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Problems named after P. Cousin [[#References|[1]]], who first solved them for certain simple domains in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267901.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267902.png" />.
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Problems named after P. Cousin [[#References|[1]]], who first solved them for certain simple domains in the complex $  n $-
 +
dimensional space $  \mathbf C  ^ {n} $.
  
 
==First (additive) Cousin problem.==
 
==First (additive) Cousin problem.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267903.png" /> be a covering of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267904.png" /> by open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267905.png" />, in each of which is defined a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267906.png" />; assume that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267907.png" /> are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267908.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c0267909.png" /> (compatibility condition). It is required to construct a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679010.png" /> which is meromorphic on the entire manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679011.png" /> and is such that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679012.png" /> are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679014.png" />. In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.
+
Let $  {\mathcal U} = \{ U _  \alpha  \} $
 +
be a covering of a complex manifold $  M $
 +
by open subsets $  U _  \alpha  $,  
 +
in each of which is defined a meromorphic function $  f _  \alpha  $;  
 +
assume that the functions $  f _ {\alpha \beta }  = f _  \alpha  - f _  \beta  $
 +
are holomorphic in $  U _ {\alpha \beta }  = U _  \alpha  \cap U _  \beta  $
 +
for all $  \alpha , \beta $(
 +
compatibility condition). It is required to construct a function $  f $
 +
which is meromorphic on the entire manifold $  M $
 +
and is such that the functions $  f - f _  \alpha  $
 +
are holomorphic in $  U _  \alpha  $
 +
for all $  \alpha $.  
 +
In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679015.png" />, defined in the pairwise intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679016.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679017.png" />, define a holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679019.png" />-cocycle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679020.png" />, i.e. they satisfy the conditions
+
The functions $  f _ {\alpha \beta }  $,  
 +
defined in the pairwise intersections $  U _ {\alpha \beta }  $
 +
of elements of $  {\mathcal U} $,  
 +
define a holomorphic $  1 $-
 +
cocycle for $  {\mathcal U} $,  
 +
i.e. they satisfy the conditions
  
<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/c/c026/c026790/c02679021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f _ {\alpha \beta }  +
 +
f _ {\beta \alpha }  = 0 \ \
 +
\mathop{\rm in}  U _ {\alpha \beta }  ,
 +
$$
  
<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/c/c026/c026790/c02679022.png" /></td> </tr></table>
+
$$
 +
f _ {\alpha \beta }  + f _ {\beta \gamma }  + f _ {\gamma \alpha }  = 0 \  \mathop{\rm in}  U _  \alpha  \cap U _  \beta  \cap U _  \gamma  ,
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679023.png" />. A more general problem (known as the first Cousin problem in cohomological formulation) is the following. Given holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679024.png" /> in the intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679025.png" />, satisfying the cocycle conditions (1), it is required to find functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679026.png" />, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679027.png" />, such that
+
for all $  \alpha , \beta , \gamma $.  
 +
A more general problem (known as the first Cousin problem in cohomological formulation) is the following. Given holomorphic functions $  f _ {\alpha \beta }  $
 +
in the intersections $  U _ {\alpha \beta }  $,  
 +
satisfying the cocycle conditions (1), it is required to find functions $  h _  \alpha  $,  
 +
holomorphic in $  U _  \alpha  $,  
 +
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/c/c026/c026790/c02679028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f _ {\alpha \beta }  = \
 +
h _  \beta  - h _  \alpha  $$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679029.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679030.png" /> correspond to the data of the first Cousin problem and the above functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679031.png" /> exist, then the function
+
for all $  \alpha , \beta $.  
 +
If the functions $  f _ {\alpha \beta }  $
 +
correspond to the data of the first Cousin problem and the above functions $  h _  \alpha  $
 +
exist, then the function
  
<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/c/c026/c026790/c02679032.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ f _  \alpha  + h _  \alpha  \ \
 +
\mathop{\rm in}  U _  \alpha  \}
 +
$$
  
is defined and meromorphic throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679033.png" /> and is a solution of the first Cousin problem. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679034.png" /> is a solution of the first Cousin problem with data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679035.png" />, then the holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679036.png" /> satisfy (2). Thus, a specific first Cousin problem is solvable if and only if the corresponding cocycle is a holomorphic coboundary (i.e. satisfies condition (2)).
+
is defined and meromorphic throughout $  M $
 +
and is a solution of the first Cousin problem. Conversely, if $  f $
 +
is a solution of the first Cousin problem with data $  \{ f _  \alpha  \} $,  
 +
then the holomorphic functions $  h _  \alpha  = f - f _  \alpha  $
 +
satisfy (2). Thus, a specific first Cousin problem is solvable if and only if the corresponding cocycle is a holomorphic coboundary (i.e. satisfies condition (2)).
  
