This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid $C_r(Y)(U)$ of effective cocycles on a normal, quasi-projective variety $U$ with values in a projective variety $Y$ consists of algebraic cycles on $U×Y$ equi-dimensional of relative dimension r over $U$. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion $Z_r(Y)(U)$ has “sensible” homotopy groups, the construction is contravariant with respect to $U$, convariant with respect to $Y$, and there is a natural “quality map” to the topological group of cycles on $U×Y$. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map $Z_r(Y)(U)$ to $Z_r+m(U × Y)$ is a homotopy equivalence provided that both $U$ and $Y$ are smooth (where $m=dim U$). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.

Each finite dimensional irreducible rational representation V of the symplectic group Sp$_2g$(Q) determines a generically defined local system V over the moduli space M$_g$ of genus g smooth projective curves. We study H$^2$ (M$_g$; V) and the mixed Hodge structure on it. Specifically, we prove that if g [ges ] 6, then the natural map IH$^2$(M˜$_g$; V) → H$^2$(M$_g$; V) is an isomorphism where M˜$_g$ is the Satake compactification of M$_g$. Using the work of Saito we conclude that the mixed Hodge structure on H$^2$(M$_g$; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H$^2$(M$_g$; V) for 3 [les ] g < 6. Results of this article can be applied in the study of relations in the Torelli group T$_g$.

We study here the O-dimension Chow group of some particular smooth varieties which are fibrations over curves (i.e. are equipped with a proper, surjective morphism over a smooth curve). We obtain finiteness results for fibrations, whose generic fibres are Severi-Brauer varieties of squarefree index, when the group field is a number field (or in some particular cases a finitely generated field over Q).

For a real Enriques surface $Y$ we prove that every homology class in $H_1(Y(R), Z/2)$ can be represented by a real algebraic curve if and only if all connected components of $Y(R)$ are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface $Y$.

Let Φ(x) denote the number of those integers n with φ(n)[les ] x, where φ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax [Lt] R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.