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corrected some mispellings, added links to two references mentioned

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edit approved Oct 25, 2017 at 8:33

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I am looking for a geometric and topologictopological way to make a visualization of higher dimensional berkovich spaces, statringstarting with the berkovich plane. Of course, this is just a collection of bounded semi-norms, but the question remains: is there a visualization like the infinite brnched tree

Is there a visualization possible for $\mathbb A^2_{\text{Berk}}$ like the infinite branched tree for $\mathbb A^1_{\text{Berk}}$?

(seeFor $\mathbb A^1_{\text{Berk}}$ see for example Matt Baker's Dynamics abd PotentialBaker and Rumely's [Potential Theory and Dynamics on the Berkovich projective line) possibleline] 1, Chapters 1 - 2.)

I think you get a simplicial complex, but I don't know exactly how. On the one hand (reading Favre and Johnsson's: The valuative tree The valuative tree), you have this list of valuations (thus, also of seminorms, allthough this book discusses seminorms on $\mathbb{C}^2$). On the other hand we have Berkovich's theory of Type I - Type IV points. I guess there just more Type I - Type IV points in a plane (i.e. more seminorms that occur as it were type I points), and some can be only represented by faces (two dimensional simplices), only I don't know how.

Are there any references on the visualization part ?

I am looking for a geometric and topologic way to make a visualization of higher dimensional berkovich spaces, statring with the berkovich plane. Of course, this is just a collection of bounded semi-norms, but the question remains: is there a visualization like the infinite brnched tree (see for example Matt Baker's Dynamics abd Potential Theory on the Berkovich projective line) possible.

I think you get a simplicial complex, but I don't know exactly how. On the one hand (reading Favre and Johnsson's: The valuative tree), you have this list of valuations (thus, also of seminorms, allthough this book discusses seminorms on $\mathbb{C}^2$. On the other hand we have Berkovich's theory of Type I - Type IV points. I guess there just more Type I - Type IV points in a plane (i.e. more seminorms that occur as it were type I points), and some can be only represented by faces (two dimensional simplices), only I don't know how.

Are there any references on the visualization part ?

I am looking for a geometric and topological way to make a visualization of higher dimensional berkovich spaces, starting with the berkovich plane. Of course, this is just a collection of bounded semi-norms, but the question remains:

Is there a visualization possible for $\mathbb A^2_{\text{Berk}}$ like the infinite branched tree for $\mathbb A^1_{\text{Berk}}$?

(For $\mathbb A^1_{\text{Berk}}$ see for example Baker and Rumely's [Potential Theory and Dynamics on the Berkovich projective line] 1, Chapters 1 - 2.)

I think you get a simplicial complex, but I don't know exactly how. On the one hand (reading Favre and Johnsson's The valuative tree), you have this list of valuations (thus, also of seminorms, allthough this book discusses seminorms on $\mathbb{C}^2$). On the other hand we have Berkovich's theory of Type I - Type IV points. I guess there just more Type I - Type IV points in a plane (i.e. more seminorms that occur as it were type I points), and some can be only represented by faces (two dimensional simplices), only I don't know how.

Are there any references on the visualization part?

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asked Jul 22, 2010 at 10:19

Willem Noorduin

Higher dimensional berkovich spaces

Are there any references on the visualization part ?

berkovich-geometry