Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $\left(\overline{X}\right)\oplus$ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$\left(X\right)\oplus$ Br$\left(Y\right)\to$ Br$(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the...

Let $G$ be a semisimple linear algebraic group of inner type over a field $F$, and let $X$ be a projective homogeneous $G$-variety such that $G$ splits over the function field of $X$. We introduce the $J$-invariant of $G$ which characterizes the motivic behavior of $X$, and generalizes the $J$-invariant defined by A. Vishik in the context of quadratic forms. We use this $J$-invariant to provide motivic decompositions of all generically split projective homogeneous $G$-varieties, e.g. Severi-Brauer varieties,...

Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C\left(𝒱\right)$ such that for almost all primes $p$ for all but at most $C\left(𝒱\right)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Let $K$ be a field of odd characteristic $p$, let $f\left(x\right)$ be an irreducible separable polynomial of degree $n\ge 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^2=f\left(x\right)$ and $J\left(C\right)$ its jacobian. We prove that $J\left(C\right)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n\ge 7$ or $p\ne 3$.

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

Let $q$ be a positive integer. An algebra is said to have the property $(2,q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2,q)$ if every member of $𝒱$ has the property $(2,q)$. Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2,q)$, $2<q$, blocks of Steiner system of type $(2,q)$ are obtained. The stated correspondence...

A cluster ensemble is a pair $(𝒳,𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma$. The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p:𝒜\to 𝒳$. The space $𝒜$ is equipped with a closed $2$-form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Let $p$ be a prime integer and $F$ a field of characteristic $0$. Let $X$ be theof a symbol in the Galois cohomology group $H^{n+1}(F,{\mu }_p^{\otimes n})$ (for some $n\ge 1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F\left(X\right)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\mathrm{CH}\left(Y\right)\to \mathrm{CH}\left(Y_{F\left(X\right)}\right)$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $\&lt;(dimX)/(p-1)$. One of the main ingredients of the proof is a computation...

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $ℝ$. For any $P\in {\mathbb{P}}^n\left(ℝ\right)$ the real $X$-rank $r_{X,ℝ}\left(P\right)$ is the minimal cardinality of $S\subset X\left(ℝ\right)$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...