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A generalization of the concept of a [[Residue of an analytic function|residue of an analytic function]] of one complex variable to several complex variables. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815501.png" /> be a complex-analytic manifold (cf. [[Analytic manifold|Analytic manifold]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815502.png" /> be an analytic submanifold of complex codimension one and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815503.png" /> be a closed exterior [[Differential form|differential form]] of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815504.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815505.png" /> with a first-order polar singularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815506.png" />. The last condition means that for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815507.png" />, holomorphic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815508.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r0815509.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155010.png" /> and such that
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155011.png" /></td> </tr></table>
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the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155012.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155013.png" />. Under these conditions there exist, in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155014.png" /> of an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155015.png" />, forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155017.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155018.png" /> such that
+
A generalization of the concept of a [[Residue of an analytic function|residue of an analytic function]] of one complex variable to several complex variables. Let  $  X $
 +
be a complex-analytic manifold (cf. [[Analytic manifold|Analytic manifold]]), let  $  S $
 +
be an analytic submanifold of complex codimension one and let  $  \omega ( x) $
 +
be a closed exterior [[Differential form|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155019.png" /></td> </tr></table>
+
$$
 +
S \cap U _ {y}  = \
 +
\{ {x } : {s ( x, y) = 0 } \}
 +
,\ \
 +
ds  \not\equiv  0  \textrm{ if }  x = y,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155020.png" /> is a closed form of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155021.png" /> that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155022.png" />. The closed form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155023.png" /> which is defined in a neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155024.png" /> by the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155025.png" />, is called the residue form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155026.png" />, and is denoted by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155027.png" /></td> </tr></table>
+
$$
 +
\omega ( x)  = \
  
If the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155028.png" /> is holomorphic, its residue form is holomorphic as well (cf. [[Holomorphic form|Holomorphic form]]). For instance, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155030.png" /> and the form
+
\frac{ds ( x, y) }{s ( x, y) }
 +
\wedge \psi ( x, y) + \theta ( x, y),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155031.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155033.png" /> are holomorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155036.png" />, the residue form is
+
$$
 +
\mathop{\rm res}  [ \omega ]  = \
 +
\left .  
 +
\frac{s \omega }{ds }
 +
\right | _ {S} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155037.png" /></td> </tr></table>
+
If the form  $  \omega $
 +
is holomorphic, its residue form is holomorphic as well (cf. [[Holomorphic form|Holomorphic form]]). For instance, for  $  X = \mathbf C  ^ {n} $,
 +
$  S = \{ {x \in \mathbf C  ^ {n} } : {s( x) = 0 } \} $
 +
and the form
  
at the points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155038.png" />.
+
$$
 +
\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:
 
The residue formula corresponding to residue forms is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155039.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\delta \gamma } \omega  = 2 \pi i
 +
\int\limits _  \gamma  \mathop{\rm res}  [ \omega ],
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155040.png" /> is an arbitrary cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155041.png" /> of dimension equal to the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155043.png" /> — a cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155044.png" /> — is the boundary of some chain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155045.png" /> in general position with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155046.png" /> and intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155047.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155048.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155049.png" /> is defined by induction.
+
The composite residue form $  \mathop{\rm res}  ^ {m}  [ \omega ] $
 +
is defined by induction.
  
The residue class of a closed form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155051.png" /> is the [[Cohomology|cohomology]] class on the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155052.png" /> produced by the residue forms of the forms of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155054.png" /> that are cohomologous with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155055.png" /> and have a first-order polar singularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155056.png" />. The residue class of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155057.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155058.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155059.png" /> is a [[Stein manifold|Stein manifold]]. The residue class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155060.png" /> does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155061.png" /> out of one cohomology class and realizes a homomorphism from the group of cohomology classes of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155062.png" /> to the group of cohomology classes of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155063.png" />:
+
The residue class of a closed form $  \omega $
 +
in $  X \setminus  S $
 +
is the [[Cohomology|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|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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155064.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Res} : H  ^ {*} ( X \setminus  S)  \rightarrow  H  ^ {*} ( S).
 +
$$
  
 
As for residue forms, the following residue formula is valid:
 
As for residue forms, the following residue formula is valid:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155065.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\delta \gamma } \omega  = 2 \pi i
 +
\int\limits _  \gamma  \mathop{\rm Res}  [ \omega ],
 +
$$
  
and the integral on the right-hand side of this equation is taken over any form in the residue class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155066.png" /> and is independent of it.
+
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 (, ,
 
For references, see (, ,
  
 
to) [[Residue of an analytic function|Residue of an analytic function]].
 
to) [[Residue of an analytic function|Residue of an analytic function]].
 
