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

Historical construction

When Poincaré first introduced residues he was studying period integrals of the form

$\underset\iint \omega$ for $\Gamma \in H_2(\mathbb^2 - D)$

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

$\int_\gamma \text(\omega)$ for $\gamma \in H_1(D)$

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

$\omega = \frac$

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

$\text(\omega) = -\frac = \frac$

which are both cohomologous forms.

Construction

Preliminary definition

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

:$d: A^_(X) \to A^p_k(X)$

Define

:$\mathcal_k(X) = \frac$

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

$\mathcal_1(X) \subset \mathcal_2(X) \subset \cdots \subset \mathcal_n(X) = H^(\mathbb^-X)$

corresponding to the Hodge filtration.

Definition of residue

Consider an $(n-1)$-cycle $\gamma \in H_(X;\mathbb)$. 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 a rational $n$-form $\omega$ and get a number. If we write this as

:$\int_\omega : H_(X;\mathbb) \to \mathbb$

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

:$\operatorname(\omega) \in H^(X;\mathbb)$

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 $\mathbb^1$. This definition can be summarized as the map

$\text: H^(\mathbb^\setminus X) \to H^(X)$

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

:$\operatorname\left(\frac z + a\right) = 1$

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

:$\frac\wedge \rho$

Then we can write it out as

:$\frac\left( \frac + d\left(\frac\right) \right)$

This shows that the two cohomology classes

:\left \frac\wedge \rho \right= \left \frac \right/math>

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

:$\operatorname\left( \alpha \wedge \frac w + \beta \right) = \alpha, _X$

Example

For example, consider the curve $X \subset \mathbb^2$ 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

:$\omega = \frac = \frac$

Since
:$\begin -z\,dy\wedge\left( \frac \, dx + \frac \, dy + \frac \, dz \right) &=z\frac \, dx\wedge dy - z \frac \, dy\wedge dz \\ y \, dz\wedge\left(\frac \, dx + \frac \, dy + \frac \, dz\right) &= -y\frac \, dx\wedge dz - y \frac \, dy\wedge dz \end$
and
:$3F_t - z\frac - y\frac = x \frac$

we have that

:$\omega = \frac \wedge \frac + \frac$

This implies that

:$\operatorname(\omega) = \frac$

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 geometryInfinitesimal variations of Hodge structure and the global Torelli problem
- Page 7 contains general computation formula using Cech cohomology

*
Higher Dimensional Residues - Mathoverflow

Advanced

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*

References

* Boris A. Khesin, Robert Wendt, ''The Geometry of Infinite-dimensional Groups'' (2008) p. 171
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{{DEFAULTSORT:Poincare residue
Several complex variables