Residues in several complex variables

Question

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:

0. 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:

1. (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 $$res_{\sigma_{\alpha}} f=\frac{1}{(2\pi i)^n} \int_{\sigma_{\alpha}} f dz$$

2. (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 $$\mathcal{O}_0/(f_1,...,f_n) \to \mathbb{C}$$

3. 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.

Answer 1

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.

Answer 2

I believe residue currents encompass most definitions of residues in several complex variables. Residue currents as developed in the 20th century are discussed for example in the survey "Residue currents" by Tsikh and Yger. Given a tuple $(f_1,\dots,f_p)$ defining a complete intersection, i.e., such that $\{ f_1 = \dots = f_p = 0 \}$ has codimension $p$, there is an associated residue current $\mu^f$, as first defined by Coleff and Herrera. Their definition is by taking limits of integrals similar to in 2), but where $g$ in the definition of $\omega$ should be a test form and you should consider limits where $\epsilon_1,\dots,\epsilon_p$ tend to $0$ in an appropriate way.

Just as the Grothendieck residue, these residue currents can also be defined with the help of Bochner-Martinelli forms, as was first done by Passare, Tsikh and Yger. In fact, if $B_f$ is the Bochner-Martinelli form of $f$, then the action of $\mu^f$ on a test form $\varphi$ is given by $\lim_{\epsilon \to 0} \int_{\{|f|=\epsilon\}} B_f \wedge \varphi$.

In the absolute case in 2), i.e., when $p=n$, and you take a cut-off function $\chi$ with compact support that is $\equiv 1$ at the origin, then $\chi \omega$ is a test-form, and the action of $\mu^f$ on $\chi \omega$ equals the Grothendieck residue. With the help of the representation in terms of Bochner-Martinelli forms, it is immediate that $\mu^f$ acting on $\chi \omega$ equals the Grothendieck residue of $\omega$.

I would believe that also the case 1) should be possible to represent by residue currents, by taking a holomorphic function $g$ whose zero set contains the indeterminacy locus $P$ and letting $\mu^g$ act on an appropriate form, but I'm not familiar enough with the definition of Shabat to describe this.

More recently, there are also residue currents defined more generally for coherent sheaves by Andersson and Wulcan, "Residue currents with prescribed annihilator ideals", not just complete intersections as considered above.

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Regarding how $(*)$ fits into this picture, I don't know if this has been explicitly elaborated on earlier, but at least it is discussed in "Direct images of semi-meromorphic currents" by Andersson and Wulcan.

A semi-meromorphic form $\psi$ is a form that is locally a smooth form times a meromorphic form, and one might identify the form with its corresponding principal value current. An almost semi-meromorphic form is the push-forward of a semi-meromorphic form under a modification. The Bochner-Martinelli form $B_f$ is an example of an almost semi-meromorphic form. (When $p=1$, it is indeed even meromorphic.)

If $\psi$ is an almost semi-meromorphic form on $X$ that is smooth outside a subvariety $Z$, then $\bar\partial\psi$ is a smooth form on $X \setminus Z$, and it turns out that $\bar\partial\psi|_{X\setminus Z}$ has a principal value extension to $X$ that is again an almost semi-meromorphic form. In this way, there is a $\bar\partial$-operator acting on almost semi-meromorphic currents. Andersson and Wulcan define the residue of $\psi$ as the current $R(\psi)=\bar\partial T_\psi - T_{{\bar\partial} \psi}$, see section 4.4 of their paper. The residue is thus the difference between this $\bar\partial$-operator on almost semi-meromorphic forms and the $\bar\partial$-operator acting in the sense of currents. As is basically detailed in their Example 4.18, the current $\mu^f$ is then in fact the residue of the Bochner-Martinelli form $B_f$.