I'm pretty clear in my understanding of scalar-valued differential $(p, q)$-forms (resp. holomorphic $(p,0)$-forms) on a complex manifold $M$ and the related Hodge theory. What I'm not sure about is whether there is a matrix-valued analogue of such differential forms on a complex manifold in the math literature. If there is such a thing, what are some good references that I may find useful?



matrix-valued differential forms on complex manifolds
Actually, matrix-valued differential forms are used a lot in hypercomplex analysis/hypercomplex geometry, which, as the name suggests, includes certain complex manifolds. There is a nice account of such differential forms in *Rocha-Chavez, Shapiro, and Sommen "Integral Theorems for Functions and Differential Forms in $\mathbb{C}^m$" (2001)* (~200 pages). The first chapter is a general introduction to (complex-valued) differential forms, the second chapter discusses differential forms with values in the 2x2-matrices, and then the book goes on into the realm of hyperholomorphy, including hyperholomorphic Hodge theory.


