Poincaré residue

In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions.

Given a hypersurface $X \subset \mathbb{P}^n$ defined by a degree $d$ polynomial $F$ and a rational $n$ -form $\omega$ on $\mathbb{P}^n$ with a pole of order $k > 0$ on $X$ , then we can construct a cohomology class $\operatorname{Res}(\omega) \in H^{n-1}(X;\mathbb{C})$ . If $n=1$ we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues{{Cite journal|last=Poincaré|first=H.|date=1887|title=Sur les résidus des intégrales doubles|url=https://projecteuclid.org/euclid.acta/1485888747|journal=Acta Mathematica|language=FR|volume=9|pages=321–380|doi=10.1007/BF02406742|issn=0001-5962|doi-access=free}} he was studying period integrals of the form

$\underset{\Gamma}\iint \omega$ for $\Gamma \in H_2(\mathbb{P}^2 - D)$

where

\omega

was a rational differential form with poles along a divisor

D

. He was able to make the reduction of this integral to an integral of the form

$\int_\gamma \text{Res}(\omega)$ for $\gamma \in H_1(D)$

where

\Gamma = T(\gamma)

, sending

\gamma

to the boundary of a solid

\varepsilon

-tube around

\gamma

on the smooth locus

D^*

of the divisor. If

$\omega = \frac{q(x,y)dx\wedge dy}{p(x,y)}$

on an affine chart where

p(x,y)

is irreducible of degree

N

and

\deg q(x,y) \leq N-3

(so there is no poles on the line at infinity{{Cite journal|last=Griffiths|first=Phillip A.|date=1982|title=Poincaré and algebraic geometry|url=https://www.ams.org/bull/1982-06-02/S0273-0979-1982-14967-9/|journal=Bulletin of the American Mathematical Society|language=en|volume=6|issue=2|pages=147–159|doi=10.1090/S0273-0979-1982-14967-9|issn=0273-0979|doi-access=free}} ^{page 150}). Then, he gave a formula for computing this residue as

$\text{Res}(\omega) = -\frac{qdx}{\partial p / \partial y} = \frac{qdy}{\partial p / \partial x}$

which are both cohomologous forms.

Construction

= Preliminary definition =

Given the setup in the introduction, let $A^p_k(X)$ be the space of meromorphic $p$ -forms on $\mathbb{P}^n$ which have poles of order up to $k$ . Notice that the standard differential $d$ sends

: $d: A^{p-1}_{k-1}(X) \to A^p_k(X)$

Define

: $\mathcal{K}_k(X) = \frac{A^p_k(X)}{dA^{p-1}_{k-1}(X)}$

as the rational de-Rham cohomology groups. They form a filtration

$\mathcal{K}_1(X) \subset \mathcal{K}_2(X) \subset \cdots \subset \mathcal{K}_n(X) =
H^{n+1}(\mathbb{P}^{n+1}-X)$

corresponding to the Hodge filtration.

= Definition of residue =

Consider an $(n-1)$ -cycle $\gamma \in H_{n-1}(X;\mathbb{C})$ . We take a tube $T(\gamma)$ around $\gamma$ (which is locally isomorphic to $\gamma\times S^1$ ) that lies within the complement of $X$ . Since this is an $n$ -cycle, we can integrate a rational $n$ -form $\omega$ and get a number. If we write this as

: $\int_{T(-)}\omega : H_{n-1}(X;\mathbb{C}) \to \mathbb{C}$

then we get a linear transformation on the homology classes. Homology/cohomology duality implies that this is a cohomology class

: $\operatorname{Res}(\omega) \in H^{n-1}(X;\mathbb{C})$

which we call the residue. Notice if we restrict to the case $n=1$ , this is just the standard residue from complex analysis (although we extend our meromorphic $1$ -form to all of $\mathbb{P}^1$ . This definition can be summarized as the map

$\text{Res}: H^{n}(\mathbb{P}^{n}\setminus X) \to H^{n-1}(X)$

= Algorithm for computing this class =

There is a simple recursive method for computing the residues which reduces to the classical case of $n=1$ . Recall that the residue of a $1$ -form

