Calibrations in hyperkahler geometry

Abstract

We describe a family of calibrations arising naturally on a hyperkähler manifold $M$. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When $M$ is an HKT (hyperkaehler with torsion) manifold with holonomy $SL(n,\mathbb{H})$, we construct another family of calibrations ${\Phi }_i$, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations ${\Phi }_i$ are (generally speaking) not parallel with respect to any torsion-free connection on $M$.

v. 3.1, 33 pages, minor corrections, final version (accepted by Commun. Contemp. Math.)