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We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of…

We prove Bloch's formula for the Chow group of 0-cycles with modulus on smooth projective varieties over finite fields. The proof relies on two new results in global ramification theory.

Let $X$ be a smooth projective real algebraic variety. We give new positive and negative results on the problem of approximating a submanifold of the real locus of $X$ by real loci of subvarieties of $X$, as well as on the problem of…

We present a relation between the classical Chow group of relative $0$-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring, and the Levine-Weibel Chow group of 0-cycles on the…

We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over $\mathbb{C}$ of genus at least 3. We show that, the Chow…

This paper answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only…

We extend the scope of Balmer's tensor triangular Chow groups to compactly generated triangulated categories $\mathcal{K}$ that only admit an action by a compactly-rigidly generated tensor triangulated category $\mathcal{T}$ as opposed to…

We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with…

We introduce a notion of `proChow group' of varieties, agreeing with the notion of Chow group for complete varieties and covariantly functorial with respect to arbitrary morphisms. We construct a natural transformation from the functor of…

Connections between the second Chow group of a smooth projective variety and its third unramified cohomology group, with coefficients the roots of unity twisted twice, feature in several recent works. In this note we revisit a 1996 paper by…

Any smooth, projective variety satisfies the Hodge conjecture in codimension one, known as the Lefschetz (1,1) theorem. Totaro formulated a version for singular varieties. He asked whether the natural Bloch-Gillet-Soul\'{e} cycle class map…

Observations on rational Chow groups and cycle class maps in equivariant contexts.

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the relative…

In this paper we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This…

We show that for a very general principally polarized complex abelian 3-fold, the Chow group of algebraic cycles is infinite modulo every prime number. In particular, this gives the first examples of complex varieties with infinite Chow…

The paper discusses four approaches to the biextension of Chow groups and their equivalences. These are the following: an explicit construction given by S.Bloch, a construction in terms of the Poincare biextension of dual intermediate…

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We…

We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional…

The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset…

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the…