In
mathematics, the Cousin problems are two questions in
several complex variables, concerning the existence of
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s that are specified in terms of local data. They were introduced in special cases by
Pierre 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
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
of ''M'' by sets ''U
i'' is given, along with a meromorphic function ''f
i'' on each ''U
i''.
First Cousin problem
The first Cousin problem or additive Cousin problem assumes that each difference
:
is a
holomorphic function, where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that
:
is ''holomorphic'' on ''U
i''; in other words, that ''f'' shares the
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
behaviour of the given local function. The given condition on the
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 In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed pole (complex analysis), poles. Conversely, it can be used to express any meromorphic function as a sum of Partial fractions in complex ana ...
on prescribing poles, when ''M'' is an open subset of the
complex plane.
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
theory shows that some restriction on ''M'' will be required. The problem can always be solved on a
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Ste ...
.
The first Cousin problem may be understood in terms of
sheaf cohomology as follows. Let K be the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of meromorphic functions and O the sheaf of holomorphic functions on ''M''. A global section
of K passes to a global section
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 ''H''
1(''M'',O) vanishes. In particular, by
Cartan's theorem B
In mathematics, Cartan's theorems A and B are two results mathematical proof, proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to Function of several complex variables, seve ...
, 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
:
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.
The attack on this problem by means of taking
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s, to reduce it to the additive problem, meets an obstruction in the form of the first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
(see also
exponential sheaf sequence In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.
Let ''M'' be a complex manifold, and write ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be ...
). In terms of sheaf theory, let
be the sheaf of holomorphic functions that vanish nowhere, and
the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of
abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s, and the quotient sheaf
is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map
:
The long exact sheaf cohomology sequence associated to the quotient is
:
so the second Cousin problem is solvable in all cases provided that
The quotient sheaf
is the sheaf of germs of
Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
s 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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
.
The cohomology group
for the multiplicative structure on
can be compared with the cohomology group
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 ''H''
1 is in
, from the long exact cohomology sequence
:
When ''M'' is a Stein manifold, the middle arrow is an isomorphism because
for
so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that
See also
*
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 on a Stein manifold . They are significant both as applied to several complex variables, and in the general development o ...
References
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{{Refend
Complex analysis
Several complex variables
Sheaf theory