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 d polynomial F and a rational n-form \omega on with a pole of order k > 0 on X, then we can construct a cohomology class. If n=1 we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues he was studying period integrals of the form"for"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"for"where, sending \gamma to the boundary of a solid \varepsilon-tube around \gamma on the smooth locus D^*of the divisor. If""on an affine chart where p(x,y) is irreducible of degree N 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 A^p_k(X) be the space of meromorphic p-forms on which have poles of order up to k. Notice that the standard differential d sends Define as the rational de-Rham cohomology groups. They form a filtration corresponding to the Hodge filtration.

Definition of residue

Consider an (n-1)-cycle. We take a tube T(\gamma) around \gamma (which is locally isomorphic to ) 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 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 n=1, this is just the standard residue from complex analysis (although we extend our meromorphic 1-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 n=1. Recall that the residue of a 1-form 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 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 1 and define the residue of \omega 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

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