 
  
 
====Comments====
 
====Comments====
 
A [[Differential form|differential form]] whose coefficients are distributions (generalized functions) is called a current. The theory of currents was developed largely by H. Federer [[#References|[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., [[#References|[a6]]]–[[#References|[a8]]].
 
A [[Differential form|differential form]] whose coefficients are distributions (generalized functions) is called a current. The theory of currents was developed largely by H. Federer [[#References|[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., [[#References|[a6]]]–[[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155067.png" /> be holomorphic on a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155068.png" /> except for a (finite) set of singularities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155069.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155070.png" /> be a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155071.png" /> with smooth boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155072.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155073.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155074.png" /> be smooth, compactly supported on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155075.png" /> and holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155076.png" />, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\mathop{\rm Res} ( g)( \psi )  = \
 +
\sum _ { i } \int\limits _ {\partial  D _ {j} }
 +
g( z) \psi ( z)  dz  = - \int\limits _ { D } g \overline \partial \; \psi  dz
 +
$$
  
is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155078.png" /> as long as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155079.png" /> are contained in the neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155080.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155081.png" /> is holomorphic. If one takes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155082.png" /> a function that equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155083.png" /> in a small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155084.png" />, one obtains the usual residue. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155085.png" /> represents a germ of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155086.png" />-closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155087.png" />-form at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155089.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155090.png" />-closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155091.png" />-form. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155092.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155093.png" /> denotes Dolbeault cohomology of germs of forms at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155094.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155095.png" /> is called the cohomological residue. This can be generalized to several variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155096.png" /> will be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155098.png" /> a closed subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r08155099.png" />, to obtain a homomorphism
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550100.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Res} : H ^ {p, q+ 1 } ( D \setminus  S)  \rightarrow \
 +
\mathop{\rm Hom} ( H ^ {n- p, n- q- 1 } ( S), \mathbf C ) .
 +
$$
  
In another direction one would like to have an interpretation of (a1) for smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550101.png" />, not necessarily closed. This can be done if one imposes the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550102.png" /> is meromorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550103.png" />. One may write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550104.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550105.png" /> holomorphic, and assume by a partition of unity that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550106.png" /> is supported on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550107.png" /> only. Then the following limit exists independently of the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550108.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\lim\limits _ {\epsilon \rightarrow 0 }  \int\limits _ {| g _ {2} | = \epsilon }
 +
g( z) \psi ( z) dz .
 +
$$
  
It defines a current supported on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550110.png" />. To obtain a sensible analogue of this for several variables is much harder.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550111.png" /> is a smooth differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550112.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550113.png" /> that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550114.png" /> admits a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550115.png" /> defined on some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550116.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550117.png" /> is smooth at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550118.png" />. A good generalization of (a2) should yield "residues" of a semi-meromorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550119.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550120.png" />, which should be currents supported on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550121.png" />. One needs the existence of limits of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550122.png" /></td> </tr></table>
+
$$
 +
R _ {I , J }  ^ {\omega ,f }
 +
( \psi )  = \lim\limits _ {\delta \rightarrow 0 }  \int\limits _
 +
{D _ {I, J }  ^  \delta  ( \epsilon , f  ) } \omega \wedge \psi ,
 +
$$
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550123.png" /></td> </tr></table>
+
$$
 +
D _ {I ,J }  ^  \delta  ( \epsilon , f  ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550124.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ {z \in D } : {| f _ {i} ( z) | = \epsilon _ {i} ( \delta ), i \in I,
 +
\  | f _ {j} ( z) | > \epsilon _ {j} ( \delta ) , j \in J }
 +
\} .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550126.png" /> are disjoint subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550128.png" /> is a holomorphic mapping with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550130.png" /> is an arbitrary compactly-supported smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550131.png" />-form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550132.png" /> is an admissible path, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550134.png" /> tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550135.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550136.png" />. In fact, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550137.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081550/r081550138.png" />-currents. For these two approaches, see [[#References|[a4]]].
+
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 [[#References|[a4]]].
  
 
A third direction towards residue currents is by analytic continuation of holomorphic current-valued mappings. See [[#References|[a2]]].
 
A third direction towards residue currents is by analytic continuation of holomorphic current-valued mappings. See [[#References|[a2]]].

Latest revision as of 08:11, 6 June 2020


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
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How to Cite This Entry:
Residue form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residue_form&oldid=48525
This article was adapted from an original article by A.P. Yuzhakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article