Open problemGeometry · Graduate → research

The Hodge conjecture

Can equations recover a space’s hidden geometry?

Reviewed: 2026-10-20Editorial introduction · not original research

H^{2p}(X,\mathbb Q)\cap H^{p,p}(X)\stackrel{?}{=}\operatorname{span}_{\mathbb Q}\{[Z]:\operatorname{codim}Z=p\}

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01 / The question

For smooth complex projective varieties, the conjecture asks whether rational Hodge classes come from rational combinations of algebraic cycles. It links an abstract description of a space to geometric objects defined by equations.

02 / Why it is hard

The exact hypotheses matter: changing coefficients or the class of spaces changes the claim. A diagram of a familiar surface offers intuition, not a proof in arbitrary dimension.

03 / Progress & scope

Clay lists the general conjecture as unsolved. Results for special families of varieties must retain their scope when reported.

OpenAI Math · GitHub

Family 032 · Hodge conjecture for CM abelian varieties

This manuscript concerns CM abelian varieties; it does not by itself settle the conjecture for all smooth complex projective varieties.

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Sources & further reading

Clay Mathematics Institute

Sources last checked 2026-10-20. Status reflects an editorial review on that date, not continuous live monitoring.

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