TITLE: Tropicalization of perfectoid spaces and their tilts
QUESTION [20 upvotes]: Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far. 
For concreteness let $K$ be the completion of $\mathbb{Q}_p(p^{\frac{1}{p^\infty}})$, and $X$ a perfectoid space over $K$. Can one tropicalize $X$ and its tilt $X^\flat$? If so, how are the two tropicalizations related? Are they isomorphic (as rational polyhedral cone complexes)?
Further, what is the information tropicalization would retain in this setting? Is there a shadow left of the Galois action on $X$ and $X^\flat$?

REPLY [11 votes]: Sorry for the late answer!
Tropicalization works for all analytic adic spaces $X$ (in particular, for perfectoid spaces). One minor variation on the usual story is that adic spaces have their valuations in abstract totally ordered abelian groups. On analytic adic spaces, these valuations always have unique rank $1$ generalizations, taking values in $\mathbb R_{>0}$ up to isomorphism. In other words, the value group is only defined up to rescaling. If you fix a nonarchimedean base field $K$, you can get rid of this ambiguity by fixing the norm on $K$. More generally, you can fix a global topologically nilpotent unit $\varpi\in \mathcal O_X^\times(X)$ and normalize the norm of $\varpi$. But that's actually just a special case of the following:
For any analytic adic space $X$ and global units $f_0,\ldots,f_n\in\mathcal O_X^\times(X)$, such that the locus $\{|f_0|=\ldots=|f_n|=1\}$ is empty, one can define a tropicalization map
$$|X|\to \mathbb P^n(\mathbb R)$$
taking any $x\in |X|$ with rank $1$-generalization $\tilde{x}$ to
$$[\log |f_0(\tilde{x})|:\ldots:\log |f_n(\tilde{x})|].$$
If $X$ is affinoid (or Stein, in a suitable sense), one can recover the maximal Hausdorff quotient of $|X|$ (i.e., "the Berkovich space") as the inverse limit of the images of all tropicalization maps (for varying $f_i$'s).
In the case of perfectoid spaces, if $X$ is perfectoid with tilt $X^\flat$, and one chooses $f_0^\flat,\ldots,f_n^\flat\in \mathcal O_{X^\flat}^\times(X^\flat)$ with $f_i=(f_i^\flat)^\sharp\in \mathcal O_X^\times(X)$, then $|X|\cong |X^\flat|$ and the corresponding tropicalization maps agree.
About the Galois action: Everything here is functorial, so you can get Galois actions, I guess.
Unfortunately, I don't really know of any published work combining perfectoid and tropical geometry, but I agree that this is a natural idea. In fact, when I tried to prove the full weight-monodromy conjecture using perfectoid spaces, the remaining step was to find algebraic varieties of the correct dimension in certain open subsets of $|\mathbb P^n_{K^\flat}|$ (as an adic space). This can (essentially) be phrased as a problem of finding algebraic varieties with given tropicalization, and fits a line of research whether tropical varieties can be realized by actual varieties. Note that the tropicalization can be realized by an algebraic variety over $\mathbb P^n_K$ by assumption, so whatever combinatorial conditions one might put on the tropicalization, they ought to be satisfied. Unfortunately, I could never really make progress on these questions.
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