Calibrations in hyperkahler geometry

Abstract

We describe a family of calibrations arising naturally on a hyperk\"ahler 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 $SL(n, {\Bbb H})$, 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 $M$.