Jean-Marc Drézet
Multiple primitive curves have been defined and studied by A. Banica and O. Forster. We introduce new invariants for coherent sheaves on these curves and define quasi locally free sheaves, which play the same role as locally free sheaves on smooth varieties. A more detailed study of torsion free sheaves is given for primitive curves of multiplicity 2.
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We give sufficient conditions for the (semi-)stability of torsion free sheaves on a primitive multiple curve. These conditions are used to prove that some moduli spaces of stable sheaves are not empty. We study mainly the quasi locally free sheaves of generic type (this includes the locally free sheaves). These sheaves are generic, i.e. for every moduli space of torsion free sheaves, the sheaves of this type correspond to an open subset of the moduli space.
This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve associated to Y, and that for every P in C, if z is an equation of C_1 in the local ring of Y at P, then (z^i) is the ideal of C_i. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e. sheaves which are locally i…
We give a new definition, simpler but equivalent, of the abelian category of Banach-Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues-Fontaine curve. One goes from one category to the other by changing the t-structure on the derived category. Along the way, we obtain a description of the pro-étale cohomology of the open disk and the affine space, of independent interest.
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the étale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or noda…
We study the behavior of integral l-adic sheaves on schemes of finite type over a local field under the six operations and the nearby cycle functor. ----- On etudie le comportement des faisceaux l-adiques entiers sur les schemas de type fini sur un corps local par les six operations et le foncteur des cycles proches.
Sur certains faisceaux de formes méromorphes sur un espace complexe réduit
本文研究约化复空间上的亚纯形式凝聚层，重点分析其在拉回和高阶直像下的行为，为复几何相关层论研究提供参考。
On revoit explicitement la construction ainsi que certaines propriétés des complexes de faisceaux étales $\mathbb{Q}/\mathbb{Z}(j)$ sur certains schémas. Le but de ces notes est d'avoir une référence précise pour la conjecture de Gersten pour les faisceaux $\mathbb{Q}/\mathbb{Z}(j)$ sur un corps imparfait ainsi que pour la comparaison entre les groupes de cohomologie $\mathrm{H}^2(\ast,\mathbb{Q}/\mathbb{Z}(1))$ et $\mathrm{H}^2(\ast,\mathbb{G}_m)$. We review explicitly the definition and some properties of étale sheaf complexes $\mathbb{Q}/\mathbb{Z}(j)$ on some schemes. The purpose of these …
Drinfeld a associe a tout systeme local irreductible de rang 2 sur une surface de Riemann compacte un "faisceau automorphe". Ce faisceau automorphe vit sur l'espace de modules des fibres vectoriels de rang 2 sur cette meme surface de Riemann. En fait, Drinfeld a donne deux constructions de ce faisceau automorphe. La premiere est basee sur l'existence d'un modele de Whittaker pour toute representation automorphe cuspidale pour GL(2). Elle garde un sens en caracteristique positive et elle s'etend conjecturalement en rang n arbitraire. La seconde construction est basee sur un calcul explicite de …
New updated edition by Yves Laszlo of the book ``Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)'', Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company - Amsterdam, 1968. Published by the Societe Mathematique de France http://smf.emath.fr/Publications/DocumentsMathematiques/ Original text also available in the LaTeX file. Dans cet ouvrage, on montre des théorèmes d'algébrisation et de pureté qui permettent d'obtenir des théorèmes de type Lefschetz pour le groupe fondamental ou de Picard. In this monograph algebraization and pu…
In this article the étale cohomology of constructible torsion sheaves on the étale site of the algebraic resp. adic Fargues-Fontaine curve is analyzed. In the $\ell\neq p$-torsion case, two conjectures of Fargues are verified: vanishing in degrees greater than two and the comparison between the étale cohomology of the adic and the algebraic curve. In the $p$-torsion case, under a certain assumption, the vanishing of the étale cohomology in degrees greater than two of those Zariski-constructible sheaves on the adic curve that come via pullback from the algebraic curve is proven.
Soit $X$ un schéma quasi-compact et séparé et soit $α\in \check{C}^2(X, \mathcal O_X^*)$ un cocycle de Cěch. Nous considérons la catégorie dérivée $D(QCoh(X,α))$ des faisceaux quasi-cohérents sur $X$ tordu par $α$. Soit $Δ(X,α)$ la plus petite sous-catégorie triangulée de $D(QCoh(X,α))$ contenant tous les objets $u_*\mathcal F^\bullet$, où $u:U\longrightarrow X$ est une immersion ouverte avec $U$ affine et $\mathcal F^\bullet\in D(QCoh(U,u^*(α)))$. Alors, le but de cet article est de montrer que $Δ(X,α)=D(QCoh(X,α))$. -- Let $X$ be a quasi-compact and separated scheme and let $α\in \check{C}^2…
We give a generalization, in the context of sheaves, of a classical result of Grothendieck concerning the integrability of connections of type $(0,1)$ over a ${\cal C}^{\infty}$ vector bundle over a complex manifold. We introduce the notion of $\bar{\partial}$-coherent sheaf, which is a ${\cal C}^{\infty}$ notion, and we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\partial}$-coherent sheaves. The principal difficulty of the proof is the solution of a quasi-linear differential equation with standard $\bar{\partial}$ a…
The ordinary algebraic curves of maximal rank are also the arithmetically Cohen-Maccaulay curves of minimal rank. We give sufficient conditions for such curves to exist as well as examples, generalizing results of [GHL] in the dimension three.
