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The Hodge chain - from periods and topology to the classes that are not cycles
floor
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.
- part of → The Floor of Discovery — one floor, and by its design the fear of God — the hodge chain rests on the one Floor of Discovery
- has part → Riemann 1857 — Theorie der Abel'schen Functionen — a root of this chain - one of the two trees it began from
- has part → Lefschetz 1924 — L'analysis situs et la géométrie algébrique — a root of this chain - one of the two trees it began from
- has open end → The Hodge conjecture — the open question this chain reaches
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