1 Answers
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 ⊂ P n {\displaystyle X\subset \mathbb {P} ^{n}} defined by a degree d {\displaystyle d} polynomial F {\displaystyle F} and a rational n {\displaystyle n} -form ω {\displaystyle \omega } on P n {\displaystyle \mathbb {P} ^{n}} with a pole of order k > 0 {\displaystyle k>0} on X {\displaystyle X} , then we can construct a cohomology class Res ∈ H n − 1 {\displaystyle \operatorname {Res} \in H^{n-1}}. If n = 1 {\displaystyle n=1} we recover the classical residue construction.