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Let $k$ be a perfect field of virtual cohomological dimension $\leq 2$. Let $G$ be a connected linear algebraic group over $k$ such that $G^{sc}$ satisfies a Hasse principle over $k$. Let $X$ be a principal homogeneous space under $G$ over…

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible…

In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any…

Let U:=L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively…

We study points and 0-cycles on del Pezzo surfaces defined over a field K of characteristic 0, with emphasis on cubic surfaces. We prove that a cubic surface that admits a point defined over a field extension of K of degree coprime to 3…

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles…

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p>0. Here we classify all irreducible kG-modules for which the principal A1 has no repeated composition factors, extending the work of…

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of…

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety…

Let $k$ be a field of arbitrary characteristic. Let $S$ be a singular surface defined over $k$ with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation $\tilde{S}$ is finite dimensional.…

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…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational…

We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of…

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative…

Let $k$ be the function field of a complex curve or the field $C((t))$. We show that for a smooth complete intersection $X$ of $r$ hypersurfaces in $P^n_k$ of respective degrees $d_1,...,d_r$ with $\sum d_i^2\leq n+1$ the R-equivalence on…

Let $G$ be a semisimple affine algebraic group defined over a field $k$ of characteristic zero. We describe all the maximal connected solvable subgroups of $G$, defined over $k$, up to conjugation by rational points of $G$.

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…