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 VII0 surfaces?