OP-0003OpenGeometry and topology$1,000,000 prize

The Hodge conjecture

Is every Hodge class on a projective complex manifold a rational combination of algebraic cycle classes?

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Problem statement

Let $X$ be a non-singular complex projective variety. For each $k$ , the Hodge decomposition gives

H^{2k}(X,\mathbb{Q}) \otimes \mathbb{C} = \bigoplus_{p+q=2k} H^{p,q}(X).

A Hodge class is an element of $H^{2k}(X,\mathbb{Q}) \cap H^{k,k}(X)$ . The conjecture states that every Hodge class is a $\mathbb{Q}$ -linear combination of the cohomology classes of algebraic subvarieties of $X$ .

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Background

Formulated by W. V. D. Hodge in 1950. The integral version is false — Atiyah and Hirzebruch produced counterexamples in 1961 — which is why the statement is over $\mathbb{Q}$ .

Why it matters

It is the central conjecture connecting the topology of a variety to its algebraic geometry, and would supply algebraic cycles that are currently only known to exist cohomologically.

Known partial results

True for $k = 1$ by the Lefschetz $(1,1)$ -theorem, which predates the conjecture.
Known for abelian varieties in many cases, and for varieties of small dimension.
Cattani–Deligne–Kaplan (1995) proved that loci of Hodge classes are algebraic, a strong structural consequence that the conjecture would explain.

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