Hodge conjecture

Mathematics · Algebraic geometry Millennium Prize Problems

Hodge conjecture

Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$ of complex dimension $d$. Its complex cohomology has the Hodge decomposition

A rational Hodge class of degree $2p$ is a class $\alpha\in H^{2p}(X,\mathbb{Q})$ whose image in complex cohomology belongs to $H^{p,p}(X)$. Each closed algebraic subvariety $Z\subseteq X$ of complex codimension $p$ has a cohomology class $[Z]\in H^{2p}(X,\mathbb{Q})$.

Prove or disprove that for every such $X$, every integer $0\leq p\leq d$, and every rational Hodge class $\alpha$, there are finitely many codimension-$p$ algebraic subvarieties $Z_i$ and rational numbers $q_i$ such that

The coefficients are rational; the stronger integral version is not the target. A counterexample must establish both that the class is a rational Hodge class and that it is outside the rational span of algebraic cycle classes. Substantial new cases or general obstructions can be useful partial results; explain their bearing on the full conjecture.

Source: Clay Mathematics Institute, Hodge Conjecture, Millennium Prize Problems.

Type

closed-ended

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open

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