Cousin problems
Encyclopedia
In mathematics
, the Cousin problems are two questions in several complex variables
, concerning the existence of meromorphic function
s that are specified in terms of local data. They were introduced in special cases by P. Cousin in 1895. They are now posed, and solved, for any complex manifold
M, in terms of conditions on M.
For both problems, an open cover of M by sets Ui is given, along with a meromorphic function fi on each Ui.
is a holomorphic function
, where it is defined. It asks for a meromorphic function f on M such that
is holomorphic on Ui; in other words, that f shares the singular
behaviour of the given local function. The given condition on the fi − fj is evidently necessary for this; so the problem amounts to asking if it is sufficient. The case of one variable is the Mittag-Leffler theorem on prescribing poles, when M is an open subset of the complex plane
. Riemann surface
theory shows that some restriction on M will be required. The problem can always be solved on a Stein manifold
.
The first Cousin problem may be understood in terms of sheaf cohomology
as follows. Let K be the sheaf
of meromorphic functions and O the sheaf of holomorphic functions on M. A global section ƒ of K passes to a global section φ(ƒ) of the quotient sheaf K/O. The converse question is the first Cousin problem: given a global section of K/O, is there a global section of K from which it arises? The problem is thus to characterize the image of the map
By the long exact cohomology sequence,
is exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,O) vanishes. In particular, by Cartan's theorem B
, the Cousin problem is always solvable if M is a Stein manifold.
is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that
is holomorphic and non-vanishing. The second Cousin problem is a multi-dimensional generalization of the Weierstrass theorem on the existence of a holomorphic function of one variable with prescribed zeros and poles.
The attack on this problem by means of taking logarithm
s, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class
. In terms of sheaf theory, let O∗ be the sheaf of holomorphic functions that vanish nowhere, and K∗ the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian group
s, and the quotient sheaf K∗/O∗ is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map φ
The long exact sheaf cohomology sequence associated to the quotient is
so the second Cousin problem is solvable in all cases provided that H1(M,O∗) = 0. The quotient sheaf K∗/O∗ is the sheaf of germs of Cartier divisors on M. The question of whether every global section is generated by a meromorphic function is thus equivalent to determining whether every line bundle
on M is trivial.
The cohomology group H1(M,O∗),
for the multiplicative structure on O∗, can be compared with the cohomology group H1(M,O) with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves
where the leftmost sheaf is the locally constant sheaf with fiber
. The obstruction to defining a logarithm at the level of H1 is in 
, from the long exact cohomology sequence
When M is a Stein manifold, the middle arrow is an isomorphism because Hq(M,O) = 0, for
so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Cousin problems are two questions in several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, concerning the existence of meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s that are specified in terms of local data. They were introduced in special cases by P. Cousin in 1895. They are now posed, and solved, for any complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
M, in terms of conditions on M.
For both problems, an open cover of M by sets Ui is given, along with a meromorphic function fi on each Ui.
First Cousin problem
The first Cousin problem or additive Cousin problem assumes that each difference- fi − fj
is a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
, where it is defined. It asks for a meromorphic function f on M such that
- f − fi
is holomorphic on Ui; in other words, that f shares the singular
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
behaviour of the given local function. The given condition on the fi − fj is evidently necessary for this; so the problem amounts to asking if it is sufficient. The case of one variable is the Mittag-Leffler theorem on prescribing poles, when M is an open subset of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
theory shows that some restriction on M will be required. The problem can always be solved on a Stein manifold
Stein manifold
In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
.
The first Cousin problem may be understood in terms of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
as follows. Let K be the sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of meromorphic functions and O the sheaf of holomorphic functions on M. A global section ƒ of K passes to a global section φ(ƒ) of the quotient sheaf K/O. The converse question is the first Cousin problem: given a global section of K/O, is there a global section of K from which it arises? The problem is thus to characterize the image of the map
By the long exact cohomology sequence,
is exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,O) vanishes. In particular, by Cartan's theorem B
Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.Theorem A states that F is spanned...
, the Cousin problem is always solvable if M is a Stein manifold.
Second Cousin problem
The second Cousin problem or multiplicative Cousin problem assumes that each ratio- fi/fj
is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that
- f/fi
is holomorphic and non-vanishing. The second Cousin problem is a multi-dimensional generalization of the Weierstrass theorem on the existence of a holomorphic function of one variable with prescribed zeros and poles.
The attack on this problem by means of taking logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
. In terms of sheaf theory, let O∗ be the sheaf of holomorphic functions that vanish nowhere, and K∗ the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, and the quotient sheaf K∗/O∗ is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map φ
The long exact sheaf cohomology sequence associated to the quotient is
so the second Cousin problem is solvable in all cases provided that H1(M,O∗) = 0. The quotient sheaf K∗/O∗ is the sheaf of germs of Cartier divisors on M. The question of whether every global section is generated by a meromorphic function is thus equivalent to determining whether every line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
on M is trivial.
The cohomology group H1(M,O∗),
for the multiplicative structure on O∗, can be compared with the cohomology group H1(M,O) with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves
where the leftmost sheaf is the locally constant sheaf with fiber
. The obstruction to defining a logarithm at the level of H1 is in , from the long exact cohomology sequenceWhen M is a Stein manifold, the middle arrow is an isomorphism because Hq(M,O) = 0, for
so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that .