Poincaré residue explained

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 P ^{n} defined by a degree d polynomial F and a rational n -form \omega on 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; C ) . If n =1 we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues^[1] he was studying period integrals of the form

\underset{\Gamma}\iint \omega for \Gamma \in H ^{2} _{2 ( P} - 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} 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 = (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^{[2] page 150}). Then, he gave a formula for computing this residue as

Res (\omega) =- (qdx)/(\partial p / \partial y) = (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 P ^{n} which have poles of order up to k . Notice that the standard differential d sendsd: A ^{p -1} _{k -1} (X) \to A ^{p} _{k(X)}

Define

l{K} _{k(X)} = (A ^{p} _{k(X)})/(dA ^{p -1} _{k -1} (X))

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

l{K} _{1(X)} \subset l{K} _{2(X)} \subset … \subset l{K} _{n(X)} = H ^{n+1} ( 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 x 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

\operatorname{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 P ^{1} . This definition can be summarized as the map

Res : H ^{n} ( 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( (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(dw)/(w ^{k}) \wedge \rho

Then we can write it out as

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

This shows that the two cohomology classes

\left[ (dw)/(w ^{k}) \wedge \rho \right] = \left[ (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 (dw)/(w) + \beta \right) = \alpha| _{X}

Example

For example, consider the curve

X \subset P ^{2} defined by the polynomialF _{t(x,y,z)} = t(x ^{3} + y ^{3} + z ^{3)} - 3 xyz

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

\omega = (\Omega)/(F _{t}) = (xdy\wedge dz - y dx\wedge dz + z dx\wedge dy)/(t(x ^{3} + y ^{3} + z ^{3)} -3 xyz)

Since

\begin{align} - zdy\wedge\left( (\partial F _{t})/(\partial x) dx + (\partial F _{t})/(\partial y) dy + (\partial F _{t})/(\partial z) dz \right) & = z (\partial F _{t})/(\partial x) dx\wedge dy - z (\partial F _{t})/(\partial z) dy\wedge dz \\ y dz\wedge\left( (\partial F _{t})/(\partial x) dx + (\partial F _{t})/(\partial y) dy + (\partial F _{t})/(\partial z) dz\right) & =- y (\partial F _{t})/(\partial x) dx\wedge dz - y (\partial F _{t})/(\partial y) dy\wedge dz \end{align} and3 F _{t} - z (\partial F _{t})/(\partial x) - y (\partial F _{t})/(\partial y) = x (\partial F _{t})/(\partial x)

we have that

\omega = (ydz - zdy)/(\partial F _{t} / \partial x) \wedge (dF _{t})/(F _{t}) + (3 dy\wedge dz)/(\partial F _{t/\partial} x)

This implies that

\operatorname{Res}(\omega) = (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

1. Poincaré. H.. 1887. Sur les résidus des intégrales doubles. Acta Mathematica. FR. 9. 321–380. 10.1007/BF02406742. 0001-5962. free.
2. Griffiths. Phillip A.. 1982. Poincaré and algebraic geometry. Bulletin of the American Mathematical Society. en. 6. 2. 147–159. 10.1090/S0273-0979-1982-14967-9. 0273-0979. free.

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

References

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