Let X be a smooth or proper variety defined over a finite field. The geometric etale fundamental group of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Q_l-algebraic completion of the fundamental group as a non-abelian Weil representation. Lafforgue's Theorem and Deligne's Weil II theorems imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When X is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of groups which cannot arise as etale fundamental groups of smooth varieties.
In this article, we generalize the Hodge-Tate decomposition of p-adic étale cohomology to non-smooth rigid spaces. Our strategy is to study pro-étale cohomology of rigid spaces introduced by Scholze, using the resolution of singularities and the simplicial method.
Étale difference algebraic groups are a difference analog of étale algebraic groups. Our main result is a Jordan-Hölder type decomposition theorem for these groups. Roughly speaking, it shows that any étale difference algebraic group can be build up from simple étale algebraic groups and two finite étale difference algebraic groups. The simple étale algebraic groups occurring in this decomposition satisfy a certain uniqueness property.
Let X be a smooth, geometrically connected variety over a p-adic local field. We show that the pro-unipotent fundamental group of X (in both the etale and crystalline settings) satisfies the weight-monodromy conjecture, following Vologodsky. We deduce (in the etale setting) that Frobenii act semisimply on the Lie algebra of the pro-unipotent fundamental group of X, and (in the crystalline setting) that the same is true for a K-linear power of the crystalline Frobenius. We give applications to the representability and geometry of the Selmer varieties appearing in the Chabauty-Kim program, even …
Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (…
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
In this paper we partly extend the Beauville-Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible holomorphic symplectic varieties.
In a work of T. Saito, the action of algebraic correspondences on the etale cohomology of varieties over local fields with semistable reduction is related to correspondences on smaller strata via weight spectral sequences. We give an intersection theoretic construction of these correspondences. Under a finiteness condition this enables us to compute them without involving the blow-ups of products, and prove their compatibility with compositions. These features are essential for the application to Shimura varieties.
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more re…
We study certain triangulated categories of $K$-motives $DK(-)$ over a wide class of base schemes, and define certain "weights" for them. We relate the weights of particular $K$-motives to (negative) homotopy invariant $K$-groups (tensored by $\mathbb{Z}[S^{-1}]$ for $S$ being the set of "non-invertible primes") $\mathcal{K}_*(-)$. Our results yield a (new) result on the vanishing of $\mathcal{K}_i(-)$ and of relative $\mathcal{K}$-groups for $i$ being "too negative"; this statement is closely related to a question of Ch. Weibel. We also prove that $\mathcal{K}_i(-)$ for $i<0$ is "supported in…
A p-typical cover of a connected scheme on which p=0 is a finite etale cover whose monodromy group (i.e., the Galois group of its normal closure) is a p-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the p-typical quotients of the etale fundamental groups, and a decomposition theorem for p-typical covers of polynomial rings over an algebraically closed field.
Our main goal in the present article is to explain how one can reconstruct the decomposition subgroups and norms of points on an arithmetic curve inside its fundamental group if the following data are given: the fundamental group, a part of the cyclotomic character and the family of the regulators of the fields corresponding to the generic points of all étale covers of the given curve. The approach is inspired by that of Tamagawa for curves over finite fields but uses Tsfasman-Vlăduţ theorem instead of Lefschetz trace formula. To the authors' knowledge, this is a new technique in the anabelian…
Let $X$ be a $\mathbb Q$-Fano variety admitting a Kähler-Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler-Einstein $\mathbb Q$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by $\mathscr O_X$ is semistable with respect to the anticanonical polarization.
In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $π_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary p…
For $X$ a complete, reduced, geometrically connected scheme over a perfect field of characteristic $p>0$, we analyze the decomposition of Nori's fundamental group scheme into its local and étale parts and raise the question of the relation between the geometry and the splitting of the group scheme. We also describe in categorial terms the functor which corresponds to the inclusion of the maximal reduced subgroup scheme.
We construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over non-Archimedean fields embeddable into $\mathbf{C}_p$, which generalizes a construction of Berkovich and solves a question raised by him. We then investigate complexes of real tropical differential forms and currents introduced by Chambert-Loir and Ducros, by establishing a relation with the weight decomposition and defining tropical cycle maps with values in the corresponding Dolbeault cohomology. As an application, we show that algebraic cycles that are cohomolo…
We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tangent sheaf are established. As a consequence, known decompositions for tangent sheaves of varieties with trivial canonical divisor are refined. In particular, we show that up to finite quasi-étale covers, varieties with strongly stable tangent sheaf are either Calabi-Yau or irreduc…
Hassett constructed a class of modular compactifications of the moduli space of pointed curves by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights where the moduli space and universal family remain constant inside a chamber, and may change upon crossing a wall. The goal of this paper is to count the number of chambers in this decomposition. We relate these chambers to a class of boolean functions known as linear threshold functions (LTFs), and discover a subclass of LTFs which are in bijection with the chambers. Usi…
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition t…
We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $Obj DM^{eff}_{gm,R}(r)=Obj DM^{eff}_{gm,R} \otimes L^{\otimes r}$ of effective Voevodsky motives (for $r\ge 1$; $R$ is the coefficient ring). In particular, $M$ is 1-effective if and only if a complex whose terms are certain Chow groups of zero-cycles is acyclic. The dual to this statement checks whether an effective motif $M$ belongs to the subcategory of $DM^{eff}_{gm,R}$ generated by motives of varieties of dimension $\le r$. These criteria are formulated in terms of the Chow-weight (c…
