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

Construction

Preliminary definition

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

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

Define

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

as the rational de-Rham cohomology groups.

Definition of residue

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

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

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

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

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

Algorithm for computing this class

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

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

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

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

Then we can write it out as

{\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

\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 $1$ and define the residue of $\omega$ as

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

Example

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

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

\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

{\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

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

\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

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

References

Introductory

Advanced

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

Reference

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

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