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

Historical construction

When Poincaré first introduced residues^[1] he was studying period integrals of the form

${\displaystyle {\underset {\Gamma }{\iint }}\omega }$ for ${\displaystyle \Gamma \in H_{2}(\mathbb {P} ^{2}-D)}$

where ${\displaystyle \omega }$ was a rational differential form with poles along a divisor ${\displaystyle D}$. He was able to make the reduction of this integral to an integral of the form

${\displaystyle \int _{\gamma }{\text{Res}}(\omega )}$ for ${\displaystyle \gamma \in H_{1}(D)}$

where ${\displaystyle \Gamma =T(\gamma )}$, sending ${\displaystyle \gamma }$ to the boundary of a solid ${\displaystyle \varepsilon }$-tube around ${\displaystyle \gamma }$ on the smooth locus ${\displaystyle D^{*}}$of the divisor. If

${\displaystyle \omega ={\frac {q(x,y)dx\wedge dy}{p(x,y)}}}$

on an affine chart where ${\displaystyle p(x,y)}$ is irreducible of degree ${\displaystyle N}$ and ${\displaystyle \deg q(x,y)\leq N-3}$ (so there is no poles on the line at infinity^{[2] page 150}). Then, he gave a formula for computing this residue as

${\displaystyle {\text{Res}}(\omega )=-{\frac {qdx}{\partial p/\partial y}}={\frac {qdy}{\partial p/\partial x}}}$

which are both cohomologous forms.

Construction

Example

For example, consider the curve ${\displaystyle X\subset \mathbb {P} ^{2}}$ defined by the polynomial

${\displaystyle F_{t}(x,y,z)=t(x^{3}+y^{3}+z^{3})-3xyz}$

Then, we can apply the previous algorithm to compute the residue of

${\displaystyle \omega ={\frac {\Omega }{F_{t}}}={\frac {x\,dy\wedge dz-y\,dx\wedge dz+z\,dx\wedge dy}{t(x^{3}+y^{3}+z^{3})-3xyz}}}$

Since

${\displaystyle {\begin{aligned}-z\,dy\wedge \left({\frac {\partial F_{t}}{\partial x}}\,dx+{\frac {\partial F_{t}}{\partial y}}\,dy+{\frac {\partial F_{t}}{\partial z}}\,dz\right)&=z{\frac {\partial F_{t}}{\partial x}}\,dx\wedge dy-z{\frac {\partial F_{t}}{\partial z}}\,dy\wedge dz\\y\,dz\wedge \left({\frac {\partial F_{t}}{\partial x}}\,dx+{\frac {\partial F_{t}}{\partial y}}\,dy+{\frac {\partial F_{t}}{\partial z}}\,dz\right)&=-y{\frac {\partial F_{t}}{\partial x}}\,dx\wedge dz-y{\frac {\partial F_{t}}{\partial y}}\,dy\wedge dz\end{aligned}}}$

and

${\displaystyle 3F_{t}-z{\frac {\partial F_{t}}{\partial x}}-y{\frac {\partial F_{t}}{\partial y}}=x{\frac {\partial F_{t}}{\partial x}}}$

we have that

${\displaystyle \omega ={\frac {y\,dz-z\,dy}{\partial F_{t}/\partial x}}\wedge {\frac {dF_{t}}{F_{t}}}+{\frac {3\,dy\wedge dz}{\partial F_{t}/\partial x}}}$

This implies that

${\displaystyle \operatorname {Res} (\omega )={\frac {y\,dz-z\,dy}{\partial F_{t}/\partial x}}}$

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