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 $n$ 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