Sophie Térouanne
In this note we consider a quotient π: X-> X/G, with G a finite group and X a smooth projective scheme such that X/G is smooth. We compare the equivariant derived category D^G(X) and the derived category of the quotient D(X/G) by giving the image of the functor π^*: D(X/G)-> D^G(X).
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In order to have cohomological operations for de Rham p-adic cohomology with coefficients as manageable as possible, the main purpose of this paper is to solve intrinsically and from a cohomological point of view the lifting problem of smooth schemes and their morphisms from characteristic p > 0 to characteristic zero which has been one of the fundamental difficulties in the theory of de Rham cohomology of algebraic schemes in positive characteristic since the beginning. We show that although smooth schemes and morphisms fail to lift geometrically, it is as if this was the case within the coho…
The monodromy groups of lisse sheaves and overconvergent $F$-isocrystals
本研究将有限域光滑曲线上相容半单纯光滑层单值群的相关结果，推广到任意光滑簇上的半单光滑层与超收敛F-等晶相容系统，拓展了Frobenius环面定理，并用Tannakian形式明确了两类系数对象的相似性。
Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a category of complexes. Roughly speaking, the criterion shows that if such a functor satisfies the standard exact sequence of a blowing-up, then the functor can be extended to all separated k-schemes of finite type. The result is applied to the Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an analytic space, and to the rational homotopy of k-schemes in algebraic De Rham theory.
In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks which are locally of finite type over suitable regular basis of dimension at most 1.
Greenberg's $μ=0$ conjecture for lisse sheaves over global function fields
本文在正特征下证明了Greenberg的μ=0猜想的类比，针对全局函数域上的lisheaves，展示了Selmer群的结构及相关变形环的性质。
Let F be a finite field and let C be a smooth projective curve over F. For some smooth projective surfaces X over F we establish that the third unramified cohomology of the product of X and C vanishes. This applies in particular to geometrically rational surfaces. Soit F un corps fini et soit C une courbe projective et lisse sur F. Pour certaines surfaces projectives et lisses X sur F on établit la nullité du troisième groupe de cohomologie non ramifiée du produit de X et C. Cela s'applique en particulier aux surfaces géométriquement rationnelles.
We provide an effective version of Katz' criterion for finiteness of the monodromy group of a lisse, pure of weight zero, $\ell$-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf
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…
For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. ---- Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée …
Soit $X$ une variété projective et lisse, définie sur un corps de nombres. Sous l'hypothèse $H^2(X,\mathcal O_X)=0,$ Colliot-Thélène et Raskind ont démontré que le sous-groupe de torsion $CH^2(X)_{tors}$ du groupe de Chow en codimension $2$ est fini. Dans cette note, on donne des bornes uniformes pour le groupe fini $CH^2(X)_{tors}$ quand $X$ varie en famille. Let $X$ be a smooth projective variety defined over a number field. Assuming $H^2(X,\mathcal O_X)=0,$ Colliot-Thélène and Raskind proved that the torsion subgroup $CH^2(X)_{tors}$ in the Chow group of cycles of codimension $2$ is finite.…
One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.
B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-c…
Admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le même résultat vaut pour n=4, c'est-à-dire pour les surfaces de del Pezzo de degré 4, lorsque le groupe de Brauer est réduit aux constantes et que la surface est suffisamment générale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics i…
Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une variété propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des zé…
Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Thélène, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the i…
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…
Dualité étale à la Poitou-Tate pour les tores sur des variétés définies sur un corps fini
本文研究特征正整体域上光滑概形上环面的 Tate-Shafarevich 群，建立了推广的 Poitou-Tate 对偶，将其从经典环面情形扩展到任意维数有限域上的簇。
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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