Editing Cousin problems

{{Short description|Make a meromorphic function from local data in multiple variables}}
In [[mathematics]], the '''Cousin problems''' are two questions in [[Function of several complex variables|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 [[Pierre Cousin (mathematician)|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]] of ''M'' by sets ''U<sub>i</sub>'' is given, along with a meromorphic function ''f<sub>i</sub>'' on each ''U<sub>i</sub>''.

==First Cousin problem==
The '''first Cousin problem''' or '''additive Cousin problem''' assumes that each difference

:<math>f_i-f_j</math>

is a [[holomorphic function]], where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that

:<math>f-f_i</math>

is ''holomorphic'' on ''U<sub>i</sub>''; in other words, that ''f'' shares the [[mathematical singularity|singular]] behaviour of the given local function. The given condition on the <math>f_i-f_j</math> 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 (mathematics)|sheaf]] of meromorphic functions and '''O''' the sheaf of holomorphic functions on ''M''. A global section <math>f</math> of '''K''' passes to a global section <math>\phi(f)</math> 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

:<math>H^0(M,\mathbf{K}) \, \xrightarrow{\phi} \, H^0(M,\mathbf{K}/\mathbf{O}).</math>

By the [[long exact sequence in homology|long exact cohomology sequence]],

:<math>H^0(M,\mathbf{K}) \,\xrightarrow{\phi}\, H^0(M,\mathbf{K}/\mathbf{O})\to H^1(M,\mathbf{O})</math>

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group ''H''<sup>1</sup>(''M'','''O''') vanishes.  In particular, by [[Cartan's theorems A and B|Cartan's theorem B]], 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

:<math>f_i/f_j</math>

is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that

:<math>f/f_i</math>

is holomorphic and non-vanishing. The second Cousin problem is a multi-dimensional generalization of the [[Weierstrass factorization theorem|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]]s, to reduce it to the additive problem, meets an obstruction in the form of the first [[Chern class]] (see also [[exponential sheaf sequence]]). In terms of sheaf theory, let <math>\mathbf{O}^*</math> be the sheaf of holomorphic functions that vanish nowhere, and <math>\mathbf{K}^*</math> the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of [[abelian group]]s, and the quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map <math>\phi</math>

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*).</math>

The long exact sheaf cohomology sequence associated to the quotient is

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*)\to H^1(M,\mathbf{O}^*)</math>

so the second Cousin problem is solvable in all cases provided that <math>H^1(M,\mathbf{O}^*)=0.</math> The quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is the sheaf of germs of [[Cartier divisor]]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 bundle|trivial]].

The cohomology group <math>H^1(M,\mathbf{O}^*),</math> for the multiplicative structure on <math>\mathbf{O}^*</math> can be compared with the cohomology group <math>H^1(M,\mathbf{O})</math> with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

:<math>0\to 2\pi i\Z\to \mathbf{O} \xrightarrow{\exp} \mathbf{O}^* \to  0</math>

where the leftmost sheaf is the locally constant sheaf with fiber <math>2\pi i\Z</math>.  The obstruction to defining a logarithm at the level of ''H''<sup>1</sup> is in <math>H^2(M,\Z)</math>, from the long exact cohomology sequence

:<math>H^1(M,\mathbf{O})\to H^1(M,\mathbf{O}^*)\to 2\pi i H^2(M,\Z) \to H^2(M, \mathbf{O}).</math>

When ''M'' is a Stein manifold, the middle arrow is an isomorphism because <math>H^q(M,\mathbf{O}) = 0</math> for <math>q > 0</math> so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that <math>H^2(M,\Z)=0.</math>

== See also ==
*[[Cartan's theorems A and B]]

==References==
{{Refbegin}}
* {{cite journal |doi=10.24033/bsmf.1409|title=Idéaux et modules de fonctions analytiques de variables complexes|year=1950|last1=Cartan|first1=Henri|journal=Bulletin de la Société Mathématique de France|volume=2|pages=29–64|doi-access=free}}
* {{springer|first=E.M.|last=Chirka|oldid = 46538|title=Cousin problems}}.
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* {{cite journal |last1=Hitotumatu |first1=Sin |title=Cousin problems for ideals and the domain of regularity |journal=Kodai Mathematical Seminar Reports |date=1951 |volume=3 |issue=1–2 |pages=26–32|doi=10.2996/kmj/1138843066|doi-access=free }}
* {{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles|journal=Journal of Science of the Hiroshima University|volume=6|year=1936|pages=245–255|doi=10.32917/hmj/1558749869|doi-access=free}}
* {{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie|journal=Journal of Science of the Hiroshima University|volume=7|year=1937|pages=115–130|doi=10.32917/hmj/1558576819|doi-access=free}}
* {{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin|journal=Journal of Science of the Hiroshima University|volume=9|year=1939|pages=7–19|doi=10.32917/hmj/1558490525|doi-access=free|url=https://projecteuclid.org/journals/journal-of-science-of-the-hiroshima-university-series-a-mathematics-physics-chemistry/volume-9/issue-none/Sur-les-fonctions-analytiques-de-plusieurs-variables-IIIDeuxi%c3%a8me-probl%c3%a8me-de/10.32917/hmj/1558490525.pdf}}
* {{citation|last1=Gunning | first1=Robert C. | last2=Rossi | first2=Hugo | title=Analytic Functions of Several Complex Variables | publisher=[[Prentice Hall]] | year=1965}}.
* {{cite journal |last1=Chorlay |first1=Renaud |title=From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept |journal=Archive for History of Exact Sciences |date=January 2010 |volume=64 |issue=1 |pages=1–73 |doi=10.1007/s00407-009-0052-3|jstor=41342411|s2cid=73633995 }}
{{Refend}}
[[Category:Complex analysis]]
[[Category:Several complex variables]]
[[Category:Sheaf theory]]