$– Schemer1Mar 10, 2023 at 7:56
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3$\begingroup$ There are many conditions. For example, if $Z$ is smooth and the complement $D$ of the torus is a divisor with normal crossings, then $Z$ is toric (in the sense that the action of the torus extends) if and only if the log cotangent bundle $\Omega_Z(\log D)$ is trivial, by a theorem of Winkelmann. $\endgroup$– Piotr AchingerMar 10, 2023 at 9:53
2 Answers
Even in the case when the image of the morphism is open, there are plenty of non-toric examples. Consider the standard toric action on $\mathbb{CP}^3$ and consider one of the torus invariant divisors $H \cong \mathbb{CP}^2$, then blow up any smooth curve in $H$ which is not a line. The result has automorphism group of rank $<3$ hence non-toric, and contains a copy of $(\mathbb{C}^*)^3$.
In these examples, the non-toricness is measured by the rank of the automorphism group. So perhaps a reasonable condition to add is that it has rank $3$, although maybe in that case there are also counter-examples.
If you consider the pair $(Z,\Delta)$, where $\Delta=Z\setminus i(\mathbb{G}_m)$ is the divisor given as the complement of the image of the torus, then there is an numerical invariant of $(Z,\Delta)$ called the complexity which can be used to characterise whether or not $Z$ is toric. It was conjectured by Shokurov and proved in this paper by Brown, M$^{\text{c}}$Kernan, Svaldi & Zong.