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The Hodge conjecture
A question
The Hodge conjecture. On a projective non-singular algebraic variety over the complex numbers, every Hodge class is a rational linear combination of classes of algebraic cycles. Its chart is the tick stick stick_hodge_conjecture: what is sealed, what is cited and what stays open are read live at /stick?id=stick_hodge_conjecture. Open from dimension four; closed in dimension three and below (Lefschetz, hard Lefschetz).
source
card id
card_question_hodge
address
SCI.millennium.FCT/the-hodge-conjecture/REF.WITNESSED@clay-mathematics-institu
adjoining cards
- on the shelf of → The Millennium sources — the papers behind the seven sticks — one of the seven questions the Millennium shelf is about
- open end of → The Millennium floor - seven open questions, and where they connect — an open question hanging off the floor of what is proven
- connects at → The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem — The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem.
- connects at → The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields — The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields. tate 1965; art
- connects at → Positive Ricci curvature kills harmonic one-forms — Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982
- open end of → The logarithm - the instrument the joints share — the open question this chain reaches
- connects at → Deligne 1971 — Théorie de Hodge, II — where the logarithm enters: mixed Hodge theory is built on forms with logarithmic poles, d
- builds on → P. Deligne 2000 — The Hodge conjecture — a later work standing on an earlier one
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