Quaternionic plurisubharmonic function on a hypercomplex manifold

Definition

Let $(M,I,J,K)$ be a hypercomplex manifold. A continuous real-valued function $u$ is quaternionic plurisubharmonic if the generalized real $(2,0)$ -form

\partial\partial_Ju

is nonnegative. For a smooth $u$ , nonnegativity means that the associated hyperhermitian form $t(\partial\partial_Ju)$ is positive semidefinite.

Positivity map

In the right-action convention, the identification $t$ between real $(2,0)$ -forms and hyperhermitian forms is characterized by

t(\eta)(X,X)=\eta(X,XJ).

This formula translates to the corresponding left-action convention used for the chosen hypercomplex triple. It makes the positivity condition intrinsic and independent of coordinates.

Flat model

On $M=\mathbb H^n$ with its standard hypercomplex structure, $t(\partial\partial_Ju)$ is, up to the fixed normalization, the quaternionic Hessian. Hence this definition recovers the linewise flat-space definition.

Approximation class

Wedge products of generalized $\partial\partial_Ju$ forms are first defined for functions locally approximable, uniformly on compact subsets, by smooth quaternionic PSH functions. In flat space every continuous quaternionic PSH function belongs to this class. On a general hypercomplex manifold the approximation issue is an additional hypothesis, not automatic from the symbol $\partial\partial_Ju$ .

References

Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §6.