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
defined by a degree
polynomial
and a rational
-form
on
with a pole of order
on
, then we can construct a cohomology class
. If
we recover the classical residue construction.
Historical construction
[edit]
When Poincaré first introduced residues[1] he was studying period integrals of the form
for ![{\displaystyle \Gamma \in H_{2}(\mathbb {P} ^{2}-D)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f72a885cc2e1fc604c6ce3da6b9f391d1486b2e)
where
was a rational differential form with poles along a divisor
. He was able to make the reduction of this integral to an integral of the form
for ![{\displaystyle \gamma \in H_{1}(D)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b22cb291632389152597881c876be72e320c309f)
where
, sending
to the boundary of a solid
-tube around
on the smooth locus
of the divisor. If
![{\displaystyle \omega ={\frac {q(x,y)dx\wedge dy}{p(x,y)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43b3d0c86de837c787420f0a5c7982b39c53a72)
on an affine chart where
is irreducible of degree
and
(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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9c3f3a5bac6ea4a7d941dd1408632e388f18bf)
which are both cohomologous forms.
Preliminary definition
[edit]
Given the setup in the introduction, let
be the space of meromorphic
-forms on
which have poles of order up to
. Notice that the standard differential
sends
![{\displaystyle d:A_{k-1}^{p-1}(X)\to A_{k}^{p}(X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7f3a5b57a30d25876a07d18ecd8571932af3dc)
Define
![{\displaystyle {\mathcal {K}}_{k}(X)={\frac {A_{k}^{p}(X)}{dA_{k-1}^{p-1}(X)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc46b2d54aaac600871a382ec894d8ea0bfb4a2b)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb8b9372ee56a7c7a14ad568662d80ce3e3a2ce)
corresponding to the Hodge filtration.
Definition of residue
[edit]
Consider an
-cycle
. We take a tube
around
(which is locally isomorphic to
) that lies within the complement of
. Since this is an
-cycle, we can integrate a rational
-form
and get a number. If we write this as
![{\displaystyle \int _{T(-)}\omega :H_{n-1}(X;\mathbb {C} )\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a2058554ed5e86a32e4767a4dcf4c2533aa7d5)
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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc2403182ed66a346adf11ad95e6ece17dae37e)
which we call the residue. Notice if we restrict to the case
, this is just the standard residue from complex analysis (although we extend our meromorphic
-form to all of
. This definition can be summarized as the map
![{\displaystyle {\text{Res}}:H^{n}(\mathbb {P} ^{n}\setminus X)\to H^{n-1}(X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/372fa148d97326025fd19f65e4e6dd3bbd26abba)
Algorithm for computing this class
[edit]
There is a simple recursive method for computing the residues which reduces to the classical case of
. Recall that the residue of a
-form
![{\displaystyle \operatorname {Res} \left({\frac {dz}{z}}+a\right)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e44dc1daa754bd6ca50afcec3eb3e0826480e580)
If we consider a chart containing
where it is the vanishing locus of
, we can write a meromorphic
-form with pole on
as
![{\displaystyle {\frac {dw}{w^{k}}}\wedge \rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cb35568c79314590dfa52358250640ba4e3e26)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe925bc7e67322cb72c21812b60daa9126ed9b8)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d43255b479c72a827155e292d29e75b40a85d2)
are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order
and define the residue of
as
![{\displaystyle \operatorname {Res} \left(\alpha \wedge {\frac {dw}{w}}+\beta \right)=\alpha |_{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47905e0aefbe2576104e5c559c5de337a29e6f1)
For example, consider the curve
defined by the polynomial
![{\displaystyle F_{t}(x,y,z)=t(x^{3}+y^{3}+z^{3})-3xyz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ad3fb14149c95c7543955c71247c293506c759)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9158cf657841778fe35976eba7446e4d164bdec)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c28c35fdf8b479f0377d4d781d82e66aaa10b66)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/600d6150ade63e5aceab22e8e6383fdc8181163b)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5639529712ab1ed6c86da92ec967a4319f00d98e)
This implies that
![{\displaystyle \operatorname {Res} (\omega )={\frac {y\,dz-z\,dy}{\partial F_{t}/\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8698906b2042b2e16e4c70b85898641993641258)