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
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the form belongs to the class
. Under these conditions there exist, in a neighbourhood
of an arbitrary point
, forms
,
of class
such that
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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
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If the form is holomorphic, its residue form is holomorphic as well (cf. Holomorphic form). For instance, for
,
and the form
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where and
are holomorphic functions in
,
on
, the residue form is
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at the points where .
The residue formula corresponding to residue forms is:
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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
:
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As for residue forms, the following residue formula is valid:
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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
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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
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with
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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) MR0735793 |
[a2] | C.A. Berenstein, R. Gay, A. Yger, "Analytic continuation of currents and division problems" Forum Math. (1989) pp. 15–51 MR0978974 Zbl 0651.32005 |
[a3] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001 |
[a4] | M. Passare, "Residues, currents and their relation to ideals of holomorphic functions" Math. Scand. , 62 (1988) pp. 75–152 MR0961584 Zbl 0633.32005 |
[a5] | H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108 MR0257325 Zbl 0176.00801 |
[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 MR0447619 Zbl 0374.32002 |
[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 MR0740881 Zbl 0539.32006 |
[a8] | E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002 |
Residue form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residue_form&oldid=23956