Do you agree that something like $z_1^2z_2 dz_1 \wedge d\bar{z}_3 + z_3 dz_1 \wedge dz_2$ is a natural sort of thing to integrate over a complex 2 dimensional submanifold of $C^3$? I think that these forms are "intrinsically interesting", just because we want to do integral calculus.

Closed forms are interesting because their integrals are $0$. Closed forms are always locally exact, but may or may not be globally exact. It seems intrinsically interesting to investigate those forms which are locally but not globally exact.

I guess I am just saying that these are the natural questions you would ask when you start doing integral calculus on complex spaces?

What sort of application are you looking for?

Do you agree that something like $z_1^2z_2 dz_1 \wedge d\bar{z}_3 + z_3 dz_1 \wedge dz_2$ is a natural sort of thing to integrate over a complex 2 dimensional submanifold of $C^3$? I think that these forms are "intrinsically interesting", just because we want to do integral calculus.

Closed forms are always locally exact, but may or may not be globally exact. It seems intrinsically interesting to investigate those forms which are locally but not globally exact.

I guess I am just saying that these are the natural questions you would ask when you start doing integral calculus on complex spaces?

What sort of application are you looking for?