The Hodge Conjecture

en · official · sha256:28acf9735d664d5da39431958284a22593d60b703c4242b5d3a067df9ac563f7

If Z is a closed analytic subspace of X, of complex codimension p, Z is an integral cycle and, by Poincare duality, defines a class cl(Z) in H^{2p}(X,Z). The integration current on Z is a closed (p,p)-form with generalized function coefficients, representing the image of cl(Z) in H^{2p}(X,C).

Referencing norms: defines-algebraic-cycle

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: IntegralHodgeFailsClaim, integral-fails-identifies-hodge, integral-fails-obtained-by-weakening, names-atiyah-hirzebruch, source-asserts-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: IntegralHodgeRestrictedOpenClaim, integral-restricted-open-identifies-hodge, source-asserts-integral-restricted-open

en · official · sha256:c7568838485cf2bf7e642bb82f91296db8da12d85781e2b315a29e57b983b46a

(i) By Chow's theorem, on a complex projective variety, algebraic cycles are the same as closed analytic subspaces.

Referencing norms: defines-chow-equivalence

en · official · sha256:7fa4f2120a32fe06a830e2253b0e4e3d159f3ddea6bff217f9db245e1168818e

The class cl(Z) in H^{2p}(X,Z) is hence of type (p,p), in the sense that its image in H^{2p}(X,C) is. Rational (p,p)-classes are called Hodge classes. They form the group H^{2p}(X,Q) intersect H^{p,p}(X) = H^{2p}(X,Q) intersect F^p, a subgroup of H^{2p}(X,C).

Referencing norms: defines-hodge-classes

en · official · sha256:f697e809ddf0721dc7c344c7d725ba8db84241b6dd4058d94b6709975dcd8524

(v) The assumption in the Hodge conjecture that X be algebraic cannot be weakened to X being merely Kahler. See Zucker's appendix to [11] for counterexamples where X is a complex torus.

Referencing norms: defines-kahler-boundary

en · official · sha256:97e5081e7f24ba889b78ff4d737db1b771947d1e8b99cbc304a0d905e2f290f7

(iii) A particular case of (ii) is that the integral linear combinations of classes of divisors (= codimension 1 cycles) are simply the first Chern classes of line bundles. If Z+ - Z- is the divisor of a meromorphic section of L, c1(L) = cl(Z+) - cl(Z-). This is the starting point of the proof given by Kodaira and Spencer [7] of the Hodge conjecture for H^2: a class c in H^2(X,Z) of type (1,1) has image 0 in the quotient H^{0,2} = H^2(X,O) of H^2(X,C), and the long exact sequence of cohomology defined by the exponential exact sequence 0 -> Z -> O -[exp(2 pi i .)]-> O* -> 0 shows that c is the first Chern class of a line bundle.

Referencing norms: HodgeConjectureDegree2Claim, degree2-identifies-hodge, names-kodaira-spencer, source-asserts-degree2

en · official · sha256:3d08180b98f5b6c584b490c76c7d6fff71b4af2eed1755c4ab067725166d97b0

(vi) When Hodge formulated his conjecture, he had not realized it could hold only rationally (i.e. after tensoring with Q). He also proposed a further conjecture, characterizing the subspace of H^n(X,Z) spanned by the images of cohomology classes with support in a suitable closed analytic subspace of complex codimension k. Grothendieck observed that this further conjecture is trivially false, and gave a corrected version of it in [5].

Referencing norms: names-grothendieck-correction

en · official · sha256:656166dc3cb62c39f701b64cd3a2eec53e7b37440290275a55969abca673eaf4

Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.

Referencing norms: HodgeConjectureRational, rational-identifies-hodge, source-asserts-rational

en · official · sha256:9ea8fe5bc25c2b43ec571a815f7d809c489569c650158ebd445ab9a230d1ae2b

THE HODGE CONJECTURE

Referencing norms: CmiHodgeSource, classified_as_millennium_problem, defines_notion, names_historical_result, rational-classified