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

Historical construction

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

[math]\displaystyle{ \underset{\Gamma}\iint \omega }[/math] for [math]\displaystyle{ \Gamma \in H_2(\mathbb{P}^2 - D) }[/math]

where [math]\displaystyle{ \omega }[/math] was a rational differential form with poles along a divisor [math]\displaystyle{ D }[/math]. He was able to make the reduction of this integral to an integral of the form

[math]\displaystyle{ \int_\gamma \text{Res}(\omega) }[/math] for [math]\displaystyle{ \gamma \in H_1(D) }[/math]

where [math]\displaystyle{ \Gamma = T(\gamma) }[/math], sending [math]\displaystyle{ \gamma }[/math] to the boundary of a solid [math]\displaystyle{ \varepsilon }[/math]-tube around [math]\displaystyle{ \gamma }[/math] on the smooth locus [math]\displaystyle{ D^* }[/math]of the divisor. If

[math]\displaystyle{ \omega = \frac{q(x,y)dx\wedge dy}{p(x,y)} }[/math]

on an affine chart where [math]\displaystyle{ p(x,y) }[/math] is irreducible of degree [math]\displaystyle{ N }[/math] and [math]\displaystyle{ \deg q(x,y) \leq N-3 }[/math] (so there is no poles on the line at infinity ^{page 150}). Then, he gave a formula for computing this residue as

[math]\displaystyle{ \text{Res}(\omega) = -\frac{qdx}{\partial p / \partial y} = \frac{qdy}{\partial p / \partial x} }[/math]

which are both cohomologous forms.

Construction

Preliminary definition

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

[math]\displaystyle{ d: A^{p-1}_{k-1}(X) \to A^p_k(X) }[/math]

Define

[math]\displaystyle{ \mathcal{K}_k(X) = \frac{A^p_k(X)}{dA^{p-1}_{k-1}(X)} }[/math]

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

[math]\displaystyle{ \mathcal{K}_1(X) \subset \mathcal{K}_2(X) \subset \cdots \subset \mathcal{K}_n(X) = H^{n+1}(\mathbb{P}^{n+1}-X) }[/math]

corresponding to the Hodge filtration.

Definition of residue

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

[math]\displaystyle{ \int_{T(-)}\omega : H_{n-1}(X;\mathbb{C}) \to \mathbb{C} }[/math]

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

[math]\displaystyle{ \operatorname{Res}(\omega) \in H^{n-1}(X;\mathbb{C}) }[/math]

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

[math]\displaystyle{ \text{Res}: H^{n}(\mathbb{P}^{n}\setminus X) \to H^{n-1}(X) }[/math]

Algorithm for computing this class

There is a simple recursive method for computing the residues which reduces to the classical case of [math]\displaystyle{ n=1 }[/math]. Recall that the residue of a [math]\displaystyle{ 1 }[/math]-form

[math]\displaystyle{ \operatorname{Res}\left(\frac{dz} z + a\right) = 1 }[/math]

If we consider a chart containing [math]\displaystyle{ X }[/math] where it is the vanishing locus of [math]\displaystyle{ w }[/math], we can write a meromorphic [math]\displaystyle{ n }[/math]-form with pole on [math]\displaystyle{ X }[/math] as

[math]\displaystyle{ \frac{dw}{w^k}\wedge \rho }[/math]

Then we can write it out as

[math]\displaystyle{ \frac{1}{(k-1)}\left( \frac{d\rho}{w^{k-1}} + d\left(\frac{\rho}{w^{k-1}}\right) \right) }[/math]

This shows that the two cohomology classes

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

are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order [math]\displaystyle{ 1 }[/math] and define the residue of [math]\displaystyle{ \omega }[/math] as

[math]\displaystyle{ \operatorname{Res}\left( \alpha \wedge \frac{dw} w + \beta \right) = \alpha|_X }[/math]

Example

For example, consider the curve [math]\displaystyle{ X \subset \mathbb{P}^2 }[/math] defined by the polynomial

[math]\displaystyle{ F_t(x,y,z) = t(x^3 + y^3 + z^3) - 3xyz }[/math]

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

[math]\displaystyle{ \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} }[/math]

Since

[math]\displaystyle{ \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} }[/math]

and

[math]\displaystyle{ 3F_t - z\frac{\partial F_t}{\partial x} - y\frac{\partial F_t}{\partial y} = x \frac{\partial F_t}{\partial x} }[/math]

we have that

[math]\displaystyle{ \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} }[/math]

This implies that

[math]\displaystyle{ \operatorname{Res}(\omega) = \frac{y\,dz - z\,dy}{\partial F_t / \partial x} }[/math]

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 Cech cohomology

• Introduction to residues and resultants, http://people.math.umass.edu/~cattani/chapter1.pdf 
• Higher Dimensional Residues - Mathoverflow

Advanced

• Nicolaescu, Liviu, Residues and Hodge Theory, https://www3.nd.edu/~lnicolae/residues.pdf 
• Schnell, Christian, On Computing Picard-Fuchs Equations, 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
• Weber, Andrzej, Leray Residue for Singular Varieties, http://matwbn.icm.edu.pl/ksiazki/bcp/bcp44/bcp44123.pdf