# Elementary Complex Differential Forms

I learned that an elementary differential form was one like $$dx^{i_1}\wedge\cdots\wedge dx^{i_n}$$ where $$1\leq i_1<\cdots < i_n\leq n$$ if we are on an real $$n$$-dimensional manifold. I'm trying to learn about complex differential forms, but I am getting confused about the introduction of $$d\overline{z}$$. The crux of my question is if I want to take a complex contour integral on a complex analytic $$n$$-dimensional (with respect to $$\mathbb{C}$$ basis) manifold what does it look like? I had imagined that if I had a complex function, $$f$$, defined on my complex analytic $$n$$-dimensional manifold, then the integral would look like $$\int_C fdz^1\wedge\cdots\wedge dz^n$$. But I keep seeing use of $$d\overline{z}$$, and I don't know how this fits in.

I've also seen a few complexification arguments in my reading, but if this can be done without considering the real space, I would prefer to do it that way. Any advice is much appreciated!

First, in your first line, you needn't have an $$n$$-form on an $$n$$-dimensional manifold. I would write a general $$k$$-form as a linear combination of $$dx^{i_1}\wedge\dots\wedge dx^{i_k}$$ with $$1\le i_1<\dots.
Next, if you want to think about integrating a top-degree form over an $$n$$-dimensional complex manifold, you will want $$f(z)dz^1\wedge d\bar z^1\wedge\dots\wedge dz^n\wedge d\bar z^n,$$ often referred to as a form of type $$(n,n)$$. In general, a form of type $$(p,q)$$ will be a linear combination of forms of the type $$f(z)dz^{j_1}\wedge\dots\wedge dz^{j_p}\wedge d\bar z^{k_1}\wedge\dots\wedge d\bar z^{k_q}.$$ If you're thinking about the analog of residue computations in one complex variable, what you'll actually want to do is integrate a meromorphic $$n$$-form $$\omega = h(z)dz^1\wedge\dots\wedge dz^n$$ (where $$h(z)$$ is a meromorphic function, typically with poles along a divisor with normal crossings) over an $$n$$-cycle $$\Gamma$$. For example, if $$h(z) = \dfrac{f(z)}{g_1(z)\dots g_n(z)}$$, with $$f$$ and all $$g_j$$ holomorphic, then the immediate generalization of the classical residue computation will be $$\displaystyle\int_\Gamma \omega$$ for $$\Gamma = \{|g_j|=\varepsilon, j=1,\dots,n\}$$. You can see more about such things in, e.g., Griffiths and Harris, Chapter 5.