{"id":"openai-math/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus","pdf":"https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026","tags":["algebraic-geometry","hodge-theory","k3-surfaces"],"score":0,"title":"Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus","domain":"Algebraic and complex geometry","family":"032","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AG","summary":"We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in $`\\mathbb U_{\\mathbb Q}^{\\oplus2}\\perp\\langle-1\\rangle^4`$. This includes every ample P -polarized K3 surface, including all Picard jumps, for $`P=\\mathbb U\\oplus D_8(-1)\\oplus D_4(-1)`$. On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-30","reactions":{},"formalized":false,"family_title":"The rational Hodge conjecture for CM abelian varieties"}