Cousin problems
Problems named after P. Cousin [1], who first solved them for certain simple domains in the complex -
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 |
Cousin problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cousin_problems&oldid=53392