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. Ifon 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 aswhich 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 filtrationcorresponding 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
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
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 Čech cohomology
Higher Dimensional Residues - Mathoverflow
Advanced
References Boris A. Khesin, Robert Wendt, The Geometry of Infinite-dimensional Groups (2008) p. 171