Restriction, corestriction and the characteristic homomorphism for 0-cycles of degree 0

Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois coverings

Let f: S' --> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps α_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and β_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that α_{r}oβ_{r}=β_{r}oα_…

The norm map and the capitulation kernel

Let f: S'--> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n > 1 and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. Under certain restrictions on f and G, we relate the kernel of the restriction map in degree r>0 étale cohomology Res_{G}^{(r)}: H^{r}(S_{\et},G)--> H^{r}(S'_{\et},G) to a certain quotient of the kernel of the mod n corestriction map in degree r-1, namely Cores_{G}^{(r-1)}/n: H^{r-1}(S'_{\et},G)/n\to H^{r-1}\lbe(S_{\et},G)/n. When r=1 and f is a Galois covering with Galois group D, our main the…

Weil transfer of noncommutative motives

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integral coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with i…

Invariants de classes : exemples de non-annulation en dimension supérieure

The so-called class-invariant homomorphism $ψ$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (Néron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $ψ$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed wit…

关于 $\hat{\mathcal{G}}^{(\lambda)}$ 被 $\hat{\mathbb{G}}_m$ 扩张的某些同态在 $\mathbb{Z}_{(p)}$-代数上的单射性

On the injectivity of certain homomorphisms between extensions of $\hat{\mathcal{G}}^{(λ)}$ by $\hat{\mathbb{G}}_m$ over a $\mathbb{Z}_{(p)}$-algebra

本文证明在 $\mathbb{Z}_{(p)}$-代数上，由 Frobenius 型同态 $\psi^{(l)}$ 诱导的第二个上同调群同态是单射，并描述了核的 Cartier 对偶。

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety $M$ and a smooth closed curve $X \subset M$ such that the homomorphism of fundamental groups $π_1(X) \rightarrow π_1(M)$ is surjective, we study the restriction map of Higgs bundles, namely from Higgs bundles on $M$ to those on $X$. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on $M$ and $X$. We also consider the set-up where a finite group is acting on $M$ via holomorphic automorphisms or anti-holomorphic involutions, and the curve $X$ is preserved …