hodge_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Hodge Conjecture: For a non-singular complex algebraic variety $X$, every rational cohomology class of type $(p, p)$ is a linear combination (over $\mathbb{Q}$) of classes of algebraic subvarieties.

One of the 7 Millennium Prize Problems ($1M). The conjecture relates the topology of complex algebraic varieties (Hodge decomposition of cohomology) to their geometry (algebraic cycles). Proved for $p = 0, \dim X$ (trivially), and for special cases. The general case is open.

Source: Hodge, W.V.D. (1950). The topological invariants of algebraic varieties. Proc. ICM 1950, vol. 1, 182–192. Clay Mathematics Institute.

Why this node was retired

The posted statement is

theorem hodge_conjecture (n : ℕ) (p : ℕ) (hp : p ≤ n) :
    hodgeType n p = hodgeAlgebraic n p := by
  sorry

Both hodgeType and hodgeAlgebraic are constantly PUnit; there is no variety or cohomology.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For every smooth projective complex algebraic variety X and p≥0, every class in H^{2p}(X,ℚ) whose complexification has Hodge type (p,p) lies in the ℚ-span of classes of codimension-p algebraic subvarieties. Define both spaces using actual cohomology and cycle classes.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.