Calibrations in hyperkahler geometry

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 $SL(n, {\Bbb H})$, we construct another family of calibrations $Φ_i$, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations $Φ_i$ are (generally speaking) not parallel with respect to any torsion-free connection on $M$.

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