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.)

    Like

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