{"id": "card_question_hodge", "kind": "reference", "title": "The Hodge conjecture", "body": "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": {"label": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "url": "https://www.claymath.org/millennium-problems/", "ref": "stick_hodge_conjecture", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "millennium", "box": "question", "bands": ["millennium", "open question", "clay", "hodge", "algebraic geometry", "cohomology", "cycles"], "subject": "The Hodge conjecture", "connections": [{"to_card_id": "card_spine_millennium_sources", "relationship": "member_of", "evidence": "one of the seven questions the Millennium shelf is about"}, {"to_card_id": "card_floor_millennium", "relationship": "open_end_of", "evidence": "an open question hanging off the floor of what is proven"}, {"to_card_id": "card_joint_weil_conjectures", "relationship": "connects_at", "evidence": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_joint_tate_conjecture", "relationship": "connects_at", "evidence": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields. tate 1965; artin tate 1966; milne 1975. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_joint_bochner_vanishing", "relationship": "connects_at", "evidence": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)"}, {"to_card_id": "card_floor_logarithm", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chain_deligne_1971", "relationship": "connects_at", "evidence": "where the logarithm enters: mixed Hodge theory is built on forms with logarithmic poles, dz/z"}, {"to_card_id": "card_src_mill_deligne_2000", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_voisin_2002", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_atiyah_hirzebruch_1962", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_floor_hodge", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "millennium.question", "facets": {"subject": ["algebraic geometry", "clay", "cohomology", "cycles", "hodge", "millennium", "open question"]}, "presentation": {"glyph": "?", "kind_label": "A question", "by": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://www.claymath.org/millennium-problems/", "host": "claymath.org", "provider": "", "titled": "The Hodge conjecture", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "relationship": "on the shelf of", "why": "one of the seven questions the Millennium shelf is about", "href": "/card/card_spine_millennium_sources", "resolved": true}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "open end of", "why": "an open question hanging off the floor of what is proven", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_joint_weil_conjectures", "title": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "relationship": "connects at", "why": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. 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Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)", "href": "/card/card_joint_bochner_vanishing", "resolved": true}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "relationship": "open end of", "why": "the open question this chain reaches", "href": "/card/card_floor_logarithm", "resolved": true}, {"id": "card_chain_deligne_1971", "title": "Deligne 1971 — Théorie de Hodge, II", "relationship": "connects at", "why": "where the logarithm enters: mixed Hodge theory is built on forms with logarithmic poles, dz/z", "href": "/card/card_chain_deligne_1971", "resolved": true}, {"id": "card_src_mill_deligne_2000", "title": "P. 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