Residue form

A generalization of the concept of a residue of an analytic function of one complex variable to several complex variables. Let $ X $ be a complex-analytic manifold (cf. Analytic manifold), let $ S $ be an analytic submanifold of complex codimension one and let $ \omega ( x) $ be a closed exterior differential form of class $ C ^ \infty $ on $ X \setminus S $ with a first-order polar singularity on $ S $. The last condition means that for a function $ s( x, y) $, holomorphic with respect to $ x $ in a neighbourhood $ U _ {y} $ of a point $ y \in S $ and such that

$$ S \cap U _ {y} = \ \{ {x } : {s ( x, y) = 0 } \} ,\ \ ds \not\equiv 0 \textrm{ if } x = y, $$

the form $ \omega ( x) \cdot s ( x, y) $ belongs to the class $ C ^ \infty ( U _ {y} ) $. Under these conditions there exist, in a neighbourhood $ U $ of an arbitrary point $ y \in S $, forms $ \psi ( x, y) $, $ \theta ( x, y) $ of class $ C ^ \infty $ such that

$$ \omega ( x) = \ \frac{ds ( x, y) }{s ( x, y) } \wedge \psi ( x, y) + \theta ( x, y), $$

where $ \psi ( x, y) \mid _ {S \cap U _ {y} } $ is a closed form of class $ C ^ \infty $ that depends only on $ \omega $. The closed form on $ S $ which is defined in a neighbourhood of any point $ y \in S $ by the restriction $ \psi ( x, y) \mid _ {S \cap U _ {y} } $, is called the residue form of $ \omega $, and is denoted by

$$ \mathop{\rm res} [ \omega ] = \ \left . \frac{s \omega }{ds } \right | _ {S} . $$

If the form $ \omega $ is holomorphic, its residue form is holomorphic as well (cf. Holomorphic form). For instance, for $ X = \mathbf C ^ {n} $, $ S = \{ {x \in \mathbf C ^ {n} } : {s( x) = 0 } \} $ and the form

$$ \omega ( x) = \ \frac{f ( x) }{s ( x) } \ dx _ {1} \wedge \dots \wedge dx _ {n} , $$

where $ f $ and $ s $ are holomorphic functions in $ \mathbf C ^ {n} $, $ \mathop{\rm grad} s \neq 0 $ on $ S $, the residue form is

$$ \mathop{\rm res} [ \omega ] = \ \left . \left [ f( \frac{x)}{ {\partial s } / {\partial x _ {j} } } \right ] \right | _ {S} \ dx _ {1} \wedge \dots \wedge [ dx _ {j} ] \wedge \dots \wedge dx _ {m} $$

at the points where $ d s/ d x _ {j} \neq 0 $.

The residue formula corresponding to residue forms is:

$$ \int\limits _ {\delta \gamma } \omega = 2 \pi i \int\limits _ \gamma \mathop{\rm res} [ \omega ], $$

where $ \gamma $ is an arbitrary cycle in $ S $ of dimension equal to the degree of $ \mathop{\rm res} [ \omega ] $ and $ \delta \gamma $— a cycle in $ X \setminus S $— is the boundary of some chain in $ X $ in general position with $ S $ and intersecting $ S $ along $ \gamma $.

The composite residue form $ \mathop{\rm res} ^ {m} [ \omega ] $ is defined by induction.

The residue class of a closed form $ \omega $ in $ X \setminus S $ is the cohomology class on the submanifold $ S $ produced by the residue forms of the forms of class $ C ^ \infty $ in $ X \setminus S $ that are cohomologous with $ \omega $ and have a first-order polar singularity on $ S $. The residue class of a form $ \omega $ is denoted by $ \mathop{\rm Res} [ \omega ] $. 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 $ X $ is a Stein manifold. The residue class $ \mathop{\rm Res} [ \omega ] $ does not depend on the choice of $ \omega $ out of one cohomology class and realizes a homomorphism from the group of cohomology classes of the manifold $ X \setminus S $ to the group of cohomology classes of the manifold $ S $:

$$ \mathop{\rm Res} : H ^ {*} ( X \setminus S) \rightarrow H ^ {*} ( S). $$

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

$$ \int\limits _ {\delta \gamma } \omega = 2 \pi i \int\limits _ \gamma \mathop{\rm Res} [ \omega ], $$

