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

This article is devoted to studying a topos introduced by Faltings for the purpose of $p$-adic Hodge theory. We present a new approach based on a generalisation of Deligne's co-vanishing topos. Along the way, we correct Faltings' original definition.

We continue our study on the corresponding period rings with big coefficients, with the corresponding application in mind on relative $p$-adic Hodge theory and noncommutative analytic geometry. In this article, we extend the discussion of the corresponding noncommutative descent over \'etale topology to the corresponding noncommutative descent over pro-\'etale topology in both Tate and analytic setting.

We introduce a new compactification of the space of relative stable maps. This new method uses logarithmic geoemtry in the sense of Kato-Fontaine-Illusie rather than the expanded degeneration. The underlying structure of our log stable maps is stable in the usual sense.

We prove a comparison isomorphism between the De Rham rational homotopy type of a smooth proper log variety defined over a p-adic field and the crystalline rational homotopy type of a semi-stable reduction mod p.

A. Sikora has shown results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's for...

We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of cohere...

In this article, we define the logarithmic Hasse symbol in the same way as the usual one but in the context of the logarithmic ramification. We study its fondamental properties. The interesting point is that we get an expression of the defect of the \^a-adic Hasse principle. Then we study particular cases and get a logarithmic version of the principal ideal theorem.

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.

Let $X$ be a complex analytic manifold, $D\subset X$ a free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), $j: U=X-D \to X$ the corresponding open inclusion, $E$ an integrable logarithmic connection with respect to $D$ and $L$ the local system of the horizontal sections of $E$ on $U$. In this paper we prove that the canonical morphisms between the logarithmic de Rham complex of $E(kD)$ and $R j_* L$ (resp. the logarithmic de Rham c...

The main purpose of this paper is to make $\bar{C}_{n,n-1}$, which is the main theorem of [Ka1] Y.Kawamata, Addition formula of logarithmic Kodaira dimensions for morphisms of relative dimension one, more accessible.