These lectures are devoted to the interplay between cohomology and Chow groups, and also to the consequences, for the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general or very general fiber.

A crucial notion is that of theconiveauof a cohomology. A Betti cohomology class has geometric coniveau ≥cif it is supported on a closed algebraic subset of codimension ≥c. The coniveau of a class of degreekis$\le \frac{k}{2}$. As a smooth projective varietyXhas nonzero cohomology in degrees 0, 2, 4, … obtained by...

We follow [43] and [101, II]. LetXbe a scheme over a fieldK(which in practice will always be a quasi-projective scheme, that is, a Zariski open set in a projective scheme). Let${{\cal Z}_{k}}(X)$be the group ofk-dimensional algebraic cycles ofX, that is, the free abelian group generated by the (reduced and irreducible) closedk-dimensional subvarieties ofXdefined overK. IfY⊂Xis a subscheme of dimension ≤k, we can associate a cycle$c(Y)\in {{\cal Z}_{k}}(X)$to it as follows: Set\[c(Y)=\sum\limits_{W}{{{n}_{W}}}W, \caption {(2.1)}\]where the sum is taken over thek-dimensional irreducible reduced components of...

The following result is essentially due to Bloch and Srinivas [15]. Letf:X→Ybe a smooth projective morphism, whereXandYare smooth (for simplicity), and letZ∈ CH^k(X) be a cycle inX. Consider the following property.

(∗)There exists a subvariety X′⊂X such that for every y∈Y, the cycle${{Z}_{y}}:={{Z}_{\left| {{X}_{y}} \right.}}:=j_{y}^{*}Z\in \text{C}{{\text{H}}^{k}}({{X}_{y}})$vanishes in$\text{C}{{\text{H}}^{k}}({{X}_{y}}-X_{y}^{\prime})$,where j_yis the inclusion of the fiber X_y=f⁻¹(y)into X.

Theorem 3.1.If Z satisfies the property(∗),then there exist an integer m> 0and a cycle Z′ supported in X′,...

Consider a smooth complete intersection$X\subset {{\mathbb{P}}^{n}}$ofrhypersurfaces of degree${{d}_{1}}\le \cdots \le {{d}_{r}}$. By the Lefschetz hyperplane section theorem (see [101, II, 1.2.2]), the only interesting Hodge structure in the cohomology ofXis the Hodge structure on$H_{B}^{n-r}(X,\mathbb{Q})$, and in fact on the primitive part of it (that is, the orthogonal of the restriction of$H_{B}^{*}({{\mathbb{P}}^{n}},\mathbb{Q})$with respect to the intersection pairing). We will say thatXhas Hodge coniveaucif the Hodge structure on$H_{B}^{n-r}{{(X,\mathbb{Q})}_{\text{prim}}}$has coniveauc.

The Hodge coniveau of a complete intersection in projective space is computed as follows (see [48] for the...

The following theorem is proved in [11].

Theorem 5.1 (Beauville and Voisin 2004).Let S be a K3surface, D_i∈ CH¹(S)be divisors on S and n_ijbe integers. Then if the0-cycle${{\sum }_{i,j}}{{n}_{ij}}{{D}_{i}}{{D}_{j}}\in \text{C}{{\text{H}}_{0}}(S)$is cohomologous to0on S, it is equal to0inCH₀(S).

Proof. It suffices to prove that there is a 0-cycleoof degree 1, with the property that for any two line bundlesL,L′ onS,\[L\cdot {L}'=\deg (L\cdot {L}')\ o\ \text{in}\quad \text{C}{{\text{H}}_{0}}(S). \caption {(5.1)}\]

The cycleois defined to be the class of any point ofScontained in a (singular) rational curveC⊂S. We claim...

Atiyah and Hirzebruch [5] found counterexamples to the Hodge conjecture (Conjecture 2.25) stated for degree 2k integralHodge classes (as opposed torationalHodge classes) whenk≥ 2. In degree 2, the most optimistic statements are true, due to the Lefschetz theorem on (1, 1)-classes.

In [95], Totaro revisited the examples of Atiyah and Hirzebruch and reformulated more directly the obstructions they had found, using the complex cobordism graded ring MU* (X) ofX. Let us first describe this ring that is defined for all differentiable compact manifolds: Given such anX, we consider first of all the objects...