Chow Rings, Decomposition of the Diagonal, and the Topology by Claire Voisin

By Claire Voisin

During this ebook, Claire Voisin presents an creation to algebraic cycles on complicated algebraic types, to the most important conjectures pertaining to them to cohomology, or even extra accurately to Hodge buildings on cohomology. the amount is meant for either scholars and researchers, and never basically provides a survey of the geometric equipment built within the final thirty years to appreciate the well-known Bloch-Beilinson conjectures, but additionally examines fresh paintings through Voisin. The booklet specializes in critical items: the diagonal of a variety—and the partial Bloch-Srinivas variety decompositions it could have counting on the scale of Chow groups—as good as its small diagonal, that is the precise item to think about so that it will comprehend the hoop constitution on Chow teams and cohomology. An exploration of a sampling of modern works via Voisin seems to be on the relation, conjectured in most cases by means 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 via hyper-Kähler manifolds. specifically, the publication delves into arguments originating in Nori’s paintings which were extra built by way of others.

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Indeed, if we have a codimension k cycle Z ⊂ X, whose cohomology 2k class [Z] ∈ HB (X, Q) vanishes on the open set X \ Y , where codim Y ≥ k − 1, then we claim that there are Hodge classes αi ∈ Hdg2k−2ci (Yi , Q) such that ˜ji∗ αi , [Z] = i where ˜ji : Yi → X are projective desingularizations of the irreducible components Yi of Y , and ci := codim Yi . 36. 36 from a class in the pure part of H2n−2k,B (Y, Q), thus from a class in ⊕H2n−2k,B (Yi , Q). 24 to conclude that it comes from a Hodge class in ⊕H2n−2k,B (Yi , Q).

Let H be the Hermitian intersection pairing on LC defined by H(a, b) = ik (a, b), then we have the following: (i) (First Hodge–Riemann bilinear relations). The Hodge decomposition of L is orthogonal with respect to H. (ii) (Second Hodge–Riemann bilinear relations). The restriction H|Lp,q is definite, of sign (−1)p . The interest of polarized Hodge structures lies in the following semisimplicity result. 22. Let (L, Lp,q ) be a rational polarized Hodge structure and L ⊂ L be a sub-Hodge structure.

The cycle Z ⊂ T × X is of codimension (n − c + 1) and induces a morphism Z∗ : CH0 (T ) → CHc−1 (X). Now, we know by assumption that the 2n−2c+2 kernel of cl : CHc−1 (X) → HB (X, Z) is torsion. Thus, the map Z∗ maps CH0 (T )hom to the torsion of CHc−1 (X). By a Baire countability argument, it follows that there exists an integer M such that M Z∗ = 0 on CH0 (T )hom . For each component Ti of T , choose a point ti ∈ Ti as above, and let Wi = Z∗ (ti ). The cycle Z − Z =M Ti × Wi ⊂T ×X i then satisfies the property that for any t ∈ T , Z∗ (t) = 0 in CHc−1 (X).

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