The del-J operator - Knowlpedia

Definition

Let $(M,I,J,K)$ be a hypercomplex manifold, and use the type decomposition determined by $I$ . The del-J operator is

\partial_J=J^{-1}\circ\bar\partial\circ J.

It maps $I$ -type $(p,q)$ forms to type $(p+1,q)$ forms and satisfies

\partial\partial_J=-\partial_J\partial.

Action of

J

on forms

The endomorphism $J$ acts on forms by pullback on every argument. Because $IJ=-JI$ , it exchanges the $(p,q)$ and $(q,p)$ types determined by $I$ . Conjugating $\bar\partial$ by this action therefore produces another operator of $(1,0)$ -degree.

Quaternionic potentials

For a real smooth function $u$ , the form $\partial\partial_Ju$ is a real $(2,0)$ -form in the quaternionic sense. Its positivity defines quaternionic plurisubharmonicity on a hypercomplex manifold, and strict positivity produces local HKT metrics.

Convention warning

Some sources let $I,J,K$ act on tangent vectors on the right and write the same construction with the corresponding right-action signs. The invariant content is the conjugated Dolbeault operator and the anticommutation identity; formulas should not mix left- and right-action conventions.

References

Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record. Relevant: §2.