Browse issues
no. 2
Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether–Lefschetz loci
Daniel Huybrechts1
1 Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1741

Daniel Huybrechts 1

1 Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2023_6_32_2_397_0,
     author = {Daniel Huybrechts},
     title = {Brilliant families of {K3} surfaces: {Twistor} spaces, {Brauer} groups, and {Noether{\textendash}Lefschetz} loci},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {397--421},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 32},
     number = {2},
     year = {2023},
     doi = {10.5802/afst.1741},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1741/}
}
TY  - JOUR
AU  - Daniel Huybrechts
TI  - Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether–Lefschetz loci
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2023
SP  - 397
EP  - 421
VL  - 32
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1741/
DO  - 10.5802/afst.1741
LA  - en
ID  - AFST_2023_6_32_2_397_0
ER  - 
%0 Journal Article
%A Daniel Huybrechts
%T Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether–Lefschetz loci
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2023
%P 397-421
%V 32
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1741/
%R 10.5802/afst.1741
%G en
%F AFST_2023_6_32_2_397_0
Daniel Huybrechts. Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether–Lefschetz loci. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 397-421. doi : 10.5802/afst.1741. https://afst.centre-mersenne.org/articles/10.5802/afst.1741/

[1] Arend Bayer; Emanuele Macrì MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math., Volume 198 (2014) no. 3, pp. 505-590 | DOI | MR | Zbl

[2] Daniel Bragg; Max Lieblich Twistor spaces for supersingular K3 surfaces (2018) (https://arxiv.org/abs/1804.07282)

[3] Robert Friedman; John W. Morgan Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 27, Springer, 1994 | DOI

[4] Daniel Huybrechts Compact hyperkähler manifolds: Basic results, Invent. Math., Volume 135 (1999) no. 1, pp. 63-113 | DOI | Zbl

[5] Daniel Huybrechts The global Torelli theorem: classical, derived, twisted, Algebraic geometry, Seattle 2005 (Proceedings of Symposia in Pure Mathematics), Volume 80.1, American Mathematical Society, 2009, pp. 235-258 | Zbl

[6] Daniel Huybrechts Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, 158, Cambridge University Press, 2016 | DOI

[7] Daniel Huybrechts Motives of isogenous K3 surfaces, Comment. Math. Helv., Volume 94 (2019) no. 3, pp. 445-458 | DOI | MR | Zbl

[8] Daniel Huybrechts Complex multiplication in twistor spaces, Int. Math. Res. Not., Volume 2021 (2021) no. 24, pp. 18972-18999 | Zbl

[9] Daniel Huybrechts; Paolo Stellari Equivalences of twisted K3 surfaces, Math. Ann., Volume 332 (2005) no. 4, pp. 901-936 | DOI | MR | Zbl

[10] Eyal Markman Lagrangian fibrations of holomorphic-symplectic varieties of K3 [n] -type, Algebraic and complex geometry (Springer Proceedings in Mathematics & Statistics), Volume 71, Springer, 2014, pp. 241-283 | DOI | MR | Zbl

[11] Kieran G. O’Grady The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebr. Geom., Volume 6 (1997) no. 4, pp. 599-644 | MR | Zbl

[12] Arvid Perego; Matei Toma Moduli spaces of bundles over non-projective K3 surfaces, Kyoto J. Math., Volume 57 (2017) no. 1, pp. 107-146 | Zbl

[13] Matthias Schütt; Tetsuji Shioda; Ronald van Luijk Lines on Fermat surfaces, J. Number Theory, Volume 130 (2010) no. 9, pp. 1939-1963 | DOI | MR | Zbl

[14] Bert Van Geemen Real multiplication on K3 surfaces and Kuga–Satake varieties, Mich. Math. J., Volume 56 (2008) no. 2, pp. 375-399 | MR | Zbl

[15] Misha Verbitsky Degenerate twistor spaces for hyperkähler manifolds, J. Geom. Phys., Volume 91 (2015), pp. 2-11 | DOI | Zbl

[16] Francesco Viganò Complex multiplication and Noether–Lefschetz loci of the twistor space of a K3 surface (2021) (https://arxiv.org/abs/2102.07285)

[17] Yuriĭ G. Zarkhin Hodge groups of K3 surfaces, J. Reine Angew. Math., Volume 341 (1983), pp. 193-220 | MR | Zbl

Cited by Sources: