Related papers: Refined obstructions to local-global principles fo…

We give formulas for calculating the unramified Brauer group of a homogeneous space $X$ of a semisimple simply connected group $G$ with finite geometric stabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$ is a…

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the Hasse principle for Kummer varieties associated to a 2-covering of a principally…

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step…

We discuss the role of primes of good reduction in the existence of the Brauer--Manin obstruction to weak approximation for varieties defined over number fields. Following Bright and Newton, we give some necessaries conditions on the…

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes $Z^\bullet$ of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic…

In this paper, we study weak approximation with Brauer--Manin obstruction with respect to extensions of number fields. For any nontrivial extension $L/K,$ assuming a conjecture of M. Stoll, we prove that there exists a $K$-threefold…

We consider arithmetic analogs of the relative Langlands program and applications of new non-reductive geometry. Firstly, we introduce mirabolic special cycles, which produce special cycles on many Hodge type Rapoport-Zink spaces via…

For a quasi-projective smooth geometrically integral variety over a number field $k$, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an…

The \'etale Brauer\textendash Manin obstruction is equivalent to each other among Weil restrictions of an arbitrarily given quasi-projective algebraic variety defined over a number field.

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil $L$-series…

Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of density properties of rational points on the Kummer variety…

Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$.…

Strong approximation with Brauer-Manin obstruction is established for smooth varieties containing a connected linear algebraic group with a compatible action.

We extend the methods of Davis-Januszkiewicz-Lafont to provide a new obstruction to smooth Riemannian metric with non-positive sectional curvature. We construct examples of locally CAT(0) 4-manifolds $M$, whose universal covers satisfy…

We prove that, on a smooth projective variety over an algebraically closed field of characteristic 0, the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable…

This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-K\"ahler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a…

We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $\alpha$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized…

In this paper, we investigate the local Euler obstruction and the relative local Euler obstruction in terms of constructible complexes of sheaves, characteristic cycles, and vanishing cycles. The fundamental tool that we use is the notion…

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…