Problemas con los primos

En matemáticas , los problemas de Cousin son dos cuestiones en varias variables complejas que tratan sobre la existencia de funciones meromorfas especificadas en términos de datos locales. Fueron introducidos en casos particulares por Pierre Cousin en 1895. Actualmente se plantean y resuelven para cualquier variedad compleja M , en términos de condiciones sobre M.

Para ambos problemas, se da una cubierta abierta de M por conjuntos U _i , junto con una función meromorfa f _i en cada U _i .

El problema del primo primero o problema del primo aditivo supone que cada diferencia${\displaystyle f_{i}-f_{j}}$es una función holomorfa , donde está definida. Pide una función meromorfa f en M tal que${\displaystyle f-f_{i}}$es holomorfa en U _i ; en otras palabras, que f comparte el comportamiento singular de la función local dada. La condición dada en la${\displaystyle f_{i}-f_{j}}$Es evidentemente necesario para esto; por lo tanto, el problema se reduce a preguntar si es suficiente. El caso de una variable es el teorema de Mittag-Leffler sobre la prescripción de polos, cuando M es un subconjunto abierto del plano complejo . La teoría de superficies de Riemann muestra que se requerirá alguna restricción sobre M. El problema siempre se puede resolver en una variedad de Stein .

El problema del primo primero puede entenderse en términos de cohomología de haces de la siguiente manera. Sea K el haz de funciones meromorfas y O el haz de funciones holomorfas en M. Una sección global${\displaystyle f}$de K pasa a una sección global${\displaystyle \phi (f)}$del haz cociente K / O . La pregunta inversa es el primer problema del primo: dada una sección global de K / O , ¿existe una sección global de K de la que surja? El problema consiste, por lo tanto, en caracterizar la imagen del mapa

${\displaystyle H^{0}(M,\mathbf {K} )\,\xrightarrow {\phi } \,H^{0}(M,\mathbf {K} /\mathbf {O} ).}$

Por la larga secuencia de cohomología exacta ,

${\displaystyle H^{0}(M,\mathbf {K} )\,\xrightarrow {\phi } \,H^{0}(M,\mathbf {K} /\mathbf {O} )\to H^{1}(M,\mathbf {O} )}$

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.

The second Cousin problem or multiplicative Cousin problem assumes that each ratio ${\displaystyle f_{i}/f_{j}}$ is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that ${\displaystyle f/f_{i}}$ 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 logarithms, 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 ${\displaystyle \mathbf {O} ^{*}}$ be the sheaf of holomorphic functions that vanish nowhere, and ${\displaystyle \mathbf {K} ^{*}}$ the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian groups, and the quotient sheaf ${\displaystyle \mathbf {K} ^{*}/\mathbf {O} ^{*}}$ is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map ${\displaystyle \phi }$

${\displaystyle H^{0}(M,\mathbf {K} ^{*}){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} ^{*}/\mathbf {O} ^{*}).}$

The long exact sheaf cohomology sequence associated to the quotient is

${\displaystyle H^{0}(M,\mathbf {K} ^{*}){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} ^{*}/\mathbf {O} ^{*})\to H^{1}(M,\mathbf {O} ^{*})}$

so the second Cousin problem is solvable in all cases provided that ${\displaystyle H^{1}(M,\mathbf {O} ^{*})=0.}$ The quotient sheaf ${\displaystyle \mathbf {K} ^{*}/\mathbf {O} ^{*}}$ 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 ${\displaystyle H^{1}(M,\mathbf {O} ^{*}),}$ for the multiplicative structure on ${\displaystyle \mathbf {O} ^{*}}$ can be compared with the cohomology group ${\displaystyle H^{1}(M,\mathbf {O} )}$ with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

${\displaystyle 0\to 2\pi i\mathbb {Z} \to \mathbf {O} {\xrightarrow {\exp }}\mathbf {O} ^{*}\to 0}$

where the leftmost sheaf is the locally constant sheaf with fiber ${\displaystyle 2\pi i\mathbb {Z} }$. The obstruction to defining a logarithm at the level of H¹ is in ${\displaystyle H^{2}(M,\mathbb {Z} )}$, from the long exact cohomology sequence

${\displaystyle H^{1}(M,\mathbf {O} )\to H^{1}(M,\mathbf {O} ^{*})\to 2\pi iH^{2}(M,\mathbb {Z} )\to H^{2}(M,\mathbf {O} ).}$

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

• Cartan's theorems A and B