{"schema_version":"dataset-v1","id":"hodge-diamond-abelian-fourfold","version":1,"title":"Hodge diamond of an abelian fourfold","description":"Hodge numbers h^{p,q} = C(4,p)·C(4,q) of a complex abelian variety of dimension four. Hodge classes of codimension k lie in H^{k,k}.","license":"CC0-1.0 (exact values)","problems":["hodge-conjecture"],"statements":[{"problem":"hodge-conjecture","statement":"hodge-divisor-classes","role":"Codimension one (H^{1,1}): Lefschetz (1,1) theorem."},{"problem":"hodge-conjecture","statement":"hodge-abelian-fourfolds","role":"Codimension two (H^{2,2}) on abelian fourfolds."}],"figure":{"family":"diamond","title":"Where the fourfold results live","caption":"The Hodge diamond of an abelian fourfold. Hodge classes of codimension k sit in the middle column H^{k,k}; the recorded results cover H^{1,1} for every smooth projective variety and H^{2,2} for abelian fourfolds.","grid":"hodge_numbers","row_symbol":"p","column_symbol":"q","highlights":[{"row":1,"column":1,"label":"H^{1,1} · divisors","status":"proved","statement":"hodge-divisor-classes"},{"row":2,"column":2,"label":"H^{2,2} · codimension 2","status":"proved","statement":"hodge-abelian-fourfolds"}],"boundary":"The numbers are dimensions of cohomology groups, not counts of Hodge classes. Highlighting marks where recorded results apply; it does not cover arbitrary fourfolds or other codimensions."},"provenance":{"method":"h^{p,q} = C(4,p)·C(4,q) for a complex torus of dimension 4 (exact).","software":{"python":"3.12.13"},"precision":"Exact integers.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-20"},"data":{"series":[],"marks":[],"grids":[{"key":"hodge_numbers","label":"h^{p,q}","values":[[1,4,6,4,1],[4,16,24,16,4],[6,24,36,24,6],[4,16,24,16,4],[1,4,6,4,1]]}]},"sha256":"694ed642eddba682f1088f2e6772920869767f2b958edba31d739c943f01cca0"}