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 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 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.

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


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

Introductory

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


 * Higher Dimensional Residues - Mathoverflow
 * Higher Dimensional Residues - Mathoverflow