and the integral on the right-hand side of this equation is taken over any form in the residue class $ \mathop{\rm Res} [ \omega ] $ 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 $ g $ be holomorphic on a bounded domain $ D \subset \mathbf C $ except for a (finite) set of singularities $ S = \{ a _ {1} \dots a _ {m} \} $. Let $ D _ {j} $ be a neighbourhood of $ a _ {j} $ with smooth boundary, $ a _ {i} \notin D _ {j} $ if $ i \neq j $. Let $ \psi $ be smooth, compactly supported on $ D $ and holomorphic in a neighbourhood of $ S $, then

$$ \tag{a1 } \mathop{\rm Res} ( g)( \psi ) = \ \sum _ { i } \int\limits _ {\partial D _ {j} } g( z) \psi ( z) dz = - \int\limits _ { D } g \overline \partial \; \psi dz $$

is independent of $ D _ {j} $ as long as the $ D _ {j} $ are contained in the neighbourhood of $ S $ where $ \psi $ is holomorphic. If one takes for $ \psi $ a function that equals $ 1 $ in a small neighbourhood of $ a _ {j} $, one obtains the usual residue. Note that $ \psi dz $ represents a germ of a $ \overline \partial \; $- closed $ ( 1, 0) $- form at $ S $ and $ g $ is a $ \overline \partial \; $- closed $ ( 0, 0) $- form. Thus $ \mathop{\rm Res} : H ^ {0,0 } ( D \setminus S) \rightarrow \mathop{\rm Hom} ( H ^ {1, 0 } ( S), \mathbf C ) $. Here $ H ^ {\star, \star } ( S) $ denotes Dolbeault cohomology of germs of forms at $ S $. $ \mathop{\rm Res} ( g) $ is called the cohomological residue. This can be generalized to several variables, $ D $ will be a domain in $ \mathbf C ^ {n} $, $ S $ a closed subvariety of $ D $, to obtain a homomorphism

$$ \mathop{\rm Res} : H ^ {p, q+ 1 } ( D \setminus S) \rightarrow \ \mathop{\rm Hom} ( H ^ {n- p, n- q- 1 } ( S), \mathbf C ) . $$

In another direction one would like to have an interpretation of (a1) for smooth $ \psi $, not necessarily closed. This can be done if one imposes the condition that $ g $ is meromorphic on $ D $. One may write $ g = g _ {1} / g _ {2} $, with $ g _ {j} $ holomorphic, and assume by a partition of unity that $ \psi $ is supported on $ D _ {j} $ only. Then the following limit exists independently of the representation of $ g $:

$$ \tag{a2 } \lim\limits _ {\epsilon \rightarrow 0 } \int\limits _ {| g _ {2} | = \epsilon } g( z) \psi ( z) dz . $$

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

A semi-meromorphic form on $ D \setminus S $ is a smooth differential form $ \omega $ on $ D \setminus S $ that for every point $ z \in D $ admits a holomorphic function $ g $ defined on some neighbourhood of $ z $ such that $ g \omega $ is smooth at $ z $. A good generalization of (a2) should yield "residues" of a semi-meromorphic $ ( q, r) $- form $ \omega $, which should be currents supported on $ S $. One needs the existence of limits of the form

$$ R _ {I , J } ^ {\omega ,f } ( \psi ) = \lim\limits _ {\delta \rightarrow 0 } \int\limits _ {D _ {I, J } ^ \delta ( \epsilon , f ) } \omega \wedge \psi , $$

with

$$ D _ {I ,J } ^ \delta ( \epsilon , f ) = $$

$$ = \ \{ {z \in D } : {| f _ {i} ( z) | = \epsilon _ {i} ( \delta ), i \in I, \ | f _ {j} ( z) | > \epsilon _ {j} ( \delta ) , j \in J } \} . $$

Here $ I $ and $ J $ are disjoint subsets of $ 1 \dots p $, $ f = ( f _ {1} \dots f _ {p} ) : D \rightarrow \mathbf C ^ {p} $ is a holomorphic mapping with $ S \subset \cup _ {k \in I \cup J } \{ f _ {k} = 0 \} $, $ \psi $ is an arbitrary compactly-supported smooth $ ( 2n- | I | - q- r) $- form and $ \epsilon ( \delta ) = ( \epsilon ( \delta ) _ {1} \dots \epsilon ( \delta ) _ {p} ) : ( 0, 1] \rightarrow \mathbf R _ {+} ^ {p} $ is an admissible path, that is, $ \epsilon _ {j} ( \delta ) $ and $ \epsilon _ {j} / \epsilon _ {j+} 1 $ tend to $ 0 $ with $ \delta $. In fact, the $ R _ {I , J } ^ {\omega , f } $ are $ ( q , r + | I |) $- 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