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  "body": "\n## Statement\n\nLet $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] over $\\C$, and let $p \\ge 0$.\nLet $H^{p,p}(X)$ be the $(p, p)$ part of the [[Hodge decomposition|HABCJ4Y0SR]] of $H^{2p}(X, \\C)$.\nHere $H^{2p}(X, \\Q)$ is [[singular cohomology|6N7BC70CX0]], viewed inside $H^{2p}(X, \\C)$.\nLet $\\mathrm{Hdg}^p(X) = H^{2p}(X, \\Q) \\cap H^{p,p}(X)$ be the group of rational Hodge classes.\nThen every class in $\\mathrm{Hdg}^p(X)$ is a $\\Q$-linear combination of classes of closed subvarieties of $X$ of codimension $p$.\nIn other words, it is the class of an [[algebraic cycle|9VFWPZEVGN]] with rational coefficients.\n\n## Remarks\n\nThe conjecture is open.\nIt is one of the Millennium Prize Problems of the Clay Mathematics Institute.\nThe class of a subvariety of codimension $p$ always lies in $\\mathrm{Hdg}^p(X)$, so the conjecture asks for the converse.\n\nThe case $p = 1$ is true: it is the Lefschetz theorem on $(1,1)$-classes, which follows from the exponential sequence.\nIt shows that every class in $H^{2}(X, \\Z)$ whose image lies in $H^{1,1}$ is the first Chern class of a line bundle.\nBy the hard Lefschetz theorem, the case $p = 1$ implies the case $p = n - 1$, where $n = \\dim X$.\nSo the conjecture holds for varieties of dimension at most $3$.\n\nThe version with integer coefficients is false: Atiyah and Hirzebruch found torsion classes that are not algebraic, and Kollár found non-torsion counterexamples.\nThe analogous statement for compact Kähler manifolds, with analytic cycles or Chern classes of coherent sheaves, is false by an example of Voisin.\n\n## Explanations\n\n### L1 human\n\nThe Hodge conjecture asks which cohomology classes of a complex projective variety come from geometry, namely from subvarieties.\nHodge theory singles out a natural candidate set, the rational classes of type $(p, p)$, and every subvariety gives such a class.\nThe conjecture says that there are no others: every class that looks algebraic from the point of view of Hodge theory is algebraic.\n",
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      "first_stated": "1950",
      "posed_by": "W. V. D. Hodge"
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