Related papers: Surjectivity of cycle maps for singular varieties

For specific classes of smooth, projective varieties $X$ over a field $k$, we compare two cycle maps on the torsion subgroup $CH^2(X)_{\text{tors} }$ of the second Chow group. The first one goes back to work of S. Bloch (1981), the second…

Jannsen asked whether the rational cycle class map in continuous $\ell$-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by…

Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} :…

We study the injectivity of the cycle class map with values in Jannsen's continuous \'etale cohomology, by using refinements that go through \'etale motivic cohomology and the ``tame'' version of Jannsen's cohomology. In particular, we use…

Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is…

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the…

Let U be a smooth quasiprojective complex variety and CH^r(U,1) a special instance of Bloch's higher Chow groups. Jannsen was the first to show that the cycle class map cl_{r,1} from CH^r(U,1) (tensored with Q) to hom_{MHS}(Q(0),…

The theorem of Mather on generic projections of smooth algebraic varieties is also proved for the singular ones.

In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a…

A conjecture of Colliot-Th\'{e}l\`{e}ne predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a…

We investigate the surjectivity of the real cycle class map from $I$-cohomology to classical intergral cohomology for some real smooth varieties, in particular surfaces. This might be considered as one of several possible incarnations of…

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of…

The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of…

We show the surjectivity of a specialisation map on higher $(0,1)$-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture stated in an article of Kerz, Esnault and…

In this short note we prove a version of Bertini's theorem for unipotent rigid fundamental groups, stating that for every smooth, projective, geometrically connected variety $X$ over an infinite perfect field $k$ of characteristic $p>0$,…

Consider a finite morphism f:X -> Y of smooth projective varieties over a finite field k. Suppose X is the vanishing locus in projective N-space of at most r forms of degree at most d. We show there is a constant C, depending only on N, r,…

One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on \'etale…

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

We prove that a jointly conservative family of geometric functors between rigidly-compactly generated tensor triangulated categories induces a surjective map on Balmer spectra. From this we deduce a fiberwise criterion for Balmer's…

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its…