Related papers: Parshin's conjecture revisited

Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\'el\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of…

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a…

Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several…

Let $k$ be a field and $X$ a smooth projective variety over $k$. When $k$ is a number field, the Beilinson-Bloch conjecture relates the ranks of the Chow groups of $X$ to the order of vanishing of certain $L$-functions. We consider the same…

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the…

For quasi-projective varieties over a higher local field $k_N$, we prove that its $K$-groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to-$p$ torsion subgroup of certain higher Chow groups…

A proof based on reduction to finite fields of Esnault-Viehweg's stronger version of Sommese Vanishing Theorem for $k$-ample line bundles is given. This result is used to give different proofs of isotriviality results of A. Parshin and L.…

We prove that a strengthened version of Minac-Tan's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-$2$ Galois group is of elementary type (as defined by I. Efrat). In particular,…

We present a new perspective on the weak approximation conjecture of Hassett and Tschinkel: formal sections of a rationally connected fibration over a curve can be approximated to arbitrary order by regular sections. The new approach…

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta…

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable…

We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finally…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

In this paper I verify Manin's conjecture for a class of rational projective toric varieties with a large class of heights other than the usual one that comes from the standard metric on projective space.

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…

The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not…

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. Andr\'{e} proved this conjecture for…

We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D \subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally isotropic,…

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic $p > 5$. As a consequence, we deduce the Grauert-Riemenschneider vanishing theorem for excellent divisorial log terminal…

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both…