Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether-Lefschetz loci

TL;DR

This paper explores the Hodge theory of special K3 surface families, linking their geometric and algebraic properties through Noether-Lefschetz loci, Brauer groups, and twistor deformations.

Contribution

It introduces a unified framework connecting Hodge structures, Brauer groups, and twistor spaces in the context of brilliant K3 surface families.

Findings

Hodge structures of fibers are closely linked over Noether-Lefschetz points

Brauer groups are characterized via Noether-Lefschetz loci and twistor spaces

Deformations between algebraic and transcendental fibers are systematically described

Abstract

We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction is a transcendental operation, is here viewed as first deforming along the more algebraic Brauer family and then along a family of algebraic K3 surfaces.