What does a general proper rigid space look like?

As the title says. Consider proper rigid spaces X over some nonarchimedean field K. The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface (\mathbf{A}^2 - 0)/p^\mathbf{Z}, ii) non-algebraizable deformations of K3 surfaces over the residue field of K, 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 n \geq 2, does there exist a proper n-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 \mathrm{tr.deg}K(X)/K is small compared to \dim X.

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

-Do there exist rigid analytic analogues of Kodaira’s class VII0 surfaces?

2 thoughts on “What does a general proper rigid space look like?”

  1. 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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