The first Cousin problem may also be formulated in a local version. To each set of data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679037.png" /> satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679040.png" /> are the sheaves of germs of meromorphic and holomorphic functions, respectively; the correspondence is such that any global section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679041.png" /> corresponds to some first Cousin problem (the value of the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679042.png" /> corresponding to data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679043.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679044.png" /> is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679045.png" /> with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679046.png" />). The mapping of global sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679047.png" /> maps each meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679049.png" /> to a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679052.png" /> is the class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679053.png" /> of the germ of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679054.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679056.png" />. The localized first Cousin problem is then: Given a global section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679057.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679058.png" />, to find a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679060.png" /> (i.e. a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679061.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679062.png" />.
+
The first Cousin problem may also be formulated in a local version. To each set of data $  \{ U _  \alpha  , f _  \alpha  \} $
 +
satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf $  {\mathcal M} / {\mathcal O} $,  
 +
where $  {\mathcal M} $
 +
and $  {\mathcal O} $
 +
are the sheaves of germs of meromorphic and holomorphic functions, respectively; the correspondence is such that any global section of $  {\mathcal M} / {\mathcal O} $
 +
corresponds to some first Cousin problem (the value of the section $  \kappa $
 +
corresponding to data $  \{ f _  \alpha  \} $
 +
at a point $  z \in U _  \alpha  $
 +
is the element of $  {\mathcal M} _ {z} / {\mathcal O} _ {z} $
 +
with representative $  f _  \alpha  $).  
 +
The mapping of global sections $  \phi : \Gamma ( {\mathcal M} ) \rightarrow \Gamma ( {\mathcal M} / {\mathcal O} ) $
 +
maps each meromorphic function $  f $
 +
on $  {\mathcal M} $
 +
to a section $  \kappa _ {f} $
 +
of $  {\mathcal M} / {\mathcal O} $,  
 +
where $  \kappa _ {f} ( z) $
 +
is the class in $  {\mathcal M} _ {z} / {\mathcal O} _ {z} $
 +
of the germ of $  f $
 +
at the point $  z $,  
 +
$  z \in M $.  
 +
The localized first Cousin problem is then: Given a global section $  \kappa $
 +
of the sheaf $  {\mathcal M} / {\mathcal O} $,  
 +
to find a meromorphic function $  f $
 +
on $  M $(
 +
i.e. a section of $  {\mathcal M} $)  
 +
such that $  \phi ( f) = \kappa $.
  
Theorems concerning the solvability of the first Cousin problem may be regarded as a multi-dimensional generalization of the [[Mittag-Leffler theorem|Mittag-Leffler theorem]] on the construction of a meromorphic function with prescribed poles. The problem in cohomological formulation, with a fixed covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679063.png" />, is solvable (for arbitrary compatible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679064.png" />) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679065.png" /> (the Čech cohomology for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679066.png" /> with holomorphic coefficients is trivial).
+
Theorems concerning the solvability of the first Cousin problem may be regarded as a multi-dimensional generalization of the [[Mittag-Leffler theorem|Mittag-Leffler theorem]] on the construction of a meromorphic function with prescribed poles. The problem in cohomological formulation, with a fixed covering $  {\mathcal U} $,  
 +
is solvable (for arbitrary compatible $  \{ f _  \alpha  \} $)  
 +
if and only if $  H  ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $(
 +
the Čech cohomology for $  {\mathcal U} $
 +
with holomorphic coefficients is trivial).
  
A specific first Cousin problem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679067.png" /> is solvable if and only if the corresponding section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679068.png" /> belongs to the image of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679069.png" />. An arbitrary first Cousin problem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679070.png" /> is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679071.png" /> is surjective. On any complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679072.png" /> one has an exact sequence
+
A specific first Cousin problem on $  M $
 +
is solvable if and only if the corresponding section of $  {\mathcal M} / {\mathcal O} $
 +
belongs to the image of the mapping $  \phi $.  
 +
An arbitrary first Cousin problem on $  M $
 +
is solvable if and only if $  \phi $
 +
is surjective. On any complex manifold $  M $
 +
one has an exact sequence
  
<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/c/c026/c026790/c02679073.png" /></td> </tr></table>
+
$$
 +
\Gamma ( {\mathcal M} )  \mathop \rightarrow \limits ^  \phi  \
 +
\Gamma ( {\mathcal M} / {\mathcal O} )  \rightarrow \
 +
H  ^ {1} ( M, {\mathcal O} ).
 +
$$
  
If the Čech cohomology for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679074.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679075.png" /> is trivial (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679076.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679077.png" /> is surjective and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679078.png" /> for any covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679080.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679081.png" />, any first Cousin problem is solvable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679082.png" /> (in the classical, cohomological and local version). In particular, the problem is solvable in all domains of holomorphy and on Stein manifolds (cf. [[Stein manifold|Stein manifold]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679083.png" />, then the first Cousin problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679084.png" /> is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679085.png" /> is a domain of holomorphy. An example of an unsolvable first Cousin problem is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679090.png" />.
+
If the Čech cohomology for $  M $
 +
with coefficients in $  {\mathcal O} $
 +
is trivial (i.e. $  H  ^ {1} ( M , {\mathcal O} ) = 0 $),  
 +
then $  \phi $
 +
is surjective and $  H  ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $
 +
for any covering $  {\mathcal U} $
 +
of $  M $.  
 +
Thus, if $  H  ^ {1} ( M, {\mathcal O} ) = 0 $,  
 +
any first Cousin problem is solvable on $  M $(
 +
in the classical, cohomological and local version). In particular, the problem is solvable in all domains of holomorphy and on Stein manifolds (cf. [[Stein manifold|Stein manifold]]). If $  D \subset  \mathbf C  ^ {2} $,  
 +
then the first Cousin problem in $  D $
 +
is solvable if and only if $  D $
 +
is a domain of holomorphy. An example of an unsolvable first Cousin problem is: $  M = \mathbf C  ^ {2} \setminus  \{ 0 \} $,  
 +
$  U _  \alpha  = \{ z _  \alpha  \neq 0 \} $,
 +
$  \alpha = 1, 2 $,  
 +
$  f _ {1} = ( z _ {1} z _ {2} )  ^ {-} 1 $,  
 +
$  f _ {2} = 0 $.
  
