Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degr
ee $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism between the Chow group $CH_r(X)_0$ of degree $0$ dimension $r$ algebraic cycles modulo rational equivalence on $X$, and the group of sections of the sheaf of $mathbb A^1$-path connected components of the loop space of $BC_r^{infty }(X)$ at $Spec(k)$. Equivalently, $CH_r(X)_0$ is isomorphic to the group of sections of the $S^1wedge mathbb A^1$-fundamental group $Pi _1^{S^1wedge mathbb A^1}(BC_r^{infty }(X))$ at $Spec(k)$.
Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^
p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^p(Y)$. In the paper we prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_{rm van}$. Then we apply these general results to sections of a smooth cubic fourfold in $mathbb P^5$.
We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorns sense, of a ceratin positive
projective model structure on spectra, where positivity basically means the truncation of the zero slice. The localization above is by the set of stabilizing morphisms, or their truncated version.
We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Blochs conjecture, especially for Godeaux surface
s, when the surface is given as a finite quotient of a suitable quintic in P^3.
