arXiv subjects
Publications and source records attributed to Humberto A. Diaz.
We consider a weak version of the Rost Nilpotence Principle. For characteristic zero fields k, we show that if it holds for all smooth projective schemes over k, then the Rost Nilpotence Principle does also. We also correct the proof of a lemma in arXiv:1807.08163(3).
We obtain examples of smooth projective varieties over $\mathbb{C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.
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-Thélène and Skorobogatov to higher Brauer groups.
In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of "field of definition" for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles we call "primitive cycles," the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori's isogeny method \cite{N} (modified by Totaro \cite{T}) to produce infinitely many non-divisible cycles. As an application, we prove the Chow group modulo a prime of a product of $3$ or more very general elliptic curves is infinite, generalizing work of Schoen.
We develop an approach to proving the Rost nilpotence principle involving higher unramified cohomology. We use this to prove the principle for certain varieties of dimension $\leq 3$ over a perfect field.
In this note, we consider unramified cohomology with $\mathbb{Z}/2$ coefficients for some (degree two) quotient varieties and describe a method that allows one to prove the non-vanishing of these groups under certain conditions. We apply this method to prove a non-vanishing statement in the case of Kummer varieties. Combining this with work of Colliot-Thélène and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture.
We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.
Using fundamental results of Deligne, we prove a nilpotence theorem for algebraic cycles and use this to prove a torsion nilpotence result for correspondences on surfaces.
Generalizing work of Schoen, we prove that the Chow group modulo $\ell$ of a product of $3$ or more very general complex elliptic curves is infinite.
The purpose of this note is to prove Grothendieck's standard conjectures for the Fano variety of lines on a smooth cubic hypersurface in projective space.
We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $Θ$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $Θ$.