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

A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log ...

In this paper, we construct a comparison map from the topological fundamental group to the pro-\'etale fundamental group for a complex variety.

We prove criteria for a presheaf on logarithmic schemes to be a sheaf in the full logarithmic \'etale topology and describe several situations where the structure sheaf and logarithmic structure are logarithmic \'etale sheaves. We deduce that the logarithmic Picard group is a stack in the full logarithmic \'etale topology on logarithmic schemes whose structure sheaves satisfy logarithmic \'etale descent.

We compute, in a stable range, the arithmetic p-adic etale cohomology of smooth rigid analytic and dagger varieties (without any assumption on the existence of a nice integral model) in terms of differential forms using syntomic methods. The main technical input is a construction of a Hyodo-Kato cohomology and a Hyodo-Kato isomorphism with de Rham cohomology.

We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.

An introduction to geography of log models with applications to positive cones of FT varieties and to geometry of minimal models and Mori fibrations.

We propose a method for constructing cohomology theories of logarithmic schemes with strict normal crossing boundaries by employing techniques from logarithmic motivic homotopy theory over $\mathbb{F}_1$. This method recovers the K-theory of the open complement of a strict normal crossing divisor from the K-theory of schemes as well as logarithmic topological Hochschild homology from the topological Hochschild homology of schemes. In our applications, we establish that the K-...

The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-\'etale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over $\bf{B}^+_{\rm dR}$). The key computation is the passage from absolute crystalline cohomology to Hyodo-Kato cohomology and the construction of the related Hyodo-Kato isomorphi...

In this paper we will study the moduli spaces of log Hodge structures introduced by Kato-Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. We treat the following 2 cases: (A) the case where the period domain is Hermitian symmetric, (B) the case where the Hodge structures are of the mirror quintic type. Especially we study a property of the torsor.

This paper presents an extended version of lecture notes for an introductory course on Berkovich analytic spaces that I gave in 2010 at Summer School "Berkovich spaces" at Institut de Mathmatiques de Jussieu.