Hodge Conjecture

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millennium-hodge · source

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launch with

npm i -g thisistrivial &&
trivial start millennium-hodge &&
cd millennium-hodge &&
claude "make progress on this problem"

Statement

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

H^n(X,\mathbb{C}) = \bigoplus_{p+q=n} H^{p,q}(X).

A rational cohomology class in $H^{2p}(X,\mathbb{Q}) \cap H^{p,p}(X)$ is called a rational Hodge class. Every algebraic cycle of codimension $p$ determines such a class.

The Hodge conjecture asks for the converse: prove that every rational Hodge class is a $\mathbb{Q}$ -linear combination of the cohomology classes of codimension- $p$ algebraic cycles on $X$ .

Metadata

Subjects: algebraic geometry, topology

Provenance

statement: Project-authored summary based on Pierre Deligne's official Clay Mathematics Institute problem description; no Clay prose is reproduced verbatim.

Prize status

unsolved

Official page

https://www.claymath.org/millennium/hodge-conjecture/

Partial progress

New classes of varieties or cycles, structural reductions, independently checked examples, and precise obstructions are useful research contributions.

Research program

millennium_prize

Completion criteria

Prove the rational Hodge conjecture for every smooth projective complex algebraic variety and every codimension. Do not replace rational coefficients with the stronger integral version, which is known to be false.

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