Atiyah-Hirzebruch theorem (1962): the Hodge conjecture cannot hold integrally, as reproduced in Deligne's official CMI Hodge conjecture description

en · official · sha256:8037ed020bdf50ba9bd54da7ba562a12d671fa8f2996b1bd145ec78f194e2860

(iv) The relation between algebraic cycles and algebraic vector bundles is also the basis of the Atiyah and Hirzebruch theorem [2] that the Hodge conjecture cannot hold integrally. In the Atiyah-Hirzebruch spectral sequence from cohomology to topological K-theory, E_2^{pq} = H^p(X, K^q(pt)) => K^{p+q}(X); the resulting filtration of K^n(X) is by the F^p K^n(X) = Ker(K^n(X) -> K^n((p-1)-skeleton, in any triangulation)). Equivalently, a class c is in F^p if for some topological subspace Y of codimension p, c is the image of a class c-tilde with support in Y. If Z is an algebraic cycle of codimension p, a resolution of O_Z defines a K-theory class with support in Z: c_Z in K^0(X, X-Z). Its image in F^p K^0(X) agrees with the class of Z in H^{2p}(X,Z). The latter hence is in the kernel of the successive differentials d_r of the spectral sequence.

Referencing norms: AtiyahHirzebruchProof, AtiyahHirzebruchSource, atiyah-hirzebruch-proves-integral-fails

en · official · sha256:4dc590b520c952cd262b07891811c747caa522173aa2012baf3897c50ec5c5c5

No counterexample is known to the statement that integral (p,p) classes killed by all d_r are integral linear combinations of classes cl(Z). One has no idea of which classes should be effective, that is, of the form cl(Z), rather than a difference of such.

Referencing norms: atiyah-hirzebruch-asserts-restricted-open