Related papers: Local-global principle and integral Tate conjectur…

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such…

In the 1980's Colliot-Th\'{e}l\`{e}ne, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for $0$-cycles on arbitrary smooth projective varieties over a number field. We give some evidence for these…

We extend the descent theory of Colliot-Th\'el\`ene and Sansuc to arbitrary smooth algebraic varieties by removing the condition that every invertible regular function is constant. This links the Brauer--Manin obstruction for integral…

The Tate conjecture predicts that Galois-invariant classes in $\ell$-adic cohomology, and Frobenius-invariant classes in crystalline cohomology, arise from algebraic cycles. We prove an unconditional p-adic analogue of this principle in the…

Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overline{\rho}: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the…

We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and…

We study for rationally connected varieties $X$ the group of degree 2 integral homology classes on $X$ modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies…

Embeddings of maximal tori into classical groups over global fields of characteristic not 2 are the subject matter of several recent papers, with special attention to the Hasse principle. The present paper gives necessary and sufficient…

We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over open varieties under a connected linear algebraic group or a group of multiplicative type is given. Such a relation is further refined for…

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the…

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…

We construct an Enriques surface X over Q with empty \'etale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on…

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers…

We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an \(n\)-fold cup-product Bockstein mechanism. Diaz's dimension-four example is \(V=S_1\times…

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular…

Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret…

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$-arithmetic manifold is purely local, i.e., only depends on the Galois representation at places above $p$. This is a…

In this paper, for a smooth variety equiped with an action of a connected algebraic group (not necessary linear), we introduce the notion of invariant Brauer sub-group and the notion of invariant \'etale Brauer-Manin obstruction. Then we…

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth…

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.