As the title says. Consider proper rigid spaces over some nonarchimedean field . The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface , ii) non-algebraizable deformations of K3 surfaces over the residue field of , and iii) generic abeloid varieties (which are analogous to generic compact complex tori). But there must be gazillions of other examples, right? A “random” proper rigid space is hard to write down, sort of by definition. But there are certainly some natural questions one can ask:

-For every , does there exist a proper -dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that is small compared to .

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII_{0} surfaces?

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This article: https://surfacescompactes.files.wordpress.com/2013/09/classification-of-rigid-analytic-surfaces.pdf might already have answered the last question. (But to figure out the answer, one has to read that article….)

Is it clear if the transcendence degree of K(X) over K is finite when X is proper over K?

I was wondering whether every, say, isolated singularity in rigid geometry is algebraizable. (so “arbitrarily bad” could mean even worse than algebraic ones, namely really really bad.)

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I cannot find an actual “classification” in that paper!

Yes, it’s always bounded above by , see Conrad’s article on Moishezon spaces.

I think isolated singularities can be algebraized, cf. Theorem 3.8 in (http://www.numdam.org/article/PMIHES_1969__36__23_0.pdf).

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