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In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. They show that if the ambient cycle is $\mathbb{R}^n$ (= the tropicalization of the $n$-torus), then there is a well-defined intersection product and we in fact have a Chow ring. A similar procedure is used in Francois-Rau '10 to show that there is a Chow ring on matroid varieties (= tropicalizations of linear subspaces of the torus).

The standard notion of tropicalization transforms a subvariety $X$ of a torus into a pure-dimensional polyhedral complex $\text{Trop}(X)$ in Euclidean space. There is a notion of "extended tropicalization" (see Payne '09, section 3) which is applied to a toric variety $Y$. The result, $\mathbf{Trop}(Y)$ is a partial compactification of the ordinary tropicalization $\text{Trop}(Y \cap T)$, where $T$ is the dense torus.

My question is, has anyone defined the Chow groups of the extended tropicalization of a toric variety? If so, are there certain conditions under which we can define an intersection product?

Moreover, if we can define the Chow ring of an extended tropicalization, is there any reason to expect it to be isomorphic to the Chow ring of the original toric variety?

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