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:
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(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 $$res_{\sigma_{\alpha}} f=\frac{1}{(2\pi i)^n} \int_{\sigma_{\alpha}} f dz$$
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(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 $$\mathcal{O}_0/(f_1,…,f_n) \to \mathbb{C}$$
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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 plane 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 ask.
Best Answer
There is a gentle introduction, starting with the single variable case before cranking up the dimension: "Introduction to residues and resultants" by Cattani and Dickenstein. There are also very abstract formulations that I am not familiar with (by e.g., Hartshorne "Residues and Duality", Joseph Lipman "Residues and Traces of Differential Forms Via Hochschild Homology", Amnon Yekutieli "An Explicit Construction of the Grothendieck Residue Complex (with appendix by P. Sastry)", etc.), but in down-to-earth terms the idea is: given a system of equations $F(x)=0$, and some other function $G$, how do you compute $\sum_z G(z)$ where the sum is over all solutions of $F(x)=0$. You may or not include division by the Jacobian of the $F$'s in the function $G$. Multidimensional residues answer this question. Resultants appear as denominators of residues. Moreover, taking logarithms, and by the Poisson formula, a resultant can be computed by a residue. So the two concepts are tightly related. In good cases, taking the residue seen as a linear form on the algebra of $G$'s mod the ideal of the $F$'s, gives a nondegenerate trace, hence the "duality" associated with residues.