By Claire Voisin
During this ebook, Claire Voisin offers an advent to algebraic cycles on complicated algebraic forms, to the main conjectures bearing on them to cohomology, or even extra accurately to Hodge buildings on cohomology. the quantity is meant for either scholars and researchers, and never purely offers 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 via Voisin. The booklet specializes in primary items: the diagonal of a variety—and the partial Bloch-Srinivas sort decompositions it can have reckoning on the scale of Chow groups—as good as its small diagonal, that is the ideal item to contemplate for you to comprehend the hoop constitution on Chow teams and cohomology. An exploration of a sampling of contemporary works by means of Voisin seems on the relation, conjectured normally by way of Bloch and Beilinson, among the coniveau of common entire intersections and their Chow teams and a really specific estate chuffed via the Chow ring of K3 surfaces and conjecturally by way of hyper-Kähler manifolds. specifically, the ebook delves into arguments originating in Nori’s paintings which were extra constructed by way of others.
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Extra resources for Chow Rings, Decomposition of the Diagonal, and the Topology of Families
Moreover, Z is supported in T × X with codim T ≥ c. Let l : T → X be a desingularization of T , and let Z in CHn (T ×X) be such that (l, Id)∗ (Z ) = Z . The above decomposition gives a decomposition of the corresponding morphisms of Hodge structures for every k: k k m[∆X ]∗ = m Id = [Z0 ]∗ + · · · + [Zk ]∗ + [Z ]∗ : HB (X, Q) → HB (X, Q). 17) j where , ∗ (X, Q). In particular, we have is the intersection form on HB [Zi ]∗ (α) = 0 ∀ α ∈ H p,q (X), p = q. For α satisfying this hypothesis, we thus have mα = [Z ]∗ (α) = l∗ ([Z ]∗ (α)) in H p,q (X).
This is the integral Betti cycle class of Z. In many places we will use the 2k rational cycle class [Z] ∈ HB (X, Q). We extend the above cycle class by linearity to any cycle Z = i ni Zi ∈ Z k (X). 16. If Z is rationally equivalent to 0, then [Z] = 0 in HB (X, Z). The map Z → [Z] thus gives the “cycle class” map 2n−2l cl : CHl (X) → HB (X, Z). The cycle class map is compatible with the intersection product · : CHl (X) × CHk (X) → CHk+l (X), weyllecturesformat September 3, 2013 6x9 REVIEW OF HODGE THEORY AND ALGEBRAIC CYCLES 23 and the cup-product 2k+2l 2l 2k ∪ : HB (X, Z) × HB (X, Z) → HB (X, Z).
This correspondence can be spread out over a finite cover of a Zariski open set of Y , and this can be done as well for the curve Cη and the cycle z, giving rise to families over a finite cover of a Zariski dense open set Y 0 of the base. The problem is that the spread-out curve C is not in general isotrivial on Y 0 , that is, isomorphic to C0 × Y 0 , maybe up to passing to a finite ´etale cover of Y 0 . Even if it was isotrivial, the codimension 1 cycle z would spread to a codimension 1 cycle Z on C0 × Y 0 , which might not be cohomologous to 0 over any Zariski open set of Y 0 , if g(C0 ) > 0.
Chow Rings, Decomposition of the Diagonal, and the Topology of Families by Claire Voisin
Categories: Algebraic Geometry