 
==Second (multiplicative) Cousin problem.==
 
==Second (multiplicative) Cousin problem.==
Given an open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679091.png" /> of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679092.png" /> and, in each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679093.png" />, a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679095.png" /> on each component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679096.png" />, with the assumption that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679097.png" /> are holomorphic and nowhere vanishing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679098.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c02679099.png" /> (compatibility condition). It is required to construct a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790100.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790101.png" /> such that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790102.png" /> are holomorphic and nowhere vanishing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790103.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790104.png" />.
+
Given an open covering $  {\mathcal U} = \{ U _  \alpha  \} $
 +
of a complex manifold $  M $
 +
and, in each $  U _  \alpha  $,  
 +
a meromorphic function $  f _  \alpha  $,  
 +
$  f _  \alpha  \not\equiv 0 $
 +
on each component of $  U _  \alpha  $,  
 +
with the assumption that the functions $  f _ {\alpha \beta }  = f _  \alpha  f _  \beta  ^ {-} 1 $
 +
are holomorphic and nowhere vanishing in $  U _ {\alpha \beta }  $
 +
for all $  \alpha , \beta $(
 +
compatibility condition). It is required to construct a meromorphic function $  f $
 +
on $  M $
 +
such that the functions $  ff _  \alpha  ^ {-} 1 $
 +
are holomorphic and nowhere vanishing in $  U _  \alpha  $
 +
for all $  \alpha $.
  
The cohomological formulation of the second Cousin problem is as follows. Given the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790105.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790106.png" />, holomorphic and nowhere vanishing in the intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790107.png" />, and forming a multiplicative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790109.png" />-cocycle, i.e.
+
The cohomological formulation of the second Cousin problem is as follows. Given the covering $  {\mathcal U} $
 +
and functions $  f _ {\alpha \beta }  $,  
 +
holomorphic and nowhere vanishing in the intersections $  U _ {\alpha \beta }  $,  
 +
and forming a multiplicative $  1 $-
 +
cocycle, i.e.
  
<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/c/c026/c026790/c026790110.png" /></td> </tr></table>
+
$$
 +
f _ {\alpha \beta }  f _ {\beta \alpha }  = \
 +
1 \  \mathop{\rm in} \
 +
U _ {\alpha \beta }  ,
 +
$$
  
<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/c/c026/c026790/c026790111.png" /></td> </tr></table>
+
$$
 +
f _ {\alpha \beta }  f _ {\beta \gamma }  f _ {\gamma \alpha }
 +
= 1 \  \mathop{\rm in}  U _  \alpha  \cap U _  \beta  \cap U _  \gamma  ,
 +
$$
  
it is required to find functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790112.png" />, holomorphic and nowhere vanishing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790113.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790114.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790115.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790116.png" />. If the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790117.png" /> corresponds to the data of a second Cousin problem and the required <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790118.png" /> exist, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790119.png" /> is defined and meromorphic throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790120.png" /> and is a solution to the given second Cousin problem. Conversely, if a specific second Cousin problem is solvable, then the corresponding cocycle is a holomorphic coboundary.
+
it is required to find functions $  h _  \alpha  $,  
 +
holomorphic and nowhere vanishing in $  U _  \alpha  $,  
 +
such that $  f _ {\alpha \beta }  = h _  \beta  h _  \alpha  ^ {-} 1 $
 +
in $  U _ {\alpha \beta }  $
 +
for all $  \alpha , \beta $.  
 +
If the cocycle $  \{ f _ {\alpha \beta }  \} $
 +
corresponds to the data of a second Cousin problem and the required $  h _  \alpha  $
 +
exist, then the function $  f = \{ f _  \alpha  h _ {\alpha }  \mathop{\rm in}  U _  \alpha  \} $
 +
is defined and meromorphic throughout $  M $
 +
and is a solution to the given second Cousin problem. Conversely, if a specific second Cousin problem is solvable, then the corresponding cocycle is a holomorphic coboundary.
  
The localized second Cousin problem. To each set of data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790121.png" /> for the second Cousin problem there corresponds a uniquely defined global section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790122.png" /> (in analogy to the first Cousin problem), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790123.png" /> (with 0 the null section) is the multiplicative sheaf of germs of meromorphic functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790124.png" /> is the subsheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790125.png" /> in which each stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790126.png" /> consists of germs of holomorphic functions that do not vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790127.png" />. The mapping of global sections
+
The localized second Cousin problem. To each set of data $  \{ U _  \alpha  , f _  \alpha  \} $
 +
for the second Cousin problem there corresponds a uniquely defined global section of the sheaf $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $(
 +
in analogy to the first Cousin problem), where $  {\mathcal M}  ^ {*} = {\mathcal M} \setminus  \{ 0 \} $(
 +
with 0 the null section) is the multiplicative sheaf of germs of meromorphic functions and $  {\mathcal O}  ^ {*} $
 +
is the subsheaf of $  {\mathcal O} $
 +
in which each stalk $  {\mathcal O} _ {z}  ^ {*} $
 +
consists of germs of holomorphic functions that do not vanish at $  z $.  
 +
The mapping of global sections
  
