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In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed subvarieties $T_i \cong \mathbb{G}_m^{r_i}$ isomorphic to algebraic tori such that $\bigsqcup_{i \in I} T_i(K) = X(K)$ for every algebraically closed field $K$. I'm trying to prove that $\Omega_{\overline{T_i}}^1(\text{log}D_i)$, the differential forms on $\overline{T_i}$ with logarithmic poles along $D_i = \overline{T_i} \setminus T_i$, is trivial for each $T_i$. I'm actually not sure if this is true but that's my hope. A similar result is true for toric varieties. If $Y$ is a toric variety with open orbit $B$, then $\Omega_{Y}^1(\text{log}(Y \setminus B))$ is trivial (proven in "Toric varieties" by Fulton, pg 87).

My specific question is this: are the $\overline{T_i}$ in $X$ toric varieties? I know this would be true if the action of $T_i$ on itself extended to an action of $T_i$ on $\overline{T_i}$ but I'm not sure if this is possible in general or if not when this is possible. Thanks so much for the help!

Edit: For completeness, the definition of toric variety I'm using is a variety $X$ with an open dense subset $B$ isomorphic to a torus such that the action of $B$ on itself extends to $X$. Thus since each $T_i$ above is locally closed by assumption, then it is an open dense subsets of $\overline{T_i}$ so I only need to guarantee that the action of $T_i$ on itself extends to the closure.

Edit 2: I'll rephrase my question in a more general way so that it maybe easier to approach: if $X$ and $Y$ are varieties, $\overline{Y}$ is a closure of $Y$ such that the inclusion of $Y$ is an open immersion, and $f:X \times Y \to Y$ is a morphism, under what conditions if any does $f$ extend to a morphism $f_* : X \times \overline{Y} \to \overline{Y}$? Furthermore, does the special case of $X$ and $Y$ being a torus $T$, and $f$ being the map $T \times T \to T, \enspace (t,s) \mapsto ts$ as above satisfy these conditions?

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EDIT: Added one more example.

I don't think there are simple conditions which can guarantee that the action extends. Let me give two simple examples:

1) Consider $\mathbb{A}^2$ with its standard toric structure, so $(0,0)$ is a fixed point for the torus action. Let $X_1$ be the blowup of $\mathbb{A}^2$ at $(0,0)$; this gets an induced toric structure such that the exceptional divisor $E$ contains two fixed points $x_1,x_2$. Let $X$ be obtained from $X_1$ by gluing together any two points $p,q$ on $E$ so that $p\neq x_1,x_2$. Then $X$ has a decomposition as a finite union of tori but the torus action doesn't extend to all of $X$.

2) Let $X = \mathbb{P}^2$ and consider $\mathbb{A}^2 \subset X$. As above, $\mathbb{A}^2$ is a toric variety. We now change the toric structure of $\mathbb{A}^2$ by conjugating with a non-affine automorphism, say the map given by $(x,y) \mapsto (x,y + x^2)$. A small computation shows that the resulting toric structure is non-affine (with respect to the standard coordinates). This toric structure cannot extend to an action on $X$ since the automorphism group of $\mathbb{P}^2$ is $PGL_3$ so the automorphisms preserving $\mathbb{A}^2$ are exactly the affine automprphisms.

3) There exist irreducible torified varieties which do not admit the structure of a toric variety. Most grassmannians and flag varieties are of this type since their automorphism groups do not contain a torus of dimension equal to the dimension of the variety. The simplest I can think of is a smooth quadric hypersurface in $\mathbb{P}^4$; I don't know if there is a two dimensional example.

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In example 1), instead of gluing points together you could also blow up one or two non-fixed points on the exceptional divisor. – ulrich Aug 31 '11 at 8:24
Thanks! I'm probably not understanding something in your above examples, but I'm not sure that they fit the thing I would like to be true. I know that the action can't extend to all of X in general, but what I'm looking for is if the action can extend to the closure of the torus in X. These might still be counterexamples for what I want, I'm just not sure. – Dori Bejleri Aug 31 '11 at 8:34
The type of things I had in mind when I was thinking about this (which might mean I'm having wrong intuition about it) were things like the union of the $n$ coordinate axes in affine space. You can decompose them into the disjoint union of the origin, and $n$ copies of a 1 dimensional torus, and while the action of any of these tori on itself doesn't extend to all of $X$, the closures of any of them are just a line and the action does extend to that. – Dori Bejleri Aug 31 '11 at 8:38
In the examples I gave, by choosing a different decomposition of $X$ into tori the action of the open torus does extend to all of $X$. Maybe a relevant question for you is whether any irreducible torified variety $X$ is a toric variety (after possibly changing the decomposition of $X$ as a union of tori). – ulrich Aug 31 '11 at 9:11
A remark on your example (3): An explicit decomposition of the flag variety (and, I think, also the Grassmannian) into tori can be found in Deodhar's paper "On some geometric aspects of Bruhat orderings I" – David Speyer Sep 1 '11 at 14:27

Sorry that I reply so late, I haven't been hanging much on MO recently.

The quick answer to your question is no, the closure of tori appearing in the torification are in general not toric varieties. If that was the case, every irreducible torified variety would be toric (as it would coincide with the closure of the unique dense torus in the torification), examples are for instance the Grassmannian $Gr(2,4)$ which is torified but not toric.

Now, what can actually help you with your problem is the following. You can initially restrict yourself to study regular torified varieties, where a torification is called regular if the closure of the tori are torified with (a subset of) the same torification: $\overline{T_i}=\bigsqcup_{j \leq i} T_j$ (where the ordering is simply "being in the closure of"). Under these conditions, the complement $\overline{T_i}\setminus T_i$ is again a torified variety (which might be disconnected, but it will surely be a disjoint union of torified varieties), with strictly smaller dimension than the one of $T_i$.

It is an open problem to decide whether every torified variety can be regularly torified, but so far in the examples we worked with we have been able to find regular torifications.

Your project sounds very interesting, I am looking forward to see what you come up with!

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