Residue form

A generalization of the concept of a residue of an analytic function of one complex variable to several complex variables. Let be a complex-analytic manifold (cf. Analytic manifold), let be an analytic submanifold of complex codimension one and let be a closed exterior differential form of class on with a first-order polar singularity on . The last condition means that for a function , holomorphic with respect to in a neighbourhood of a point and such that

the form belongs to the class . Under these conditions there exist, in a neighbourhood of an arbitrary point , forms , of class such that

where is a closed form of class that depends only on . The closed form on which is defined in a neighbourhood of any point by the restriction , is called the residue form of , and is denoted by

If the form is holomorphic, its residue form is holomorphic as well (cf. Holomorphic form). For instance, for , and the form

where and are holomorphic functions in , on , the residue form is

at the points where .

The residue formula corresponding to residue forms is:

where is an arbitrary cycle in of dimension equal to the degree of and — a cycle in — is the boundary of some chain in in general position with and intersecting along .

The composite residue form is defined by induction.

The residue class of a closed form in is the cohomology class on the submanifold produced by the residue forms of the forms of class in that are cohomologous with and have a first-order polar singularity on . The residue class of a form is denoted by . The residue class of a holomorphic form need not contain a holomorphic form, since in the general case it is not permissible to restrict the considerations to the ring of holomorphic forms but one rather has to consider the ring of closed forms. It is possible, however, if is a Stein manifold. The residue class does not depend on the choice of out of one cohomology class and realizes a homomorphism from the group of cohomology classes of the manifold to the group of cohomology classes of the manifold :

As for residue forms, the following residue formula is valid:

and the integral on the right-hand side of this equation is taken over any form in the residue class and is independent of it.

For references, see (, ,

to) Residue of an analytic function.

Comments

A differential form whose coefficients are distributions (generalized functions) is called a current. The theory of currents was developed largely by H. Federer [a5]. One can define the residue of a current. Currents associated to complex-analytic varieties have attracted a great deal of attention, see, e.g., [a6]–[a8].

Residue forms are also called residue currents. As mentioned above, these arise as generalizations to several variables of the residue, or rather the principal part, of an analytic function. There are several other ways of looking at residues: Let be holomorphic on a bounded domain except for a (finite) set of singularities . Let be a neighbourhood of with smooth boundary, if . Let be smooth, compactly supported on and holomorphic in a neighbourhood of , then

(a1)

is independent of as long as the are contained in the neighbourhood of where is holomorphic. If one takes for a function that equals in a small neighbourhood of , one obtains the usual residue. Note that represents a germ of a -closed -form at and is a -closed -form. Thus . Here denotes Dolbeault cohomology of germs of forms at . is called the cohomological residue. This can be generalized to several variables, will be a domain in , a closed subvariety of , to obtain a homomorphism

In another direction one would like to have an interpretation of (a1) for smooth , not necessarily closed. This can be done if one imposes the condition that is meromorphic on . One may write , with holomorphic, and assume by a partition of unity that is supported on only. Then the following limit exists independently of the representation of :

(a2)

It defines a current supported on . To obtain a sensible analogue of this for several variables is much harder.

A semi-meromorphic form on is a smooth differential form on that for every point admits a holomorphic function defined on some neighbourhood of such that is smooth at . A good generalization of (a2) should yield "residues" of a semi-meromorphic -form , which should be currents supported on . One needs the existence of limits of the form

with

Here and are disjoint subsets of , is a holomorphic mapping with , is an arbitrary compactly-supported smooth -form and is an admissible path, that is, and tend to with . In fact, the are -currents. For these two approaches, see [a4].

A third direction towards residue currents is by analytic continuation of holomorphic current-valued mappings. See [a2].

References

[a1]	L.A. Aizenberg, A.P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis" , Transl. Math. Monogr. , 58 , Amer. Math. Soc. (1983) (Translated from Russian)
[a2]	C.A. Berenstein, R. Gay, A. Yger, "Analytic continuation of currents and division problems" Forum Math. (1989) pp. 15–51
[a3]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978)
[a4]	M. Passare, "Residues, currents and their relation to ideals of holomorphic functions" Math. Scand. , 62 (1988) pp. 75–152
[a5]	H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108
[a6]	R. Harvey, "Holomorphic chains and their boundaries" R.O. Wells jr. (ed.) , Several Complex Variables , Proc. Symp. Pure Math. , 30:1 , Amer. Math. Soc. (1977) pp. 309–382
[a7]	H. Skoda, "A survey of the theory of closed, positive currents" Y.-T. Siu (ed.) , Complex Analysis of Several Variables , Proc. Symp. Pure Math. , 41 , Amer. Math. Soc. (1984) pp. 181–190
[a8]	E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian)