By Claire Voisin
During this ebook, Claire Voisin presents an advent to algebraic cycles on complicated algebraic forms, to the key conjectures touching on them to cohomology, or even extra accurately to Hodge buildings on cohomology. the amount is meant for either scholars and researchers, and never in basic terms provides a survey of the geometric equipment built within the final thirty years to appreciate the well-known Bloch-Beilinson conjectures, but additionally examines contemporary paintings by means of Voisin. The ebook makes a speciality of important gadgets: the diagonal of a variety—and the partial Bloch-Srinivas sort decompositions it can have looking on the dimensions of Chow groups—as good as its small diagonal, that's definitely the right item to contemplate with a view 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 normally by means of Bloch and Beilinson, among the coniveau of basic entire intersections and their Chow teams and a really specific estate chuffed by means of the Chow ring of K3 surfaces and conjecturally through hyper-Kähler manifolds. specifically, the ebook delves into arguments originating in Nori’s paintings which have been extra built by means of others.
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Extra info for Chow Rings, Decomposition of the Diagonal, and the Topology of Families
Let i < n. Consider the 2n−i i (X, Q) ⊗ HB (X, Q) is the K¨ unneth component δ2n−i of ∆X , so δ2n−i ∈ HB ji class of an algebraic cycle Z on X × X. Let Yi → X be a smooth complete intersection of (n − i) ample hypersurfaces in X. 2]) says that 2n−i i ji∗ : HB (Yi , Q) → HB (X, Q) is surjective. It follows that the class of the cycle Z vanishes on X ×(X \Yi ). 29, there is an n-cycle Z supported on X × Yi such that the class (id, j)∗ [Z ] is equal to [Z]. Consider the morphism of Hodge structures induced by [Z ]: 2n−i i [Z ]∗ : HB (X, Q) → HB (Yi , Q).
The classes δi are algebraic, that is, are classes of algebraic cycles on X × X with rational coefficients. (b) Lefschetz operators and their inverses. Let L be an ample line bundle 2 on X, and l := c1 (L) ∈ HB (X, Q). 3] says that the cup-product map 2n−k k ln−k ∪ : HB (X, Q) → HB (X, Q) is an isomorphism. This is clearly an isomorphism of Hodge structure. 26 provides a Hodge class λn−k of degree 2k on X × X. The second standard conjecture we will consider (the Lefschetz conjecture or Conjecture B in the terminology of ) is the following.
19). 19, if Γ is cohomologous to 0, then Γ◦N belongs to F N CH(X × X)Q . As this filtration is conjectured to satisfy F k+1 CHk (X × X) = 0, we must have Γ◦N = 0 for N > 2n, n := dim X. For cycles algebraically equivalent to 0, the following result is proved independently in  and . 25 (Voevodsky 1995, Voisin 1994). The nilpotence conjecture holds for cycles in X × X that are algebraically equivalent to 0. Proof. Let Γ ∈ CHd (X × X), d = dim X be algebraically equivalent to 0. This means that there is a curve C that we may assume to be smooth, a 0-cycle z ∈ CH0 (C) homologous to 0, and a correspondence Z ∈ CHd (C × X × X) such that Γ = Z∗ (z) in CHd (X × X).
Chow Rings, Decomposition of the Diagonal, and the Topology of Families by Claire Voisin