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

Huybrechts, Daniel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 397-421

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–Šafarevič group, and certain 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.

On étudie la théorie de Hodge des familles brillantes des surfaces K3. Deux fibres dans le lieu de Noether–Lefschetz d’une telle famille ont des structures de Hodge très similaires. Les déformations de twisteurs, le groupe de Tate–Šafarevič et certaines courbes de Shimura donnent des exemples de telles familles. Dans ce cadre le groupe de Brauer apparaît comme lieu de Noether–Lefschetz de la famille de Brauer et aussi comme spécialisation des lieux de Noether–Lefschetz des espaces de twisteurs. Le passage transcendent d’une fibre algébrique à une autre dans l’espace de twisteur est vu comme composition de deux déformations du caractère plus algébriques.

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