Теория функций нескольких комплексных переменных - раздел математики, имеющий дело с комплексными функциями . Функцияна комплексном пространстве координат наборов комплексных чисел из n .
Как и в случае комплексного анализа функций одной переменной , который имеет место n = 1 , изучаемые функции являются голоморфными или комплексно-аналитическими, так что локально они являются степенными рядами по переменным z i . Эквивалентно, они локально равномерные пределы из многочленов ; или локальные решения n -мерных уравнений Коши – Римана . Для одной комплексной переменной произвольная область [примечание 1] () [ требуется пояснение ] всегда была областью голоморфности , но для нескольких комплексных переменных произвольная область () не всегда является областью голоморфности, поэтому область голоморфности является одной из тем в этой области. Исправление локальных данных мероморфных функций , то есть проблема создания глобальной мероморфной функции из нулей и полюсов, называется проблемой Кузена. Кроме того, интересные явления, которые происходят с несколькими комплексными переменными, фундаментально важны для изучения компактных комплексных многообразий и проективных комплексных многообразий и имеют иную окраску, чем сложная аналитическая геометрия вили на многообразиях Штейна .
Историческая перспектива
Многие примеры таких функций были известны в математике девятнадцатого века: абелевы функции , тета-функции и некоторые гипергеометрические ряды . Естественно, кандидатом также является любая функция одной переменной, которая зависит от некоторого сложного параметра . Однако теория за долгие годы не стала полноценной областью математического анализа , так как не были раскрыты ее характерные явления. Теорема Вейерштрасса препарат теперь будет классифицироваться как коммутативной алгебры ; он действительно оправдал локальную картину, разветвленность , которая обращается к обобщению точек ветвления теории римановой поверхности .
С работами Фридриха Хартогса и Киёси Ока в 1930-х годах начала появляться общая теория; в то время в этом районе работали Генрих Бенке , Питер Таллен и Карл Штайн . Гартогс доказал некоторые основные результаты, такие , как любой изолированной особенности является съемным , для любой аналитической функции
всякий раз, когда n > 1 . Естественно, с аналогами контурных интегралов будет труднее справиться: когда n = 2 , интеграл, окружающий точку, должен быть над трехмерным многообразием (поскольку мы находимся в четырех реальных измерениях), в то время как контурные (линейные) интегралы повторяются над двумя отдельными комплексными переменные должны прийти к двойному интегралу по двумерной поверхности. Это означает, что исчисление вычетов должно будет принять совсем другой характер.
После 1945 года важная работа во Франции, на семинаре Анри Картана , и в Германии с Гансом Грауэртом и Райнхольдом Реммертом , быстро изменила картину теории. Был прояснен ряд вопросов, в частности аналитического продолжения . Здесь основное различие очевидно из теории одной переменной: в то время как для любого открытого связного множества D вмы можем найти функцию, которая нигде не будет аналитически продолжаться через границу, чего нельзя сказать при n > 1 . На самом деле D такого типа довольно специфичны по своей природе (удовлетворяют условию псевдовыпуклости ). Естественные области определения функций, продолженные до предела, называются многообразиями Штейна, и их природа заключалась в том, чтобы обращать в нуль группы когомологий пучков. Кроме того, свойство исчезновения группы когомологий пучков обнаруживается и в других многомерных комплексных многообразиях, что указывает на что многообразие Ходжа проективно. Фактически, именно необходимость поставить (в частности) работу Оки на более ясную основу, быстро привела к последовательному использованию пучков для формулировки теории (что имело серьезные последствия для алгебраической геометрии , в частности, из работ Грауэрта).
С этого момента существовала основополагающая теория, которую можно было применять к аналитической геометрии , [примечание 2] автоморфных форм нескольких переменных и уравнений в частных производных . Теория деформации сложных структур и комплексных многообразий в общих чертах описана Кунихико Кодаира и Д.К. Спенсером . Знаменитая статья GAGA из Серры [ссылка 1] скованы точками кроссовера от Geometrie Аналитического до Geometrie algébrique .
Слышали, как К.Л. Сигель жаловался, что новая теория функций нескольких комплексных переменных содержит мало функций, а это означает, что специальная функциональная сторона теории подчинена пучкам. Интерес для теории чисел , конечно же, вызывают конкретные обобщения модулярных форм . Классические кандидаты являются модульными формами Гильберта и Siegel модульных форм . В наши дни они ассоциированы с алгебраическими группами (соответственно, ограничение Вейля из поля вполне вещественных чисел в GL (2) и симплектическая группа ), для которых случается, что автоморфные представления могут быть получены из аналитических функций. В некотором смысле это не противоречит Зигелю; современная теория имеет свои, разные направления.
Последующие разработки включали теорию гиперфункций и теорему о краю клина , обе из которых были в некоторой степени вдохновлены квантовой теорией поля . Есть ряд других областей, таких как теория банаховой алгебры , которые используют несколько комплексных переменных.
Комплексное координатное пространство
Комплекс координатного пространства это декартово произведение из п копий, и когда область голоморфности, можно рассматривать как многообразие Штейна . Это также n -мерное векторное пространство над комплексными числами , что дает его размерность 2 n над. [примечание 3] Следовательно, как множество и как топологическое пространство ,можно отождествить с реальным координатным пространством и его топологическая размерность , таким образом, равна 2 n .
На безкоординатном языке любое векторное пространство над комплексными числами можно рассматривать как реальное векторное пространство с вдвое большим числом измерений, где сложная структура задается линейным оператором J (таким, что J 2 = - I ), который определяет умножение на мнимую единицу i .
