|
|
| Line 1: |
Line 1: |
| 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''.
| | I am Arturo from Oberreitbach. I love to play Bass Guitar. Other hobbies are Australian Football League.<br><br>My web site :: [http://www.ferraripackaging.co.uk/ForumRetrieve.aspx?ForumID=1521&TopicID=193328&NoTemplate=False 4inkjets Discount Printer Supplies Coupon Code] |
| | |
| 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
| |
| | |
| :''f<sub>i</sub>'' − ''f<sub>j</sub>'' | |
| | |
| is a [[holomorphic function]], where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that
| |
| | |
| :''f'' − ''f<sub>i</sub>'' | |
| | |
| 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 ''f<sub>i</sub>'' − ''f<sub>j</sub>'' 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 ''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
| |
| :<math>H^0(M,\mathbf{K}) \xrightarrow{\varphi} 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{\varphi} 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
| |
| | |
| :''f<sub>i</sub>''/''f<sub>j</sub>''
| |
| | |
| is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that
| |
| | |
| :''f''/''f<sub>i</sub>''
| |
| | |
| 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]]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'''<sup>∗</sup> be the sheaf of holomorphic functions that vanish nowhere, and '''K'''<sup>∗</sup> the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of [[abelian group]]s, and the quotient sheaf '''K'''<sup>∗</sup>/'''O'''<sup>∗</sup> is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map φ
| |
| :<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 ''H''<sup>1</sup>(''M'','''O'''<sup>∗</sup>) = 0. The quotient sheaf '''K'''<sup>∗</sup>/'''O'''<sup>∗</sup> 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 ''H''<sup>1</sup>(''M'','''O'''<sup>∗</sup>),
| |
| for the multiplicative structure on '''O'''<sup>∗</sup>, can be compared with the cohomology group ''H''<sup>1</sup>(''M'','''O''') with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves
| |
| :<math>0\to 2\pi i \mathbb{Z}\to \mathbf{O} \xrightarrow{\exp} \mathbf{O}^* \to 0</math>
| |
| where the leftmost sheaf is the locally constant sheaf with fiber <math>\scriptstyle{2\pi i \mathbb{Z}}</math>. The obstruction to defining a logarithm at the level of ''H''<sup>1</sup> is in <math>\scriptstyle{H^2(M,\mathbb{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,\mathbb{Z}) \to H^2(M, \mathbf{O}).</math>
| |
| When ''M'' is a Stein manifold, the middle arrow is an isomorphism because ''H''<sup>q</sup>(''M'','''O''') = 0, 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>\scriptstyle{H^2(M,\mathbb{Z})=0}</math>.
| |
| | |
| == See also ==
| |
| *[[Cartan's theorems A and B]].
| |
| | |
| ==References==
| |
| * {{springer|first=E.M.|last=Chirka|id=c/c026790|title=Cousin problems}}.
| |
| * {{citation|doi=10.1007/BF02402869|first=P.|last=Cousin|title=Sur les fonctions de ''n'' variables|journal=Acta Math.|volume=19|year=1895|pages=1–62}}.
| |
| * {{Citation | last1=Gunning | first1=Robert C. | last2=Rossi | first2=Hugo | title=Analytic Functions of Several Complex Variables | publisher=[[Prentice Hall]] | year=1965}}.
| |
| | |
| [[Category:Complex analysis]]
| |
| [[Category:Several complex variables]]
| |
| [[Category:Sheaf theory]]
| |
I am Arturo from Oberreitbach. I love to play Bass Guitar. Other hobbies are Australian Football League.
My web site :: 4inkjets Discount Printer Supplies Coupon Code