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 ( ω ) ∈ H n − 1 ( X ; C ) {\displaystyle \operatorname {Res} (\omega )\in H^{n-1}(X;\mathbb {C} )} . If n = 1 {\displaystyle n=1} we recover the classical residue construction.
Historical construction When Poincaré first introduced residues he was studying period integrals of the form ∬ Γ ω {\displaystyle {\underset {\Gamma }{\iint }}\omega } for Γ ∈ H 2 ( P 2 − D ) {\displaystyle \Gamma \in H_{2}(\mathbb {P} ^{2}-D)} where ω {\displaystyle \omega } was a rational differential form with poles along a divisor D {\displaystyle D} . He was able to make the reduction of this integral to an integral of the form ∫ γ Res ( ω ) {\displaystyle \int _{\gamma }{\text{Res}}(\omega )} for γ ∈ H 1 ( D ) {\displaystyle \gamma \in H_{1}(D)} where Γ = T ( γ ) {\displaystyle \Gamma =T(\gamma )} , sending γ {\displaystyle \gamma } to the boundary of a solid ε {\displaystyle \varepsilon } -tube around γ {\displaystyle \gamma } on the smooth locus D ∗ {\displaystyle D^{*}} of the divisor. If ω = q ( x , y ) d x ∧ d y p ( x , y ) {\displaystyle \omega ={\frac {q(x,y)dx\wedge dy}{p(x,y)}}} on an affine chart where p ( x , y ) {\displaystyle p(x,y)} is irreducible of degree N {\displaystyle N} and deg q ( x , y ) ≤ N − 3 {\displaystyle \deg q(x,y)\leq N-3} (so there is no poles on the line at infinity page 150). Then, he gave a formula for computing this residue as Res ( ω ) = − q d x ∂ p / ∂ y = q d y ∂ p / ∂ x {\displaystyle {\text{Res}}(\omega )=-{\frac {qdx}{\partial p/\partial y}}={\frac {qdy}{\partial p/\partial x}}} which are both cohomologous forms.
Construction
Preliminary definition Given the setup in the introduction, let A k p ( X ) {\displaystyle A_{k}^{p}(X)} be the space of meromorphic p {\displaystyle p} -forms on P n {\displaystyle \mathbb {P} ^{n}} which have poles of order up to k {\displaystyle k} . Notice that the standard differential d {\displaystyle d} sends
d : A k − 1 p − 1 ( X ) → A k p ( X ) {\displaystyle d:A_{k-1}^{p-1}(X)\to A_{k}^{p}(X)}
Define
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