In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the crystalline cohomology of the curve, which is a lattice in the rigid cohomology. Following a method of Wach, we first explain how to use this lattice to compute the $(φ,Γ)$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(φ,Γ)$-module mod $p$ that relies on the Deligne-Illusie morphism.
The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety -- provided this coherent sheaf is a subquotient of a hermitian vector bundle -- and using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of t…
Cycle caractéristique sur une puissance symétrique d'une courbe et déterminant de la cohomologie étale
本文基于Beilinson和Saito发展的形式主义，计算了 tame 上同调sheaf 的对称幂的特征循环，推广了Laumon在特征0上的结果，并导致Deligne关于Abel-Jacobi映射局部空虚性的结论。
Let $k$ be a non-archimedean complete valued field and let X be a smooth Berkovich analytic $k$-curve. Let $F$ be a finite locally constant étale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the underlying topological space and by $π$ the canonical map from the étale site to $|X|$. In this text we define a triangulation of $X$, we show that it always exists and use it to compute $H^{0}(|X|,R^{q}π\_{*}F)$ and $H^{1}(|X|,R^{q}π\_{*}F)$. If $X$ is the analytification of an algebraic curve we give sufficient conditions so that those groups are isomorphic to…
We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms.
A primitive multiple curve is a Cohen-Macaulay scheme Y over the field of complex numbers such that the reduced scheme C=Y_red is a smooth curve, and that Y can be locally embedded in a smooth surface. In general such a curve Y cannot be globally embedded in a smooth surface. If Y is a primitive multiple curve of multiplicity n, then there is a canonical filtration of Y C=C_1 ... C_n=Y such that C_i is a primitive multiple curve of multiplicity i. The ideal sheaf I_C of C in Y is a line bundle on C_{n-1}. Let T be a smooth curve and t_0 a closed point of T. Let D-->T be a flat family of projec…
Let $k$ be a field, $C\to \Spec k$ be a stable curve and let $G$ be a finite group acting faithfully on the curve $C\to \Spec k$. In this article, we compute the vector space $\Ext^1_G(Ω_{C/k}, Ø_C)$, the sheaf $Ω_{C/k}$ being the sheaf of relative differentials. This vector space is naturally isomorphic to the set of first order $G$-equivariant deformations of $C$. The computation we do here will be used in a future article.
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Banica and O.Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z_n=spec(\C[t]/(t^n)). The curves of multiplicity n are obtained by taking an open cover (U_i) of a smooth curve C and by glueing schemes of type U_i X Z_n using automorphisms of U_{ij} X Z_…
Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In this article we introduce the notion of a triad, which is a generalization to families of curves in ${\bf P}^3_A$ of the notion of Rao module. A triad is a complex of graded $R_A$-modules $(L_1 \to L_0 \to L_{-1})$ with certain finiteness hypotheses on its cohomology modules. A pseudo-isomorphism between two triads is a morphism of complexes which induces an isomorphism on the functors $ h_0 (L\otimes .)$ and a monomorphism on the functors $h_{-1} (L\otimes .)$. One says that two triads are pseudo-isomorphic if they are…
Troisième groupe de cohomologie non ramifiée d'un solide cubique sur un corps de fonctions d'une variable
该论文证明了复曲线函数域上光滑三次三维流形的第三非分歧上同调群为零，并由此推出维数4的光滑射影簇上度数为4的类的整Hodge猜想成立。
This is an update of the first version. We clarify that the main results apply to more general smooth projective varieties X than products of elliptic curves (briefly: X is of "abelian type", e.g. an abelian variety or a product of curves, and the Tate conjecture holds). We also deduce the Gersten conjecture for dvrs whose residue field is the function field of such an X. In an appendix, we construct some functoriality for etale motivic cohomology. Finally, some errors which were in Section 5 of the first version are corrected.
Pour $K$ un corps global (corps de nombres ou corps de fonctions d'une variable sur un corps fini $F$), on dispose de théorèmes de dualité classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne à valeurs dans des tores et des modules abéliens finis. Nous établissons de tels théorèmes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des séries formelles en une variable sur le corps des complexes. Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a ici des diff…
We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any algebraically closed field extension, the degree map on the zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the Néron-Severi group has no torsion, we recover earlier results of A. Pirutka. The results rely on a detailed study of the third unramified cohomology group of specific products of varieties.
The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which caracterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism of coherent sheaves, which generalizes the concept of stable isomorphism. An N-type resolution for a family of curves $C$ over the local ring $A$, defined by an ideal $J$, is an exact sequence $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on $P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. We prove the two following results, when the residual field of $A$ is inf…
Gabber deduced his theorem of independence of $l$ of intersection cohomology from a general stability result over finite fields. In this article, we prove an analogue of this general result over local fields. More precisely, we introduce a notion of independence of $l$ for systems of complexes of $l$-adic sheaves on schemes of finite type over a local field, equivariant under finite groups. We establish its stability by Grothendieck's six operations and the nearby cycle functor. Our method leads to a new proof of Gabber's theorem. We also give a generalization to algebraic stacks. ----- Gabber…
Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ impl…
Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a …
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