<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/c/c026/c026790/c026790128.png" /></td> </tr></table>
+
$$
 +
\Gamma ( {\mathcal M}  ^ {*} )  \mathop \rightarrow \limits ^  \psi  \
 +
\Gamma ( {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} )
 +
$$
  
maps a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790129.png" /> to a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790130.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790131.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790132.png" /> is the class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790133.png" /> of the germ of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790134.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790136.png" />. The localized second Cousin problem is: Given a global section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790137.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790138.png" />, to find a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790139.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790141.png" /> on the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790142.png" /> (i.e. a global section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790143.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790144.png" />.
+
maps a meromorphic function $  f $
 +
to a section $  \kappa _ {f}  ^ {*} $
 +
of the sheaf $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $,  
 +
where $  \kappa _ {f}  ^ {*} ( z) $
 +
is the class in $  {\mathcal M} _ {z}  ^ {*} / {\mathcal O} _ {z}  ^ {*} $
 +
of the germ of $  f $
 +
at $  z $,  
 +
$  z \in M $.  
 +
The localized second Cousin problem is: Given a global section $  \kappa  ^ {*} $
 +
of the sheaf $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $,  
 +
to find a meromorphic function $  f $
 +
on $  M $,  
 +
$  f \not\equiv 0 $
 +
on the components of $  M $(
 +
i.e. a global section of $  {\mathcal M}  ^ {*} $),  
 +
such that $  \psi ( f  ) = \kappa  ^ {*} $.
  
The sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790145.png" /> uniquely correspond to divisors (cf. [[Divisor|Divisor]]), therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790146.png" /> is called the sheaf of germs of divisors. A divisor on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790147.png" /> is a formal locally finite sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790148.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790149.png" /> are integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790150.png" /> analytic subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790151.png" /> of pure codimension 1. To each meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790152.png" /> corresponds the divisor whose terms are the irreducible components of the zero and polar sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790153.png" /> with respective multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790154.png" />, with multiplicities of zeros considered positive and those of poles negative. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790155.png" /> maps each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790156.png" /> to its divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790157.png" />; such divisors are called proper divisors. The second Cousin problem in terms of divisors is: Given a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790158.png" /> on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790159.png" />, to construct a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790160.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790161.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790162.png" />.
+
The sections of $  M  ^ {*} / Q  ^ {*} $
 +
uniquely correspond to divisors (cf. [[Divisor|Divisor]]), therefore $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} = {\mathcal D} $
 +
is called the sheaf of germs of divisors. A divisor on a complex manifold $  M $
 +
is a formal locally finite sum $  \sum k _ {j} \Delta _ {j} $,  
 +
where $  k _ {j} $
 +
are integers and $  \Delta _ {j} $
 +
analytic subsets of $  M $
 +
of pure codimension 1. To each meromorphic function $  f $
 +
corresponds the divisor whose terms are the irreducible components of the zero and polar sets of $  f $
 +
with respective multiplicities $  k _ {j} $,  
 +
with multiplicities of zeros considered positive and those of poles negative. The mapping $  \psi $
 +
maps each function $  f $
 +
to its divisor $  ( f  ) $;  
 +
such divisors are called proper divisors. The second Cousin problem in terms of divisors is: Given a divisor $  \Delta $
 +
on the manifold $  M $,  
 +
to construct a meromorphic function $  f $
 +
on $  M $
 +
such that $  \Delta = ( f  ) $.
  
Theorems concerning the solvability of the second Cousin problem may be regarded as multi-dimensional generalizations of Weierstrass' theorem on the construction of a meromorphic function with prescribed zeros and poles. As in the case of the first Cousin problem, a necessary and sufficient condition for the solvability of any second Cousin problem in cohomological version is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790163.png" />. Unfortunately, the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790164.png" /> is not coherent, and this condition is less effective. The attempt to reduce a given second Cousin problem to a first Cousin problem by taking logarithms encounters an obstruction in the form of an integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790165.png" />-cocycle, and one obtains an exact sequence
+
Theorems concerning the solvability of the second Cousin problem may be regarded as multi-dimensional generalizations of Weierstrass' theorem on the construction of a meromorphic function with prescribed zeros and poles. As in the case of the first Cousin problem, a necessary and sufficient condition for the solvability of any second Cousin problem in cohomological version is that $  H  ^ {1} ( M, {\mathcal O}  ^ {*} ) = 0 $.  
 +
Unfortunately, the sheaf $  {\mathcal O}  ^ {*} $
 +
is not coherent, and this condition is less effective. The attempt to reduce a given second Cousin problem to a first Cousin problem by taking logarithms encounters an obstruction in the form of an integral $  2 $-
 +
cocycle, and one obtains an exact sequence
  
<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/c/c026/c026790/c026790166.png" /></td> </tr></table>
+
$$
 +
H  ^ {1} ( M, {\mathcal O} )  \rightarrow \
 +
H  ^ {1} ( M, {\mathcal O}  ^ {*} )  \mathop \rightarrow \limits ^  \alpha  \
 +
H  ^ {2} ( M, \mathbf Z ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790167.png" /> is the constant sheaf of integers. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790168.png" />, any second Cousin problem is solvable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790169.png" />, and any divisor is proper. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790170.png" /> is a Stein manifold, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790171.png" /> is an isomorphism; hence the topological condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790172.png" /> on a Stein manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790173.png" /> is necessary and sufficient for the second Cousin problem in cohomological version to be solvable. The composite mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790174.png" />,
+
where $  \mathbf Z $
 +
is the constant sheaf of integers. Thus, if $  H  ^ {1} ( M, {\mathcal O} ) = H  ^ {2} ( M, \mathbf Z ) = 0 $,  
 +
any second Cousin problem is solvable on $  M $,  
 +
and any divisor is proper. If $  M $
 +
is a Stein manifold, then $  \alpha $
 +
is an isomorphism; hence the topological condition $  H  ^ {2} ( M, \mathbf Z ) = 0 $
 +
on a Stein manifold $  M $
 +
is necessary and sufficient for the second Cousin problem in cohomological version to be solvable. The composite mapping $  c = \alpha \circ \beta $,
  
