Kuga-Satake construction on families of K3 surfaces of Picard rank 14
Flora Poon
TL;DR
This work realises a modular map from the moduli of $P$-polarised K3 surfaces with Picard rank $14$ to a moduli space of abelian $8$-folds with totally definite quaternion multiplication by exploiting the isometry between the type IV$_6$ and type II$_4$ domains. Central to the construction is the Kuga-Satake process, which associates a weight-two Hodge structure of K3 type to a weight-one Hodge structure on an abelian variety via Clifford algebras, yielding a largely explicit decomposition $\mathrm{KS}(X) \sim A_+^4 \times A_-^4$ with $\mathrm{End}_{\mathbb{Q}}(A_\pm) \simeq \mathbb{H}_{\mathbb{Q}}$. The paper provides both a general explicit framework and a worked MAGMA-based computation for a concrete rank-14 family, establishing a holomorphic, equivariant embedding of moduli spaces and describing how the map descends to the appropriate arithmetic quotients. The rank-18 specialization illustrates the geometry of the KS map on a sublattice and connects to Shioda-Inose structures, showing how $A_1$ decomposes into products of elliptic curves, thereby clarifying the interplay between K3 geometry, Clifford algebra endomorphisms, and abelian varieties with quaternion multiplication.
Abstract
The isometry between the type IV$_6$ and the type II$_4$ hermitian symmetric domains suggests a possible relation between suitable moduli spaces of K3 surfaces of Picard rank $14$ and of polarised abelian $8$-folds with totally definite quaternion multiplication. We show how this isometry induces a geometrically meaningful map between such moduli spaces using the Kuga-Satake construction. Furthermore, we illustrate how the the modular mapping can be realised for any specific families of K3 surfaces of Picard rank $14$, which can be specialised to families of K3 surfaces of higher Picard rank.