By Claire Voisin

During this publication, Claire Voisin offers an creation to algebraic cycles on advanced algebraic types, to the key conjectures pertaining to them to cohomology, or even extra accurately to Hodge constructions on cohomology. the quantity is meant for either scholars and researchers, and never in basic terms offers a survey of the geometric equipment built within the final thirty years to appreciate the well-known Bloch-Beilinson conjectures, but in addition examines fresh paintings through Voisin. The booklet specializes in crucial items: the diagonal of a variety—and the partial Bloch-Srinivas kind decompositions it might probably have reckoning on the dimensions of Chow groups—as good as its small diagonal, that's the precise item to think about so as to comprehend the hoop constitution on Chow teams and cohomology. An exploration of a sampling of modern works via Voisin appears to be like on the relation, conjectured quite often by way of Bloch and Beilinson, among the coniveau of basic whole intersections and their Chow teams and a really specific estate chuffed by means of the Chow ring of K3 surfaces and conjecturally by way of hyper-Kähler manifolds. specifically, the booklet delves into arguments originating in Nori’s paintings which were additional constructed through others.

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**Example text**

ZN of X such that L = [Z1 ], . . , [ZN ] ⊗ Q. But then L vanishes on X \ (Sup Z1 ∪ · · · ∪ Sup ZN ), as required. The next, much more sophisticated, case is that in which k = 2c + 1, so that LC = Lc+1,c ⊕ Lc,c+1 . 2]. 11), L = L(c) is a polarized Hodge structure of weight 1. We get such Hodge structures on the first cohomology groups of curves, though not every Hodge structure of weight 1 is actually a Hodge structure of a curve. However, we have the following result. 41. Any polarized Hodge structure of weight 1 arises as HB (A, Q) for some abelian variety A.

7. A divisor D ∈ CH1 (X) is algebraically equivalent to 0 if and only if it is homologous to 0. This follows from the identification between CH1 (X) and Pic(X). The exponential exact sequence and the identification ∗ Pic(X) = Pic(Xan ) = H 1 (Xan , OX ) imply that an ∗ 2 Pic0 (X) := Ker(c1 : H 1 (Xan , OX ) → HB (X, Z)) = CH1 (X)hom an is parametrized by the abelian variety which as a complex torus is the quo1 tient H 1 (Xan , OXan )/HB (X, Z). 2] for more details). 2]. 4, if we replace rational equivalence by algebraic equivalence, our assumption is (by a countability argument) equivalent to the fact that the restriction of the cycle Z to the geometric generic fiber (which is a variety Xη defined over C(Y )) is algebraically equivalent to 0.

For the motive X − , observe that the vanishing N H odd (X, Q), N > dim H odd (X, Q) says that the SN -invariant cohomology of (X − )N vanishes, or that the motive S N X − := (X N , π inv ◦ (π − )N ), where now π inv := N1 ! σ∈SN Γσ ∈ CHnN (X N × X N ) has zero cohomology. Kimura’s conjecture (presented in a simplified form) is the following. 27. For any smooth projective variety X, the motive of X is finite-dimensional, which means that it decomposes as X + + X − , where X − has only odd degree cohomology and X + has only even degree cohomology, and for large N , we have N X + = 0, S N X − = 0.