Kato-Nakayama's comparison theorem and analytic log etale topoi

We give a functorial description of the Kato-Nakayama space of a fine saturated log analytic space that is similar in spirit to the functorial description of root stacks. As a consequence we get a global description of the comparison map constructed in arXiv:1511.00037 from the Kato-Nakayama space to the (topological) infinite root stack.

We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer \'etale and pro-Kummer \'etale topology for such spaces. We also establish the primitive comparison theorem in this context, and deduce from it some related cohomological finiteness or vanishing results.

In this article, we analyze the connection between the Log De Rham Cohomology of an fs (not necessary log smooth) log scheme $Y$ over $\mathbb C$ (for $Y$ admitting an exact closed immersion into an fs log smooth log scheme over $\mathbb C$), its Log Infinitesimal Cohomology $H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}})$, and its Log Betti Cohomology, which is the Cohomology of its associated Kato-Nakayama topological space $Y^{an}_{log}$, and we prove that they are isom...

We compare the Kummer flat (resp. Kummer etale) cohomology with the flat (resp. etale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes and the logarithmic multiplicative group of Kato. We will be particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, w...

We extend the formalism of "log spaces" of arXiv:1507.06752 to topoi equipped with a sheaf of monoids, and discuss Deligne--Faltings structures and root stacks in this context.

We use the recently introduced \'etale open topology to prove several facts about large fields. We show that these facts lift to a very general topological setting.

We characterize K. Kato's log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.

Those are notes of a mini-course the author gave in July 2010 at the university Paris 6 (Jussieu) during the summer school of the ANR (Agence nationale de la recherche) BERKO.

Let D be a divisor in a complex analytic manifold X. A natural problem is to determine when the de Rham complex of meromorphic forms on X with poles along D is quasi-isomorphic to its subcomplex of logarithmic forms. In this mostly expository note, we recall the main results about this problem. In particular, we point out the relevance of the theory of D-modules to this topic.