{"id": "card_floor_hodge", "kind": "note", "title": "The Hodge chain - from periods and topology to the classes that are not cycles", "body": "Two trees. The transcendental: Riemann's periods (1857). The topological: Lefschetz's analysis situs of a variety and the (1,1) theorem (1924), de Rham's forms (1931). They become one in Hodge's harmonic integrals (1941): every class has a harmonic form, and the (p, q) decomposition. Then the conjecture as stated (Hodge 1950), the integral form refuted (Atiyah-Hirzebruch 1962), variations of Hodge structure (Griffiths 1968), mixed Hodge theory (Deligne 1971), the Kähler form refuted (Voisin 2002), the statement as the Clay Institute keeps it (Deligne 2000). The open end: the rational projective case from dimension four.", "source": {"label": "Narrow Highway - a chain on the one map (operator seed)", "url": "", "ref": "docs/MILLENNIUM_PREPAREDNESS.md", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "floor", "bands": ["floor", "chain", "two trees", "hodge", "algebraic cycles", "harmonic forms", "periods"], "subject": "The Hodge chain - from periods and topology to the classes that are not cycles", "connections": [{"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the hodge chain rests on the one Floor of Discovery"}, {"to_card_id": "card_chain_riemann_1857", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_lefschetz_1924", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_question_hodge", "relationship": "has_open_end", "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": "codex.floor", "facets": {"subject": ["algebraic cycles", "chain", "harmonic forms", "hodge", "periods", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "floor", "by": "Narrow Highway - a chain on the one map (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the hodge chain rests on the one Floor of Discovery", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_chain_riemann_1857", "title": "Riemann 1857 — Theorie der Abel'schen Functionen", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_riemann_1857", "resolved": true}, {"id": "card_chain_lefschetz_1924", "title": "Lefschetz 1924 — L'analysis situs et la géométrie algébrique", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_lefschetz_1924", "resolved": true}, {"id": "card_question_hodge", "title": "The Hodge conjecture", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_hodge", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_hodge", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_hodge"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}