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Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this construction (presumably something rigid-analytic)?

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  • $\begingroup$ I believe many people have thought about this at some point. The answer so far is no. Of course there are the algebraic analogs of the Kato-Nakayama space: the (Kummer) etale site or the infinite root stack. The following paper of Andre arxiv.org/abs/1207.3380 is the only reference I know which tries to make an analog of the 'real oriented blowup' construction. $\endgroup$ – Piotr Achinger May 11 '19 at 15:27
  • $\begingroup$ @PiotrAchinger you seem to be fairly informed about this stuff; if you made your comment an answer (maybe add some details if you wish), I would accept the answer (so this gets out of the unanswered queue). $\endgroup$ – user138661 May 11 '19 at 15:48
  • $\begingroup$ I'm working on this :-) $\endgroup$ – Konrad Voelkel Jul 6 '19 at 14:15
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I believe many people have thought about this at some point, and I don't think such a construction is known.

In the paper https://arxiv.org/abs/1207.3380 , Yves Andre considers the real blowups (in the complex plane) as a completion with respect to the "sectorial uniformity", and then considers some $p$-adic or non-archimedean variants. I am not sure if this produces a desired analog of the Kato-Nakayama space. Andre's point of view is applications to singularities of differential equations (especially irregular, i.e. worse than logarithmic poles), phenomena like overconvergence etc.

There are the algebraic analogs of the Kato-Nakayama space: the (Kummer) etale site or the infinite root stack, both equipped with a "proper" projection down to the underlying scheme, which in the log regular case are "homotopy equivalent" to the trivial locus of the log structure (in characteristic zero, or after prime to $p$ completion), and which over the complex numbers can be compared to the Kato-Nakayama space (see e.g. the paper of Carchedi-Scherotzke-Sybilla-Talpo).

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