<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/c/c026/c026790/c026790175.png" /></td> </tr></table>
+
$$
 +
\Gamma ( {\mathcal D} )  \mathop \rightarrow \limits ^  \beta  \
 +
H  ^ {1} ( M, {\mathcal O}  ^ {*} )  \mathop \rightarrow \limits ^  \alpha  \
 +
H  ^ {2} ( M, \mathbf Z )
 +
$$
  
maps each divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790176.png" /> to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790177.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790178.png" />, which is known as the Chern class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790179.png" />. The specific second Cousin problem corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790180.png" /> is solvable, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790181.png" />, if and only if the Chern class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790182.png" /> is trivial: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790183.png" />. On a Stein manifold, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790184.png" /> is surjective; moreover, every element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790185.png" /> may be expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790186.png" /> for some divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790187.png" /> with positive multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790188.png" />. Thus, the obstructions to the solution of the second Cousin problem on a Stein manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790189.png" /> are completely described by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790190.png" />.
+
maps each divisor $  \Delta $
 +
to an element c( \Delta ) $
 +
of the group $  H  ^ {2} ( M, \mathbf Z ) $,  
 +
which is known as the Chern class of $  \Delta $.  
 +
The specific second Cousin problem corresponding to $  \Delta $
 +
is solvable, assuming $  H  ^ {1} ( M, {\mathcal O} ) = 0 $,  
 +
if and only if the Chern class of $  \Delta $
 +
is trivial: $  c( \Delta ) = 0 $.  
 +
On a Stein manifold, the mapping c $
 +
is surjective; moreover, every element in $  H  ^ {2} ( M, \mathbf Z ) $
 +
may be expressed as c( \Delta ) $
 +
for some divisor $  \Delta $
 +
with positive multiplicities $  k _ {j} $.  
 +
Thus, the obstructions to the solution of the second Cousin problem on a Stein manifold $  M $
 +
are completely described by the group $  H  ^ {2} ( M, \mathbf Z ) $.
  
 
===Examples.===
 
===Examples.===
  
 +
1)  $  M = \mathbf C  ^ {2} \setminus  \{ z _ {1} = z _ {2} ,  | z _ {1} | = 1 \} $;
 +
the first Cousin problem is unsolvable; the second Cousin problem is unsolvable, e.g., for the divisor  $  \Delta = M \cap \{ z _ {1} = z _ {2} ,  | z _ {1} | < 1 \} $
 +
with multiplicity 1.
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790191.png" />; the first Cousin problem is unsolvable; the second Cousin problem is unsolvable, e.g., for the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790192.png" /> with multiplicity 1.
+
2)  $  M = \{ | z _ {1}  ^ {2} + z _ {2}  ^ {2} + z _ {3}  ^ {2} - 1 | < 1 \} \subset  \mathbf C  ^ {3} $,
 +
$  \Delta $
 +
is one of the components of the intersection of  $  M $
 +
and the plane  $  z _ {2} = iz _ {1} $
 +
with multiplicity 1. The second Cousin problem is unsolvable ( $  M $
 +
is a domain of holomorphy, the first Cousin problem is solvable).
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790194.png" /> is one of the components of the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790195.png" /> and the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790196.png" /> with multiplicity 1. The second Cousin problem is unsolvable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790197.png" /> is a domain of holomorphy, the first Cousin problem is solvable).
+
3) The first and second Cousin problems are solvable in domains $  D = D _ {1} \times \dots \times D _ {n} \subset  \mathbf C  ^ {n} $,  
 
+
where $  D _ {j} $
3) The first and second Cousin problems are solvable in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790198.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790199.png" /> are plane domains and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790200.png" />, with the possible exception of one, are simply connected.
+
are plane domains and all $  D _ {j} $,  
 +
with the possible exception of one, are simply connected.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Cousin, "Sur les fonctions de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790201.png" /> variables" ''Acta Math.'' , '''19''' (1895) pp. 1–62 {{MR|1554861}} {{ZBL|26.0456.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top"> P. Cousin, "Sur les fonctions de $n$ variables" ''Acta Math.'' , '''19''' (1895) pp. 1–62 {{MR|1554861}} {{ZBL|26.0456.02}} </TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
The Cousin problems are related to the [[Poincaré problem|Poincaré problem]] (is a meromorphic function given on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790202.png" /> globally the quotient of two holomorphic functions whose germs are relatively prime for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026790/c026790203.png" />?) and to the, more algebraic, Theorems A and B of H. Cartan and J.-P. Serre, cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].
+
The Cousin problems are related to the [[Poincaré problem|Poincaré problem]] (is a meromorphic function given on a complex manifold $  X $
 +
globally the quotient of two holomorphic functions whose germs are relatively prime for all $  x \in X $?)  
 +
and to the, more algebraic, Theorems A and B of H. Cartan and J.-P. Serre, cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.A. Cazacu, "Theorie der Funktionen mehreren komplexer Veränderlicher" , Birkhäuser (1975) (Translated from Rumanian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) {{MR|0580152}} {{ZBL|0433.32007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5 {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 6 {{MR|0847923}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.A. Cazacu, "Theorie der Funktionen mehreren komplexer Veränderlicher" , Birkhäuser (1975) (Translated from Rumanian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) {{MR|0580152}} {{ZBL|0433.32007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5 {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 6 {{MR|0847923}} {{ZBL|}} </TD></TR></table>

Latest revision as of 11:03, 26 March 2023


Problems named after P. Cousin [1], who first solved them for certain simple domains in the complex $ n $- dimensional space $ \mathbf C ^ {n} $.

