Cousin problems

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 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