Paul Sacawa

Аннотация: A rational cohomology class in a smooth complex variety X is said to have geometric coniveau $\geq k$ if its support is contained in a subset of codimension $k$. A (generalized) conjecture of Bloch proposes that if $H^*(X; Q)^{\perp alg}$ consisting of classes orthogonal under the Poincare pairing to the classes of algebraic cycles of X has coniveau $\geq k$, then the class map $cl: CH_i(X;Q) \to H^{2n-2i} (X;Q)$ is injective for $i \leq k-1$. Here, $CH_i(X;Q)$ is the rational Chow group. We sketch the proof in the case of general complete intersections, and describe how the main tool, the decomposition of the diagonal class $[\Delta] \in CH(X \times X)$, affects the relation of Chow and cohomology groups in general.

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