First (additive) Cousin problem.

Let $ {\mathcal U} = \{ U _ \alpha \} $ be a covering of a complex manifold $ M $ by open subsets $ U _ \alpha $, in each of which is defined a meromorphic function $ f _ \alpha $; assume that the functions $ f _ {\alpha \beta } = f _ \alpha - f _ \beta $ are holomorphic in $ U _ {\alpha \beta } = U _ \alpha \cap U _ \beta $ for all $ \alpha , \beta $( compatibility condition). It is required to construct a function $ f $ which is meromorphic on the entire manifold $ M $ and is such that the functions $ f - f _ \alpha $ are holomorphic in $ U _ \alpha $ for all $ \alpha $. In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.

The functions $ f _ {\alpha \beta } $, defined in the pairwise intersections $ U _ {\alpha \beta } $ of elements of $ {\mathcal U} $, define a holomorphic $ 1 $- cocycle for $ {\mathcal U} $, i.e. they satisfy the conditions

$$ \tag{1 } f _ {\alpha \beta } + f _ {\beta \alpha } = 0 \ \ \mathop{\rm in} U _ {\alpha \beta } , $$

$$ f _ {\alpha \beta } + f _ {\beta \gamma } + f _ {\gamma \alpha } = 0 \ \mathop{\rm in} U _ \alpha \cap U _ \beta \cap U _ \gamma , $$

for all $ \alpha , \beta , \gamma $. A more general problem (known as the first Cousin problem in cohomological formulation) is the following. Given holomorphic functions $ f _ {\alpha \beta } $ in the intersections $ U _ {\alpha \beta } $, satisfying the cocycle conditions (1), it is required to find functions $ h _ \alpha $, holomorphic in $ U _ \alpha $, such that

$$ \tag{2 } f _ {\alpha \beta } = \ h _ \beta - h _ \alpha $$

for all $ \alpha , \beta $. If the functions $ f _ {\alpha \beta } $ correspond to the data of the first Cousin problem and the above functions $ h _ \alpha $ exist, then the function

$$ f = \ \{ f _ \alpha + h _ \alpha \ \ \mathop{\rm in} U _ \alpha \} $$

is defined and meromorphic throughout $ M $ and is a solution of the first Cousin problem. Conversely, if $ f $ is a solution of the first Cousin problem with data $ \{ f _ \alpha \} $, then the holomorphic functions $ h _ \alpha = f - f _ \alpha $ satisfy (2). Thus, a specific first Cousin problem is solvable if and only if the corresponding cocycle is a holomorphic coboundary (i.e. satisfies condition (2)).

The first Cousin problem may also be formulated in a local version. To each set of data $ \{ U _ \alpha , f _ \alpha \} $ satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf $ {\mathcal M} / {\mathcal O} $, where $ {\mathcal M} $ and $ {\mathcal O} $ are the sheaves of germs of meromorphic and holomorphic functions, respectively; the correspondence is such that any global section of $ {\mathcal M} / {\mathcal O} $ corresponds to some first Cousin problem (the value of the section $ \kappa $ corresponding to data $ \{ f _ \alpha \} $ at a point $ z \in U _ \alpha $ is the element of $ {\mathcal M} _ {z} / {\mathcal O} _ {z} $ with representative $ f _ \alpha $). The mapping of global sections $ \phi : \Gamma ( {\mathcal M} ) \rightarrow \Gamma ( {\mathcal M} / {\mathcal O} ) $ maps each meromorphic function $ f $ on $ {\mathcal M} $ to a section $ \kappa _ {f} $ of $ {\mathcal M} / {\mathcal O} $, where $ \kappa _ {f} ( z) $ is the class in $ {\mathcal M} _ {z} / {\mathcal O} _ {z} $ of the germ of $ f $ at the point $ z $, $ z \in M $. The localized first Cousin problem is then: Given a global section $ \kappa $ of the sheaf $ {\mathcal M} / {\mathcal O} $, to find a meromorphic function $ f $ on $ M $( i.e. a section of $ {\mathcal M} $) such that $ \phi ( f) = \kappa $.

Theorems concerning the solvability of the first Cousin problem may be regarded as a multi-dimensional generalization of the Mittag-Leffler theorem on the construction of a meromorphic function with prescribed poles. The problem in cohomological formulation, with a fixed covering $ {\mathcal U} $, is solvable (for arbitrary compatible $ \{ f _ \alpha \} $) if and only if $ H ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $( the Čech cohomology for $ {\mathcal U} $ with holomorphic coefficients is trivial).

