My excuses in advance in case my question is too vague (which is mainly due to the fact that I'm not really familiar with tropical/toric geometry, but at least I still believe that the content of the question is non-empty).

I'd like to understand better the relationships between tropical and toric geometry. (The naive intersection of the two fields is of course due to the appearance of similar combinatorial structures in form of fans/polyhedra used in building the respective geometries).

More precisely, I'd like to know now the following question: Suppose we are given a (nice) toric variety. Can we recover its "invariants/properties" in terms of an associated tropical variety? (The naive way would of course be to look at the tropical variety attached to the underlying combinatorial structures of the toric variety.)

In fact, I'm most interested in describing the Todd class of a (nice) toric variety tropically.

I'd be very happy about any kind of related pointers towards the literature or (philosophical) explanations.

(Again, I'm sorry in case my question is completely off-track.)

Thanks a lot in advance!