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on The Hodge conjecture

The 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 $X$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.