A specific first Cousin problem on $ M $ is solvable if and only if the corresponding section of $ {\mathcal M} / {\mathcal O} $ belongs to the image of the mapping $ \phi $. An arbitrary first Cousin problem on $ M $ is solvable if and only if $ \phi $ is surjective. On any complex manifold $ M $ one has an exact sequence

$$ \Gamma ( {\mathcal M} ) \mathop \rightarrow \limits ^ \phi \ \Gamma ( {\mathcal M} / {\mathcal O} ) \rightarrow \ H ^ {1} ( M, {\mathcal O} ). $$

If the Čech cohomology for $ M $ with coefficients in $ {\mathcal O} $ is trivial (i.e. $ H ^ {1} ( M , {\mathcal O} ) = 0 $), then $ \phi $ is surjective and $ H ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $ for any covering $ {\mathcal U} $ of $ M $. Thus, if $ H ^ {1} ( M, {\mathcal O} ) = 0 $, any first Cousin problem is solvable on $ M $( in the classical, cohomological and local version). In particular, the problem is solvable in all domains of holomorphy and on Stein manifolds (cf. Stein manifold). If $ D \subset \mathbf C ^ {2} $, then the first Cousin problem in $ D $ is solvable if and only if $ D $ is a domain of holomorphy. An example of an unsolvable first Cousin problem is: $ M = \mathbf C ^ {2} \setminus \{ 0 \} $, $ U _ \alpha = \{ z _ \alpha \neq 0 \} $, $ \alpha = 1, 2 $, $ f _ {1} = ( z _ {1} z _ {2} ) ^ {-} 1 $, $ f _ {2} = 0 $.

Second (multiplicative) Cousin problem.

Given an open covering $ {\mathcal U} = \{ U _ \alpha \} $ of a complex manifold $ M $ and, in each $ U _ \alpha $, a meromorphic function $ f _ \alpha $, $ f _ \alpha \not\equiv 0 $ on each component of $ U _ \alpha $, with the assumption that the functions $ f _ {\alpha \beta } = f _ \alpha f _ \beta ^ {-} 1 $ are holomorphic and nowhere vanishing in $ U _ {\alpha \beta } $ for all $ \alpha , \beta $( compatibility condition). It is required to construct a meromorphic function $ f $ on $ M $ such that the functions $ ff _ \alpha ^ {-} 1 $ are holomorphic and nowhere vanishing in $ U _ \alpha $ for all $ \alpha $.

The cohomological formulation of the second Cousin problem is as follows. Given the covering $ {\mathcal U} $ and functions $ f _ {\alpha \beta } $, holomorphic and nowhere vanishing in the intersections $ U _ {\alpha \beta } $, and forming a multiplicative $ 1 $- cocycle, i.e.

$$ f _ {\alpha \beta } f _ {\beta \alpha } = \ 1 \ \mathop{\rm in} \ U _ {\alpha \beta } , $$

$$ f _ {\alpha \beta } f _ {\beta \gamma } f _ {\gamma \alpha } = 1 \ \mathop{\rm in} U _ \alpha \cap U _ \beta \cap U _ \gamma , $$

it is required to find functions $ h _ \alpha $, holomorphic and nowhere vanishing in $ U _ \alpha $, such that $ f _ {\alpha \beta } = h _ \beta h _ \alpha ^ {-} 1 $ in $ U _ {\alpha \beta } $ for all $ \alpha , \beta $. If the cocycle $ \{ f _ {\alpha \beta } \} $ corresponds to the data of a second Cousin problem and the required $ h _ \alpha $ exist, then the function $ f = \{ f _ \alpha h _ {\alpha } \mathop{\rm in} U _ \alpha \} $ is defined and meromorphic throughout $ M $ and is a solution to the given second Cousin problem. Conversely, if a specific second Cousin problem is solvable, then the corresponding cocycle is a holomorphic coboundary.

The localized second Cousin problem. To each set of data $ \{ U _ \alpha , f _ \alpha \} $ for the second Cousin problem there corresponds a uniquely defined global section of the sheaf $ {\mathcal M} ^ {*} / {\mathcal O} ^ {*} $( in analogy to the first Cousin problem), where $ {\mathcal M} ^ {*} = {\mathcal M} \setminus \{ 0 \} $( with 0 the null section) is the multiplicative sheaf of germs of meromorphic functions and $ {\mathcal O} ^ {*} $ is the subsheaf of $ {\mathcal O} $ in which each stalk $ {\mathcal O} _ {z} ^ {*} $ consists of germs of holomorphic functions that do not vanish at $ z $. The mapping of global sections

$$ \Gamma ( {\mathcal M} ^ {*} ) \mathop \rightarrow \limits ^ \psi \ \Gamma ( {\mathcal M} ^ {*} / {\mathcal O} ^ {*} ) $$

maps a meromorphic function $ f $ to a section $ \kappa _ {f} ^ {*} $ of the sheaf $ {\mathcal M} ^ {*} / {\mathcal O} ^ {*} $, where $ \kappa _ {f} ^ {*} ( z) $ is the class in $ {\mathcal M} _ {z} ^ {*} / {\mathcal O} _ {z} ^ {*} $ of the germ of $ f $ at $ z $, $ z \in M $. The localized second Cousin problem is: Given a global section $ \kappa ^ {*} $ of the sheaf $ {\mathcal M} ^ {*} / {\mathcal O} ^ {*} $, to find a meromorphic function $ f $ on $ M $, $ f \not\equiv 0 $ on the components of $ M $( i.e. a global section of $ {\mathcal M} ^ {*} $), such that $ \psi ( f ) = \kappa ^ {*} $.

