Poincaré residue

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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 Xn defined by a degree d polynomial F and a rational n-form ω on n with a pole of order k>0 on X, then we can construct a cohomology class Res(ω)Hn1(X;). If n=1 we recover the classical residue construction.

Historical construction

When Poincaré first introduced residues[1] he was studying period integrals of the form

Γω

for

ΓH2(2D)

where

ω

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

γRes(ω)

for

γH1(D)

where

Γ=T(γ)

, sending

γ

to the boundary of a solid

ε

-tube around

γ

on the smooth locus

D*

of the divisor. If

ω=q(x,y)dxdyp(x,y)

on an affine chart where

p(x,y)

is irreducible of degree

N

and

degq(x,y)N3

(so there is no poles on the line at infinity[2] page 150). Then, he gave a formula for computing this residue as

Res(ω)=qdxp/y=qdyp/x

which are both cohomologous forms.

Construction

Preliminary definition

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

d:Ak1p1(X)Akp(X)

Define

𝒦k(X)=Akp(X)dAk1p1(X)

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

𝒦1(X)𝒦2(X)𝒦n(X)=Hn+1(n+1X)

corresponding to the Hodge filtration.

Definition of residue

Consider an (n1)-cycle γHn1(X;). We take a tube T(γ) around γ (which is locally isomorphic to γ×S1) that lies within the complement of X. Since this is an n-cycle, we can integrate a rational n-form ω and get a number. If we write this as

T()ω:Hn1(X;)

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

Res(ω)Hn1(X;)

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

1

. This definition can be summarized as the map

Res:Hn(nX)Hn1(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

Res(dzz+a)=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

dwwkρ

Then we can write it out as

1(k1)(dρwk1+d(ρwk1))

This shows that the two cohomology classes

[dwwkρ]=[dρ(k1)wk1]

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 ω as

Res(αdww+β)=α|X

Example

For example, consider the curve X2 defined by the polynomial

Ft(x,y,z)=t(x3+y3+z3)3xyz

Then, we can apply the previous algorithm to compute the residue of

ω=ΩFt=xdydzydxdz+zdxdyt(x3+y3+z3)3xyz

Since

zdy(Ftxdx+Ftydy+Ftzdz)=zFtxdxdyzFtzdydzydz(Ftxdx+Ftydy+Ftzdz)=yFtxdxdzyFtydydz

and

3FtzFtxyFty=xFtx

we have that

ω=ydzzdyFt/xdFtFt+3dydzFt/x

This implies that

Res(ω)=ydzzdyFt/x

See also

References

  1. Poincaré, H. (1887). "Sur les résidus des intégrales doubles" (in FR). Acta Mathematica 9: 321–380. doi:10.1007/BF02406742. ISSN 0001-5962. https://projecteuclid.org/euclid.acta/1485888747. 
  2. Griffiths, Phillip A. (1982). "Poincaré and algebraic geometry" (in en). Bulletin of the American Mathematical Society 6 (2): 147–159. doi:10.1090/S0273-0979-1982-14967-9. ISSN 0273-0979. https://www.ams.org/bull/1982-06-02/S0273-0979-1982-14967-9/. 

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