Hodge Conjecture - Motivation

Motivation

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n. Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients:

where Hp, q(X) is the subgroup of cohomology classes which are represented by harmonic forms of type (p, q). That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z₁, ..., z_n, can be written as a harmonic function times . (See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:

Since X is a compact oriented manifold, X has a fundamental class.

Let Z be a complex submanifold of X of dimension k, and let i : Z → X be the inclusion map. Choose a differential form α of type (p, q). We can integrate α over Z:

To evaluate this integral, choose a point of Z and call it 0. Around 0, we can choose local coordinates z₁, ..., z_n on X such that Z is just z_{k + 1} = ... = z_n = 0. If p > k, then α must contain some dz_i where z_i pulls back to zero on Z. The same is true if q > k. Consequently, this integral is zero if (p, q) ≠ (k, k).

More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α. By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call, and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of X. Because is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type (p, q) ≠ (k, k), then we get zero. Because H2n(X, C) = Hn, n(X), we conclude that must lie in Hn-k, n-k(X, C). Loosely speaking, the Hodge conjecture asks:

Which cohomology classes in Hk, k(X) come from complex subvarieties Z?