Prelog Chow groups of self-products of degenerations of cubic threefolds

It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.

ON THE CHOW THEORY OF PROJECTIVIZATIONS

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$ . In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.

De Rham–Witt sheaves via algebraic cycles

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

Chow groups and pseudoffective cones of complexity-one T-varieties

We show that the pseudoeffective cone of k-cycles on a complete complexity-one T-variety is rational polyhedral for any k, generated by classes of T-invariant subvarieties. When X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the combinatorial data defining X as a T-variety.

Representability of Chow groups of codimension three cycles

Assume that we have a fibration of smooth projective varieties X → S over a surface S such that X is of dimension four and that the geometric generic fiber has finite-dimensional motive and the first étale cohomology of the geometric generic fiber with respect to ℚ l coefficients is zero and the second étale cohomology is spanned by divisors. We prove that then A 3(X), the group of codimension three algebraically trivial cycles modulo rational equivalence, is dominated by finitely many copies of A 0(S); this means that there exist finitely many correspondences Γi on S × X such that Σ i Γi is surjective from A 2(S) to A 3(X).

ISOTROPIC MOTIVES

In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$ -coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with $\mathbb{F} _p$ -coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.