Poincaré residue
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poincaré residue is a generalization, to
several complex variable The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
s and
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
theory, of the
residue at a pole In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for ...
of
complex function theory Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. 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 Leray or LeRay is a surname. Notable people with the surname include: * David Leray (born 1984), French footballer * Francis Xavier Leray (1825–1887), American prelate of the Roman Catholic Church *Jean Leray Jean Leray (; 7 November 1906 – 1 ...
*
Bott residue In mathematics, the Bott residue formula, introduced by , describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. Statement If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', t ...
*
Sheaf of logarithmic differential forms In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ '' ...
*
normal crossing singularity In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings bein ...
* Adjunction formula#Poincare residue *
Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
*
Jacobian ideal In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let \mathcal(x_1,\ldots,x_n) denote the ring of smooth functions in n variables and f a function in the ring. The Jacobi ...


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

* *


References

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