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.[edit]

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

$$\begin{array}{}\text{(1)}& f_{\alpha \beta }+f_{\beta \alpha }=0\mathrm{in}{U}_{\alpha \beta },\end{array}$$

$$f_{\alpha \beta }+f_{\beta \gamma }+f_{\gamma \alpha }=0\mathrm{in}{U}_{\alpha }\cap U_{\beta }\cap U_{\gamma },$$

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

$$\begin{array}{}\text{(2)}& f_{\alpha \beta }=h_{\beta }-h_{\alpha }\end{array}$$

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

$$f=\{f_{\alpha }+h_{\alpha }\mathrm{in}{U}_{\alpha }\}$$

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 $\varphi :\mathrm{\Gamma }(\mathcal{M})\to \mathrm{\Gamma }(\mathcal{M}/\mathcal{O})$ 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 $\varphi (f)=\kappa$.

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 $\varphi$. An arbitrary first Cousin problem on $M$ is solvable if and only if $\varphi$ is surjective. On any complex manifold $M$ one has an exact sequence

$$\mathrm{\Gamma }(\mathcal{M})\stackrel{\varphi }{\to }\mathrm{\Gamma }(\mathcal{M}/\mathcal{O})\to H^1(M,\mathcal{O}).$$

If the Čech cohomology for $M$ with coefficients in $\mathcal{O}$ is trivial (i.e. $H^1(M,\mathcal{O})=0$), then $\varphi$ 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: $M={\mathbf{C}}^2\setminus \{0\}$, $U_{\alpha }=\{z_{\alpha }\ne 0\}$, $\alpha =1,2$, $f_1=(z_1{z}_2{)}^{-}1$, $f_2=0$.

Second (multiplicative) Cousin problem.[edit]

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 $f{f}_{\alpha }^{-}1$ 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.

$$f_{\alpha \beta }{f}_{\beta \alpha }=1\mathrm{in}{U}_{\alpha \beta },$$

$$f_{\alpha \beta }{f}_{\beta \gamma }{f}_{\gamma \alpha }=1\mathrm{in}{U}_{\alpha }\cap U_{\beta }\cap U_{\gamma },$$

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 $f=\{f_{\alpha }{h}_{\alpha }\mathrm{in}{U}_{\alpha }\}$ 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 ${\mathcal{M}}^{\ast }/{\mathcal{O}}^{\ast }$( in analogy to the first Cousin problem), where ${\mathcal{M}}^{\ast }=\mathcal{M}\setminus \{0\}$( with 0 the null section) is the multiplicative sheaf of germs of meromorphic functions and ${\mathcal{O}}^{\ast }$ is the subsheaf of $\mathcal{O}$ in which each stalk ${\mathcal{O}}_z^{\ast }$ consists of germs of holomorphic functions that do not vanish at $z$. The mapping of global sections

$$\mathrm{\Gamma }({\mathcal{M}}^{\ast })\stackrel{\psi }{\to }\mathrm{\Gamma }({\mathcal{M}}^{\ast }/{\mathcal{O}}^{\ast })$$

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

The sections of $M^{\ast }/Q^{\ast }$ uniquely correspond to divisors (cf. Divisor), therefore ${\mathcal{M}}^{\ast }/{\mathcal{O}}^{\ast }=\mathcal{D}$ is called the sheaf of germs of divisors. A divisor on a complex manifold $M$ is a formal locally finite sum $\sum k_j{\mathrm{\Delta }}_j$, where $k_j$ are integers and ${\mathrm{\Delta }}_j$ 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 $\mathrm{\Delta }$ on the manifold $M$, to construct a meromorphic function $f$ on $M$ such that $\mathrm{\Delta }=(f)$.

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 $H^1(M,{\mathcal{O}}^{\ast })=0$. Unfortunately, the sheaf ${\mathcal{O}}^{\ast }$ 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

$$H^1(M,\mathcal{O})\to H^1(M,{\mathcal{O}}^{\ast })\stackrel{\alpha }{\to }{H}^2(M,\mathbf{Z}),$$

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 $c=\alpha \circ \beta$,

$$\mathrm{\Gamma }(\mathcal{D})\stackrel{\beta }{\to }{H}^1(M,{\mathcal{O}}^{\ast })\stackrel{\alpha }{\to }{H}^2(M,\mathbf{Z})$$

maps each divisor $\mathrm{\Delta }$ to an element $c(\mathrm{\Delta })$ of the group $H^2(M,\mathbf{Z})$, which is known as the Chern class of $\mathrm{\Delta }$. The specific second Cousin problem corresponding to $\mathrm{\Delta }$ is solvable, assuming $H^1(M,\mathcal{O})=0$, if and only if the Chern class of $\mathrm{\Delta }$ is trivial: $c(\mathrm{\Delta })=0$. On a Stein manifold, the mapping $c$ is surjective; moreover, every element in $H^2(M,\mathbf{Z})$ may be expressed as $c(\mathrm{\Delta })$ for some divisor $\mathrm{\Delta }$ 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.[edit]

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 $\mathrm{\Delta }=M\cap \{z_1=z_2,|z_1|<1\}$ with multiplicity 1.

2) $M=\{|z_1^2+z_2^2+z_3^2-1|<1\}\subset {\mathbf{C}}^3$, $\mathrm{\Delta }$ is one of the components of the intersection of $M$ and the plane $z_2=i{z}_1$ 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 $D=D_1\times \cdots \times D_n\subset {\mathbf{C}}^n$, where $D_j$ are plane domains and all $D_j$, with the possible exception of one, are simply connected.

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Comments[edit]

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].

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