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Hi everybody.

I wonder if there are tropical analogs of the classical notions of "metric" and of "volume" of classical complex geometry...

To be more precise, let's consider a very concrete case: the complex projective space $\mathbb P^n$. Its tropicalization $\mathbb T\mathbb P^n$ is the standard $n$-simplex.

What is (or what could be) the tropical analog of the Fubini-study metric on $\mathbb P^n$?

Same question for the invariant Fubini-Study volume form on $\mathbb P^n$.

Thank in advances.

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  • $\begingroup$ This might be problematic. The key fact about Fubini-Study that one would hope to reproduce is that it is invariant under a maximal compact subgroup of the automorphism group. But here the automorphism group is, I believe, a semidirect production of $\mathbb R^n$ and $S_{n+1}$, which means a maximal compact subgroup is $S_{n+1}$, and there are lots of invariant metrics. This is if you view the metric as just, say, a metric on the simplex as a topological space that restricts to a Riemannian metric on the subspace that's a manifold. $\endgroup$
    – Will Sawin
    Commented Jan 15, 2013 at 4:07
  • $\begingroup$ An alternate choice would be to look for a metric invariant under the full symmetry group. This would just be a flat metric on $\mathbb R^n$. Unfortunately, it would not be able to extend to the full simplex, because sequences that converge to boundary points of the simplex fail to be Cauchy in $\mathbb R^n$. $\endgroup$
    – Will Sawin
    Commented Jan 15, 2013 at 4:08

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I think in the book of Victor Guillemin, Appendix 2, you can find a correspondence between the Kähler structures on non-singular toric manifolds ( in particular $\mathbb{P}^n$) and some combinatorial data on the associated Delzant polytope (the standard simplex in the case of $\mathbb{P}^n$).

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  • $\begingroup$ Thanks a lot CofWsug for this interesting and relevant reference. This solves the case of $\mathbb P^n$ ... but just because the later is a toric manifold! But what about the tropicalization of a non-toric manifold, say the grassmanniann $G_2(\mathbb C^4)$? Of course, $G_2(\mathbb C^4)$ embeds in $\mathbb P^5$ hence $G_2^{\rm trop}(\mathbb C^4)$ embeds in $\mathbb T\mathbb P^5$ (unfortunately, this is not so simple, but let's see the things going like this...). Does it makes sense to restrict to $G_2^{\rm trop}(\mathbb C^4)$ the toric K\"ahler Fubini-Study form on $\mathbb T\mathbb P^5$? $\endgroup$
    – ElBabak
    Commented Jan 15, 2013 at 9:30
  • $\begingroup$ I think $\mathbb{T}\mathbb{P}^5$ is not a good notation for the tropicalization of $\mathbb{P}^5$, it looks like the tangent bundle of the projective space. I'm not an expert in tropical geometry, but I don't think the restriction makes sense. For exemple, the volume of Amoebas can be infinite and it depends on the codimension. $\endgroup$
    – CofWsug
    Commented Jan 15, 2013 at 11:00
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Yes, there is a direct analog (see the article "The number of vertices of a tropical curve is bounded by its area")

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