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

1. Poincaré, H. (1887). "Sur les résidus des intégrales doubles". Acta Mathematica (in French). 9: 321–380. doi:10.1007/BF02406742. ISSN 0001-5962.
2. Griffiths, Phillip A. (1982). "Poincaré and algebraic geometry". Bulletin of the American Mathematical Society. 6 (2): 147–159. doi:10.1090/S0273-0979-1982-14967-9. ISSN 0273-0979.

Introductory

• Poincaré and algebraic geometry
• - 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)