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 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

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

  for  

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  

where  , sending   to the boundary of a solid  -tube around   on the smooth locus  of the divisor. If

 

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

 

which are both cohomologous forms.

Construction

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Preliminary definition

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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

 

Define

 

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

 

corresponding to the Hodge filtration.

Definition of residue

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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

 

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

 

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

 

Algorithm for computing this class

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

 

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

 

Then we can write it out as

 

This shows that the two cohomology classes

 

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

 

Example

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For example, consider the curve   defined by the polynomial

 

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

 

Since

 

and

 

we have that

 

This implies that

 

See also

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References

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  1. ^ Poincaré, H. (1887). "Sur les résidus des intégrales doubles". Acta Mathematica (in French). 9: 321–380. doi:10.1007/BF02406742. ISSN 0001-5962.
  2. ^ Griffiths, Phillip A. (1982). "Poincaré and algebraic geometry". Bulletin of the American Mathematical Society. 6 (2): 147–159. doi:10.1090/S0273-0979-1982-14967-9. ISSN 0273-0979.

Introductory

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Advanced

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References

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