COniveaus

COniveaus is a term used to describe the coniveau filtrations in algebraic geometry and related cohomological theories. The concept provides a way to organize the cohomology of algebraic varieties by the codimension of subvarieties on which cohomology classes can be supported. In this sense, COniveaus encode information about how “algebraic” a given cohomology class is, in terms of where it can be nonzero.

Definition and construction: Let X be a smooth projective variety over a field, and fix a coefficient field such as Q. For each i and p with 0 ≤ p ≤ i, the coniveau filtration N^p H^i(X, Q) is defined as the subspace of H^i(X, Q) consisting of cohomology classes that are supported on closed subsets Z ⊆ X of codimension at least p. Equivalently, N^p H^i(X, Q) is the sum of the images of the natural maps H^i_Z(X, Q) → H^i(X, Q) over all closed Z of codimension ≥ p. The filtration is increasing in p and satisfies N^0 H^i(X, Q) = H^i(X, Q) and N^{dim X + 1} H^i(X, Q) = 0.

Key features: The coniveau filtration gives rise to a coniveau spectral sequence with E1^{p,q} = ⊕ H^{p+q}_Z(X, Q) (sums over closed Z of codimension at least p), which converges to H^{p+q}(X, Q). The filtration is functorial with respect to proper maps and interacts with Gysin and cycle-theoretic operations. It provides a bridge between algebraic cycles and topological/cohomological information.

Context and significance: COniveaus are central to discussions of the generalized Hodge conjecture and the study of how algebraic cycles influence cohomology. They are contrasted with other filtrations, such as the weight and Hodge filtrations, and with the related notion of the niveau filtration.

See also: coniveau filtration, generalized Hodge conjecture, algebraic cycles, cohomology with supports.