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!
Best Answer
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 $1\le i_1<\dots<i_k\le n$.
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.