Poincaré residue

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

Preliminary definition

Given the setup in the introduction, let ${\displaystyle A_{k}^{p}(X)}$ be the space of meromorphic ${\displaystyle p}$-forms on ${\displaystyle \mathbb {P} ^{n}}$ which have poles of order up to ${\displaystyle k}$. Notice that the standard differential ${\displaystyle d}$ sends

${\displaystyle d:A_{k-1}^{p-1}(X)\to A_{k}^{p}(X)}$

Define

${\displaystyle {\mathcal {K}}_{k}(X)={\frac {A_{k}^{p}(X)}{dA_{k-1}^{p-1}(X)}}}$

as the rational de-Rham cohomology groups. They form a filtration

${\displaystyle {\mathcal {K}}_{1}(X)\subset {\mathcal {K}}_{2}(X)\subset \cdots \subset {\mathcal {K}}_{n}(X)=H^{n+1}(\mathbb {P} ^{n+1}-X)}$

corresponding to the Hodge filtration.

Definition of residue

Consider an ${\displaystyle (n-1)}$-cycle ${\displaystyle \gamma \in H_{n-1}(X;\mathbb {C} )}$. We take a tube ${\displaystyle T(\gamma )}$ around ${\displaystyle \gamma }$ (which is locally isomorphic to ${\displaystyle \gamma \times S^{1}}$) that lies within the complement of ${\displaystyle X}$. Since this is an ${\displaystyle n}$-cycle, we can integrate a rational ${\displaystyle n}$-form ${\displaystyle \omega }$ and get a number. If we write this as

${\displaystyle \int _{T(-)}\omega :H_{n-1}(X;\mathbb {C} )\to \mathbb {C} }$

then we get a linear transformation on the homology classes. Homology/cohomology duality implies that this is a cohomology class

${\displaystyle \operatorname {Res} (\omega )\in H^{n-1}(X;\mathbb {C} )}$

which we call the residue. Notice if we restrict to the case ${\displaystyle n=1}$, this is just the standard residue from complex analysis (although we extend our meromorphic ${\displaystyle 1}$-form to all of ${\displaystyle \mathbb {P} ^{1}}$. This definition can be summarized as the map

${\displaystyle {\text{Res}}:H^{n}(\mathbb {P} ^{n}\setminus X)\to H^{n-1}(X)}$

Algorithm for computing this class

There is a simple recursive method for computing the residues which reduces to the classical case of ${\displaystyle n=1}$. Recall that the residue of a ${\displaystyle 1}$-form

${\displaystyle \operatorname {Res} \left({\frac {dz}{z}}+a\right)=1}$

If we consider a chart containing ${\displaystyle X}$ where it is the vanishing locus of ${\displaystyle w}$, we can write a meromorphic ${\displaystyle n}$-form with pole on ${\displaystyle X}$ as

${\displaystyle {\frac {dw}{w^{k}}}\wedge \rho }$

Then we can write it out as

${\displaystyle {\frac {1}{(k-1)}}\left({\frac {d\rho }{w^{k-1}}}+d\left({\frac {\rho }{w^{k-1}}}\right)\right)}$

This shows that the two cohomology classes

${\displaystyle \left[{\frac {dw}{w^{k}}}\wedge \rho \right]=\left[{\frac {d\rho }{(k-1)w^{k-1}}}\right]}$

are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order ${\displaystyle 1}$ and define the residue of ${\displaystyle \omega }$ as

${\displaystyle \operatorname {Res} \left(\alpha \wedge {\frac {dw}{w}}+\beta \right)=\alpha |_{X}}$

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

Introductory

• Poincaré and algebraic geometry
• Infinitesimal variations of Hodge structure and the global Torelli problem - Page 7 contains general computation formula using Cech cohomology

• Introduction to residues and resultants (PDF)
• Higher Dimensional Residues - Mathoverflow

Advanced

• Nicolaescu, Liviu, Residues and Hodge Theory (PDF)
• Schnell, Christian, On Computing Picard-Fuchs Equations (PDF)

References

• Boris A. Khesin, Robert Wendt, The Geometry of Infinite-dimensional Groups (2008) p. 171
• Weber, Andrzej, Leray Residue for Singular Varieties (PDF)