Abstract
We describe a family of calibrations arising naturally on a hyper-Kähler manifold M. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When M is an HKT (hyper-Kähler with torsion) manifold with holonomy SL (n, ℍ), we construct another family of calibrations Φi, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations Φi are (generally speaking) not parallel with respect to any torsion-free connection on M.
Highlights
The theory of calibrations was developed by R
Lawson in [HL], and proved to be very useful in describing the geometric structures associated with special holonomies
Since calibrations have become a central notion in many geometric developments in string physics and Mtheory
Summary
The theory of calibrations was developed by R. We show that a plane V ⊂ T M is a face of ρp,p if and only for ζ(V ) is a face of ρ for all ζ ∈ U(1), for the standard U(1)-action on T M (Theorem 5.2) Applying this result to the special Lagrangian calibration on (M, J) defined in [HL] (see [McL]), we obtain the form Ψn, n = dimH M, which calibrates complex analytic Lagrangian subvarieties on (M, I) (these subvarieties are known to be special Lagrangian on (M, J); see e.g. An interesting side effect of our construction of holomorphic Lagrangian calibrations is an appearance of a family of calibrations which are not parallel, under any torsionless connection (Claim 6.6) These calibrations are associated with the so-called HKT structures in hypercomplex geometry.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
