Poincaré residue

Template:Short description 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 $\mathrm{Res}(\omega )\in H^{n-1}(X;ℂ)$. If $n=1$ we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues he was studying period integrals of the form

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

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 $\mathrm{\Gamma }=T(\gamma )$, sending $\gamma$ to the boundary of a solid $\epsilon$-tube around $\gamma$ on the smooth locus $D^{\ast }$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 $\mathrm{deg}q(x,y)\le N-3$ (so there is no poles on the line at infinity ^{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 $A_k^p(X)$ 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

$d:A_{k-1}^{p-1}(X)\to A_k^p(X)$

Define

${𝒦}_k(X)=\frac{A_k^p(X)}{d{A}_{k-1}^{p-1}(X)}$

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

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

corresponding to the Hodge filtration.

Definition of residue

Consider an $(n-1)$-cycle $\gamma \in H_{n-1}(X;ℂ)$. 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

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

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

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

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

$\mathrm{Res}(\frac{dz}{z}+a)=1$

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

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

This shows that the two cohomology classes

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

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

$\mathrm{Res}(\alpha \wedge \frac{dw}{w}+\beta )=\alpha {|}_X$

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

$\omega =\frac{\mathrm{\Omega }}{F_t}=\frac{xdy\wedge dz-ydx\wedge dz+zdx\wedge dy}{t(x^3+y^3+z^3)-3xyz}$

Since

$\begin{array}{cc}-zdy\wedge (\frac{\partial F_t}{\partial x}dx+\frac{\partial F_t}{\partial y}dy+\frac{\partial F_t}{\partial z}dz)& =z\frac{\partial F_t}{\partial x}dx\wedge dy-z\frac{\partial F_t}{\partial z}dy\wedge dz\\ ydz\wedge (\frac{\partial F_t}{\partial x}dx+\frac{\partial F_t}{\partial y}dy+\frac{\partial F_t}{\partial z}dz)& =-y\frac{\partial F_t}{\partial x}dx\wedge dz-y\frac{\partial F_t}{\partial y}dy\wedge dz\end{array}$

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

$\omega =\frac{ydz-zdy}{\partial F_t/\partial x}\wedge \frac{d{F}_t}{F_t}+\frac{3dy\wedge dz}{\partial F_t/\partial x}$

This implies that

$\mathrm{Res}(\omega )=\frac{ydz-zdy}{\partial F_t/\partial x}$

See also

• Grothendieck residue
• Leray residue
• Bott residue
• Sheaf of logarithmic differential forms
• normal crossing singularity
• Adjunction formula#Poincare residue
• Hodge structure
• Jacobian ideal

References

Introductory

• Poincaré and algebraic geometry
• Infinitesimal variations of Hodge structure and the global Torelli problem - Page 7 contains general computation formula using Čech cohomology

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• Higher Dimensional Residues - Mathoverflow

Advanced

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References

• Boris A. Khesin, Robert Wendt, The Geometry of Infinite-dimensional Groups (2008) p. 171
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