Related papers: Parshin's conjecture revisited

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane…

Let $K\subseteq \mathbb{R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\mathbb{R}_{\exp}$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is…

For $X$ a smooth projective variety over a field $k$, we consider the problem of Galois descent for higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Th\'el\`ene and Skorobogatov to higher Brauer groups.

We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n…

In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has…

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau…

In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $\rho:\pi_1(X)\to {\rm GL}_{N}(\mathbb{C})$,…

Let $A$ be a regular ring of dimension $\le 2$. Let $G$ be a reductive group over $A$ such that its derived group is a split, i.e. a Chevalley--Demazure, semisimple group. We prove that every Zariski-locally trivial principal $G$-bundle…

By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain…

We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-$p$-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of…

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H \subseteq G$ an observable subgroup normalized by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov…

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups".…

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive…

In this article we propose a vanishing conjecture for a certain class of $\ell$-adic complexes on a reductive group $G$, which can be regraded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing…

For a certain class of compact oriented 3-manifolds, M. Goussarov and K. Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ``cut-and-paste'' operations defined by the lower…

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly, infinite-dimensional)…

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a…

We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra.…

We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two…

We discuss a possible approach to the study of the vanishing of the Kobayashi pseudometric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighborhood of X in projective space for…