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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!

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Related: mathoverflow.net/questions/53306/… –  J.C. Ottem Nov 19 '12 at 18:15
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1 Answer 1

One has to be precise when one talks about the tropicalization of a toric variety. Usually, one takes tropicalizations of subvarieties of an algebraic torus. To get around this constraint, there is a notion of extended tropicalization of subvarieties of a toric variety. Although I haven't checked the definitions, the extended tropicalization of a toric variety should contain exactly the same information as the fan.

The question should perhaps go the other way. There are lots of recipes for reading geometric invariants of toric variety from the fan which is a combinatorial object. Given a subvariety of a toric variety, can one read off geometric invariants of the subvariety from the tropicalization. The case of the toric variety is the prototypical example, but maybe other varieties act like toric varieties in this sense.

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Thank you very much for your answer! Do you have suggestions for references about "extended tropicalization"? (I'm sure my question is far from optimal in any sense, it's just that I'd like to understand if it's possible to reformulate the information given by the Todd class of a toric variety in terms of tropical data. I do completely agree that the tropical world is in a sense much more general than the toric world). –  user5831 Nov 28 '12 at 12:50
    
For extended tropicalizations, see: arxiv.org/abs/0805.1916 The Todd class of a toric variety can certainly be formulated in terms of combinatorial data. But it might be quite difficult to make sense of the Todd class of a more general tropical variety. The most simple tropical varieties after tropicalizations of toric varieties are locally matroidal varieties (see papers of Mikhalkin, my paper with Stapledon for details, or Dustin Cartwright's recent paper on the Grobner stratification for details). One can probably talk about the chern classes of their tangent bundles... –  Eric Katz Dec 16 '12 at 6:29
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