Related papers: Remarks on curve classes on rationally connected v…

We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As…

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.

We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that…

We undertake a study of conic bundle threefolds $\pi\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tilde{\Delta}\to\Delta\subset W$ are smooth and geometrically irreducible. First, we determine…

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of…

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any…

We prove that the Tate conjecture for divisors is ''generically true'' for mod p reductions of complex projective varieties with $h^{2, 0} = 1$, under a mild assumption on moduli. By refining this general result, we establish a new case of…

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the…

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first…

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-K\"ahler variety of generalized Kummer type. For an arbitrary variety $X$ of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational…

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over…

We prove the Tate conjecture for integral degree 4 classes on a smooth cubic hypersurface X of dimension 4 over an algebraic closure of a field finitely generated over its prime subfield.

Let $\Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $\Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that…

All curves on a separably rationally connected variety are rationally equivalent to a (non-effective) integral sum of rational curves, hence the first Chow group is generated by rational curves. Applying the same techniques, we also proved…

This partly expository paper investigates versions of the Tate conjecture on the cycle map for varieties defined over finite fields with values in 'etale cohomology with Z_\ell-coefficients. The bulk of the paper is an exposition of a 1998…

Given a field with a set of discrete valuations $V$, we show how the genus of a division algebra over the field is related to the genus of the residue algebras at various valuations in $V$ and the ramification data. When the division…