: $\operatorname{Res}\left(\frac{dz} z + a\right) = 1$

If we consider a chart containing $X$ where it is the vanishing locus of $w$ , we can write a meromorphic $n$ -form with pole on $X$ as

: $\frac{dw}{w^k}\wedge \rho$

Then we can write it out as

: $\frac{1}{(k-1)}\left( \frac{d\rho}{w^{k-1}} + d\left(\frac{\rho}{w^{k-1}}\right) \right)$

This shows that the two cohomology classes

: $\left[ \frac{dw}{w^k}\wedge \rho \right] = \left[ \frac{d\rho}{(k-1)w^{k-1}} \right]$

are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order $1$ and define the residue of $\omega$ as

: $\operatorname{Res}\left( \alpha \wedge \frac{dw} w + \beta \right) = \alpha|_X$

Example

For example, consider the curve $X \subset \mathbb{P}^2$ defined by the polynomial

: $F_t(x,y,z) = t(x^3 + y^3 + z^3) - 3xyz$

Then, we can apply the previous algorithm to compute the residue of

: $\omega = \frac{\Omega}{F_t} = \frac{x\,dy\wedge dz - y \, dx\wedge dz + z \, dx\wedge dy}{t(x^3 + y^3 + z^3) - 3xyz}$

Since

: $\begin{align}
-z\,dy\wedge\left( \frac{\partial F_t}{\partial x} \, dx + \frac{\partial F_t}{\partial y} \, dy + \frac{\partial F_t}{\partial z} \, dz \right) &=z\frac{\partial F_t}{\partial x} \, dx\wedge dy - z \frac{\partial F_t}{\partial z} \, dy\wedge dz \\
y \, dz\wedge\left(\frac{\partial F_t}{\partial x} \, dx + \frac{\partial F_t}{\partial y} \, dy + \frac{\partial F_t}{\partial z} \, dz\right) &= -y\frac{\partial F_t}{\partial x} \, dx\wedge dz - y \frac{\partial F_t}{\partial y} \, dy\wedge dz
\end{align}$

and

: $3F_t - z\frac{\partial F_t}{\partial x} - y\frac{\partial F_t}{\partial y} = x \frac{\partial F_t}{\partial x}$

we have that

: $\omega = \frac{y\,dz - z\,dy}{\partial F_t / \partial x} \wedge \frac{dF_t}{F_t} + \frac{3\,dy\wedge dz}{\partial F_t/\partial x}$

This implies that

: $\operatorname{Res}(\omega) = \frac{y\,dz - z\,dy}{\partial F_t / \partial x}$

References

=Introductory=

[https://www.ams.org/journals/bull/1982-06-02/S0273-0979-1982-14967-9/home.html Poincaré and algebraic geometry]
[https://web.archive.org/web/20200503190958/https://pdfs.semanticscholar.org/e6a1/9b2045dab24227affdb40fa634976b8a8125.pdf Infinitesimal variations of Hodge structure and the global Torelli problem] - Page 7 contains general computation formula using Cech cohomology

{{Citation| title=Introduction to residues and resultants |url=http://people.math.umass.edu/~cattani/chapter1.pdf}}
[https://mathoverflow.net/q/261860 Higher Dimensional Residues - Mathoverflow]

=Advanced=

{{ Citation | first=Liviu | last=Nicolaescu | title=Residues and Hodge Theory | url=https://www3.nd.edu/~lnicolae/residues.pdf}}
{{ Citation | first=Christian | last=Schnell | title=On Computing Picard-Fuchs Equations | url=https://www.math.stonybrook.edu/~cschnell/pdf/notes/picardfuchs.pdf }}

=References=

Boris A. Khesin, Robert Wendt, The Geometry of Infinite-dimensional Groups (2008) p. 171
{{ Citation | first=Andrzej | last=Weber | title=Leray Residue for Singular Varieties | url=http://matwbn.icm.edu.pl/ksiazki/bcp/bcp44/bcp44123.pdf}}

Category:Several complex variables