Calibrated geometry in hyperkähler cones, 3-Sasakian manifolds, and twistor spaces

Aslan, Benjamin ; Karigiannis, Spiro  ; Madnick, Jesse 

Canadian journal of mathematics, Tome 77 (2025) no. 4, pp. 1163-1221

Résumé

We systematically study calibrated geometry in hyperkähler cones $C^{4n+4}$, their 3-Sasakian links $M^{4n+3}$, and the corresponding twistor spaces $Z^{4n+2}$, emphasizing the relationships between submanifold geometries in various spaces. Our analysis highlights the role played by a canonical $\mathrm {Sp}(n)\mathrm {U}(1)$-structure $\gamma $ on the twistor space Z. We observe that $\mathrm {Re}(e^{- i \theta } \gamma )$ is an $S^1$-family of semi-calibrations and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri–Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler–Einstein and nearly Kähler structures.

Détail

DOI : 10.4153/S0008414X24000282

Mots-clés : Hyperkähler manifolds, calibrated submanifolds, twistor spaces, Kähler–Einstein manifolds, nearly Kähler manifolds