The sections of $ M ^ {*} / Q ^ {*} $ uniquely correspond to divisors (cf. Divisor), therefore $ {\mathcal M} ^ {*} / {\mathcal O} ^ {*} = {\mathcal D} $ is called the sheaf of germs of divisors. A divisor on a complex manifold $ M $ is a formal locally finite sum $ \sum k _ {j} \Delta _ {j} $, where $ k _ {j} $ are integers and $ \Delta _ {j} $ analytic subsets of $ M $ of pure codimension 1. To each meromorphic function $ f $ corresponds the divisor whose terms are the irreducible components of the zero and polar sets of $ f $ with respective multiplicities $ k _ {j} $, with multiplicities of zeros considered positive and those of poles negative. The mapping $ \psi $ maps each function $ f $ to its divisor $ ( f ) $; such divisors are called proper divisors. The second Cousin problem in terms of divisors is: Given a divisor $ \Delta $ on the manifold $ M $, to construct a meromorphic function $ f $ on $ M $ such that $ \Delta = ( f ) $.

Theorems concerning the solvability of the second Cousin problem may be regarded as multi-dimensional generalizations of Weierstrass' theorem on the construction of a meromorphic function with prescribed zeros and poles. As in the case of the first Cousin problem, a necessary and sufficient condition for the solvability of any second Cousin problem in cohomological version is that $ H ^ {1} ( M, {\mathcal O} ^ {*} ) = 0 $. Unfortunately, the sheaf $ {\mathcal O} ^ {*} $ is not coherent, and this condition is less effective. The attempt to reduce a given second Cousin problem to a first Cousin problem by taking logarithms encounters an obstruction in the form of an integral $ 2 $- cocycle, and one obtains an exact sequence

$$ H ^ {1} ( M, {\mathcal O} ) \rightarrow \ H ^ {1} ( M, {\mathcal O} ^ {*} ) \mathop \rightarrow \limits ^ \alpha \ H ^ {2} ( M, \mathbf Z ), $$

where $ \mathbf Z $ is the constant sheaf of integers. Thus, if $ H ^ {1} ( M, {\mathcal O} ) = H ^ {2} ( M, \mathbf Z ) = 0 $, any second Cousin problem is solvable on $ M $, and any divisor is proper. If $ M $ is a Stein manifold, then $ \alpha $ is an isomorphism; hence the topological condition $ H ^ {2} ( M, \mathbf Z ) = 0 $ on a Stein manifold $ M $ is necessary and sufficient for the second Cousin problem in cohomological version to be solvable. The composite mapping $ c = \alpha \circ \beta $,

$$ \Gamma ( {\mathcal D} ) \mathop \rightarrow \limits ^ \beta \ H ^ {1} ( M, {\mathcal O} ^ {*} ) \mathop \rightarrow \limits ^ \alpha \ H ^ {2} ( M, \mathbf Z ) $$

maps each divisor $ \Delta $ to an element $ c( \Delta ) $ of the group $ H ^ {2} ( M, \mathbf Z ) $, which is known as the Chern class of $ \Delta $. The specific second Cousin problem corresponding to $ \Delta $ is solvable, assuming $ H ^ {1} ( M, {\mathcal O} ) = 0 $, if and only if the Chern class of $ \Delta $ is trivial: $ c( \Delta ) = 0 $. On a Stein manifold, the mapping $ c $ is surjective; moreover, every element in $ H ^ {2} ( M, \mathbf Z ) $ may be expressed as $ c( \Delta ) $ for some divisor $ \Delta $ with positive multiplicities $ k _ {j} $. Thus, the obstructions to the solution of the second Cousin problem on a Stein manifold $ M $ are completely described by the group $ H ^ {2} ( M, \mathbf Z ) $.

Examples.

1) $ M = \mathbf C ^ {2} \setminus \{ z _ {1} = z _ {2} , | z _ {1} | = 1 \} $; the first Cousin problem is unsolvable; the second Cousin problem is unsolvable, e.g., for the divisor $ \Delta = M \cap \{ z _ {1} = z _ {2} , | z _ {1} | < 1 \} $ with multiplicity 1.

2) $ M = \{ | z _ {1} ^ {2} + z _ {2} ^ {2} + z _ {3} ^ {2} - 1 | < 1 \} \subset \mathbf C ^ {3} $, $ \Delta $ is one of the components of the intersection of $ M $ and the plane $ z _ {2} = iz _ {1} $ with multiplicity 1. The second Cousin problem is unsolvable ( $ M $ is a domain of holomorphy, the first Cousin problem is solvable).

3) The first and second Cousin problems are solvable in domains $ D = D _ {1} \times \dots \times D _ {n} \subset \mathbf C ^ {n} $, where $ D _ {j} $ are plane domains and all $ D _ {j} $, with the possible exception of one, are simply connected.

References

[1] P. Cousin, "Sur les fonctions de $n$ variables" Acta Math. , 19 (1895) pp. 1–62 MR1554861 Zbl 26.0456.02
[2] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[3] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601

Comments

The Cousin problems are related to the Poincaré problem (is a meromorphic function given on a complex manifold $ X $ globally the quotient of two holomorphic functions whose germs are relatively prime for all $ x \in X $?) and to the, more algebraic, Theorems A and B of H. Cartan and J.-P. Serre, cf. [a1], [a2], [a3].

References

[a1] C.A. Cazacu, "Theorie der Funktionen mehreren komplexer Veränderlicher" , Birkhäuser (1975) (Translated from Rumanian)
[a2] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) MR0580152 Zbl 0433.32007
[a3] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5 MR0344507 Zbl 0271.32001
[a4] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 MR0635928 Zbl 0471.32008
[a5] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 6 MR0847923
How to Cite This Entry:
Cousin problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cousin_problems&oldid=24406
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article