Related papers: Specialization for the pro-\'etale fundamental gro…

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in a previous paper (Conner-Herfort-Pavesic:…

Formal orbifolds are defined in higher dimension. Their \'etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to…

The paper has two parts. First we prove that the specialization maps on R-equivalence and on the Chow group of zero cycles are isomorphisms for families over a local, Henselian, Dedekind ring when the special fiber is smooth and separably…

We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of…

We compute the pro-\'etale fundamental group of a connected Nagata J-2 scheme in terms of the \'etale fundamental groups of the normalizations of its irreducible components and a discrete free group. The result generalizes a formula of E.…

The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization…

Andr\'{e} and Maulik--Poonen proved that for any smooth proper family $X\to B$ of varieties over an algebraically closed field of characteristic $0$, there is a closed fiber whose N\'{e}ron-Severi group has the same rank as that of the…

We extend the classical (connected, etale) factorization of locally connected geometric morphisms into a (terminally connected, pro-etale) factorization for all geometric morphisms between Grothendieck topoi. We discuss properties of both…

This paper establishes strong profinite rigidity results for K\"ahler groups, showing that certain groups are determined within the class of residually finite K\"ahler groups by their profinite completion. Examples include products of…

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale…

For a smooth scheme $X$ over a perfect field $k$ of positive characteristic, we define (for each $m\in\mathbb{Z}$) a sheaf of rings $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ of differential operators (of level $m$) over the Witt vectors of $X$.…

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian.…

Inspired by the recent works of M. Kontsevich--Y. Tschinkel and J. Nicaise--J. C. Ottem on specialization of birational types for smooth families (in the scheme category) and J. Koll{\'a}r's work on fiberwise bimeromorphism, we focus on…

Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham…

We introduce the specialization map in Scholzes theory of diamonds. We consider v-sheaves that behave like formal schemes and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a…

Let R be a polynomial ring over k(u), where k is a field k and u is a finite family or inderterminates. The paper introduces the specialization of an arbitrary finitely generated R-module by the substitution of u to elements of k. This…

The aim of the paper and of a wider project is to translate main notions of anabelian geometry into the language of model theory. Here we finish with giving the definition of the \'etale fundamental group $\pi^{et}_1(X,x)$ of a non-singular…

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the \'etale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed…

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) \'etale fundamental group. A key result in our investigation is that these…

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the…