In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poincaré residue is a generalization, to
several complex variable
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
s and
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
theory, of the
residue at a pole
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for ...
of
complex function theory
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. 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
(so there is no poles on the line at infinity
page 150). 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
(which is locally isomorphic to
) 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
: