すべての Hodge 類は、代数的サイクルの類の組み合わせで書けるか？（Hodge 予想）

射影的複素多様体上で、型 (p,p) の有理コホモロジー類が、常に代数的部分多様体の類の有理係数の一次結合で書けるかを決定せよ。

Decide whether, on a projective complex manifold, every rational cohomology class of type (p,p) is a rational linear combination of classes of algebraic subvarieties.

A Clay Millennium Prize Problem. The (1,1) case is a classical theorem; the integral version is false, and the extension to Kaehler manifolds is false, so the surviving statement is narrow and its two natural strengthenings are both closed roads.

解決したかどうかは、問いではなくアプローチごとに決まります。Atlas は判定しません。外部の判定者が何をしたかを記録します。

• Lefschetz (1,1) and its extensions

  査読付きの証明

  到達状況

  進行中

  判定者

  the algebraic geometry community, through journal peer review · 査読

  定式化

  Extend the classical (1,1) theorem to classes of higher type.問いより弱い

  保たれているもの

  • Proves the conjecture for divisor classes, in full generality

  弱まっているもの

  • Only the (1,1) case; no case with p at least 2 is known in general

  加えられた仮定・条件

  • Nothing

  閉じた道

  • The integral form of the conjecture is false: Atiyah and Hirzebruch exhibit integral classes of type (p,p) that are not classes of algebraic cycles, so the rational coefficients in the statement cannot be removed.無条件
    • paperdoi:10.1016/0040-9383(62)90016-6M. Atiyah, F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962) 25-45: the integral Hodge conjecture is false
  • The extension to compact Kaehler manifolds is false, so any proof must use projectivity rather than Hodge theory alone.無条件
    • paperdoi:10.4310/jdg/1090351323C. Voisin, A counterexample to the Hodge conjecture extended to Kaehler varieties, Int. Math. Res. Not. / J. Differential Geom.: the Kaehler extension fails

  証拠

  • paperS. Lefschetz, L'analysis situs et la geometrie algebrique, Gauthier-Villars (1924)The (1,1) case, the only case proved in general

• Absolute Hodge classes and motives

  査読付きの証明

  到達状況

  進行中

  判定者

  the arithmetic geometry community, through journal peer review · 査読

  定式化

  Prove Hodge classes are absolute Hodge, then algebraic, through the theory of motives.問いより強い

  保たれているもの

  • Would give the conjecture together with a structural explanation

  弱まっているもの

  • Deligne's theorem gives absolute Hodge for abelian varieties only; the last step is open in every case

  加えられた仮定・条件

  • The standard conjectures on algebraic cycles, themselves open

  閉じた道

  • The route passes through the standard conjectures, which are open; it cannot settle the Hodge conjecture without settling them first.無条件
    • standardhttps://www.claymath.org/wp-content/uploads/2022/05/hodge.pdfP. Deligne, The Hodge conjecture, Clay Mathematics Institute official problem description

• standardhttps://www.claymath.org/wp-content/uploads/2022/05/hodge.pdfP. Deligne, The Hodge conjecture, Clay Mathematics Institute official problem description
• paperS. Lefschetz, L'analysis situs et la geometrie algebrique, Gauthier-Villars (1924)The (1,1) case, the only case proved in general

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2026-09-17

最終レビュー

2026-09-19

次回レビュー期限

2027-09-19

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