Calibrations in hyperkahler geometry

TL;DR

This paper introduces new calibration forms on hyperk"ahler and HKT manifolds that identify special subvarieties, expanding understanding of geometric structures in hyperk"ahler geometry.

Contribution

It constructs families of calibrations on hyperk"ahler and HKT manifolds that target specific holomorphic subvarieties, including cases with torsion.

Findings

Calibrations for holomorphic Lagrangian, isotropic, and coisotropic subvarieties.

Construction of non-parallel calibrations on HKT manifolds with holonomy SL(n, H).

Extension of calibration theory to manifolds with torsion.

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 $M$.