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 defined by a degree polynomial and a rational -form on with a pole of order on, then we can construct a cohomology class. If we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues he was studying period integrals of the form
for
where was a rational differential form with poles along a divisor. He was able to make the reduction of this integral to an integral of the form
for
where, sending to the boundary of a solid -tube around on the smooth locus of the divisor. If
on an affine chart where is irreducible of degree and . Then, he gave a formula for computing this residue as
which are both cohomologous forms.

Construction

Preliminary definition

Given the setup in the introduction, let be the space of meromorphic -forms on which have poles of order up to. Notice that the standard differential sends
Define
as the rational de-Rham cohomology groups. They form a filtration
corresponding to the Hodge filtration.

Definition of residue

Consider an -cycle. We take a tube around that lies within the complement of. Since this is an -cycle, we can integrate a rational -form and get a number. If we write this as
then we get a linear transformation on the homology classes. Homology/cohomology duality implies that this is a cohomology class
which we call the residue. Notice if we restrict to the case, this is just the standard residue from complex analysis (although we extend our meromorphic -form to all of. This definition can be summarized as the map

Algorithm for computing this class

There is a simple recursive method for computing the residues which reduces to the classical case of. Recall that the residue of a -form
If we consider a chart containing where it is the vanishing locus of, we can write a meromorphic -form with pole on as
Then we can write it out as
This shows that the two cohomology classes
are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order and define the residue of as

Example

For example, consider the curve defined by the polynomial
Then, we can apply the previous algorithm to compute the residue of
Since
and
we have that
This implies that

Introductory

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