Remark: 032-02. Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus

Authors: OpenAI. Family 032.

Title: Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus.

Abstract: 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 complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.