I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much harder when we pass from one variable to several variables.

So for $n=1$ we have:

- For a holomorphic $f$ with an isolated singularity at point $a$, the
*residue*of $f$ at $a$ is defined as $$res_a f = \frac{1}{2\pi i} \int_{\sigma} f dz$$for a small loop $\sigma$ around $a$.

For $n>1$ we have:

(Shabat, vol. II) For a meromorphic $f$ defined on $D \subset \mathbb{C}^n$ with the indeterminacy locus $P \subset D$, choose a basis $\sigma_{\alpha}$ of $H_1(D \setminus P, \mathbb{Z})$ and define the

*residue*of $f$ with respect to $\sigma_{\alpha}$ to be $$res_{\sigma_{\alpha}} f=\frac{1}{(2\pi i)^n} \int_{\sigma_{\alpha}} f dz$$(Griffith-Harris, Chapter 5) Let $U$ be a ball $\{z\in \mathbb{C}^n \ | \ ||z||< \varepsilon\}$ and $f_1,...,f_n \in \mathcal{O}(\bar{U})$ be holomorphic functions with an isolated common zero at the origin. Take $\omega=\frac{g(z) dz_1 \wedge ... \wedge dz_n}{f_1(z)...f_n(z)}$ and $\Gamma=\{z \ : \ |f(z_i)|=\varepsilon_i\}$. The (Grothendieck)

*residue*is given by $$Res_{ \{0\}} \omega=\frac{1}{(2 \pi i)^n} \int_{\Gamma} \omega .$$It can further be viewed as a homomorphism $$\mathcal{O}_0/(f_1,...,f_n) \to \mathbb{C}$$In the "General theory of higher-dimensional residues", Dolbeault discusses

*residue-homomorphism, homological residues, cohomological residues, residue-currents*, etc.

So since there are so many various things called residue, my question is

What structure are all these things trying to capture, so that we call all these various things "residue"?

In Chapter 3, Griffiths and Harris outline a general principle when discussing distributions and currents: $$(*) \quad D T_{\psi} - T_{D \psi} = \text{"residue"},$$where $T_{\psi}$ is the current $T_{\psi}(\phi)=\int_{\mathbb{R}^n} \psi \wedge \phi$ (this discussion takes place on $\mathbb{R}^n$). They illustrate that by applying this principle to the Cauchy kernel $\psi=\frac{dz}{2 \pi i z}$: $$\phi(0)=\frac{1}{2 \pi i} \int_{\mathbb{C}} \frac{\partial \phi(z)}{\partial \bar{z}} \frac{dz \wedge d \bar{z}}{z} \ \iff \bar{\partial}(T_{\psi})=\delta_{0}.$$

This is a nice example, but later on when they discuss the Grothendieck residue (2) in Chapter 5 they do not explain how it fits into the philosophy $(*)$. I also do not see how (0), (1) and (3) fit into this philosophy. So maybe one can explain how $(*)$ might be a potential answer to the question I am asking.