Любое такое пространство, как реальное, ориентировано . На комплексной плоскости, рассматриваемой как декартова плоскость , умножение на комплексное число w = u + iv может быть представлено вещественной матрицей
с определителем
Аналогичным образом, если выразить любой конечномерный комплексный линейный оператор как вещественную матрицу (которая будет составлена из блоков 2 × 2 вышеупомянутой формы), то его определитель равен квадрату модуля соответствующего комплексного определителя. Это неотрицательное число, которое означает, что (реальная) ориентация пространства никогда не меняется на противоположную с помощью сложного оператора. То же самое относится к якобианам из голоморфных функций из к .
Подключенное пространство
Каждое произведение семейства связных (соответственно линейно связных) пространств связно (соответственно линейно связно).
Компактный
По теореме Тихонова пространство, отображаемое декартовым произведением, состоящим из любой комбинации компактных пространств, является компактным пространством.
Голоморфные функции
Функция f, определенная в областиназывается голоморфной, если f удовлетворяет следующим двум условиям. [примечание 4] [ссылка 2]
- f непрерывна [примечание 5] на D [примечание 6]
- Для каждой переменной , f голоморфна, а именно,
( 1 )
которое является обобщением уравнений Коши – Римана (с использованием частной производной Виртингера ) и берет начало в методах дифференциального уравнения Римана.
Уравнения Коши – Римана
Пусть для каждого индекса λ
и обобщив обычное уравнение Коши – Римана для одной переменной для каждого индекса λ, получим
( 2 )
Позволять
через
приведенные выше уравнения (1) и (2) оказываются эквивалентными.
Интегральная формула Коши
е удовлетворяет условиям быть непрерывным и отдельно homorphic на области D . Каждый диск имеет исправляемую кривую, кусочная гладкость, класс Замкнутая кривая Жордана. () Позволять быть областью, окруженной каждым . Замыкание декартова произведения является . Также возьмите полидиск так что становится . ( и разреши - центр каждого диска.) Повторно используя интегральную формулу Коши для одной переменной,
Так как является спрямляемой жордановой замкнутой кривой [примечание 7], а функция f непрерывна, поэтому порядок произведений и сумм можно поменять местами, чтобы повторный интеграл можно было вычислить как кратный интеграл . Следовательно,
( 3 )
Если в случае одной переменной интегральная формула Коши представляет собой интеграл по окружности диска с некоторым радиусом r , то в случае нескольких переменных по поверхности полидиска с радиусамикак в (3).
Формула оценки Коши
Поскольку порядок произведений и сумм взаимозаменяем, из ( 3 ) получаем
( 4 )
f дифференцируема любое количество раз, а производная непрерывна.
Из (4), если f голоморфна, на полидиске а также , получается следующее уравнение оценки.
Следовательно, теорема Лиувилля верна .
Разложение голоморфных функций в степенной ряд
Если функция f голоморфна, на полидиске, from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.
(5)
In addition, f that satisfies the following conditions is called an analytic function.
For each point , is expressed as a power series expansion that is convergent on D :
We have already explained that holomorphic functions are analytic. Also, from the theorem derived by Weierstrass , we can see that the analytic function (convergent power series) is holomorphic.
- If a sequence of functions which converges uniformly on compacta inside a domain D, the limit function f of also uniformly on compacta inside a domain D. Also, respective partial derivative of also compactly converges on domain D to the corresponding derivative of f.
- [ref 3]
Radius of convergence of power series
It is possible to define a combination of positive real numbers such that the power series converges uniformly at and does not converge uniformly at .
In this way it is possible to have a similar, combination of radius of convergence[note 8] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.
Identity theorem
When the function f,g is holomorphic in the concatenated domain D,[note 9] even for several complex variables, the identity theorem[note 10] holds on the domain D, because it has a power series expansion the neighbourhood of holomorphic point. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold.
Biholomorphism
From the establishment of the inverse function theorem, the following mapping can be defined.
For the domain U, V of the n-dimensional complex space , the bijective holomorphic function and the inverse mapping is also holomorphic. At this time, is called a U, V biholomorphism also, we say that U and V are biholomorphically equivalent or that they are biholomorphic.
The Riemann mapping theorem does not hold
When , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.[ref 4]
Analytic continuation
Let U, V be domain on , and . Assume that and is a connected component of . If then f is said to be connected to V, and g is said to be analytic continuation of f. From the identity theorem, if g exists, for each way of choosing w it is unique. Whether or not the definition of this analytic continuation is well-defined should be considered whether the domains U,V and W can be defined well. Several complex variables have restrictions on this domain, and depending on the shape of the domain , all holomorphic functions f belonging to U are connected to V, and there may be not exist function f with as the natural boundary. In other words, U cannot be defined. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of Several complex variables. Also, in the general dimension, there may be multiple intersections between U and V. That is, f is not connected as a single-valued holomorphic function, but as a multivalued holomorphic function. This means that W is not unique and has different properties in the neighborhood of the branch point than in the case of one variable.
Райнхардт домен
Power series expansion of several complex variables it is possible to define the combination of radius of convergence similar to that of one complex variable, but each variable cannot independently define a unique radius of convergence. The Reinhardt domain is considered in order to investigate the characteristics of the convergence domain of the power series, but when considering the Reinhardt domain, it can be seen that the convergence domain of the power series satisfies the convexity called Logarithmically-convex. There are various convexity for the convergence domain of Several complex variables.
A domain D in the complex space , , with centre at a point , with the following property: Together with any point , the domain also contains the set
A Reinhardt domain D with is invariant under the transformations , , . The Reinhardt domains constitute a subclass of the Hartogs domains [ref 5] and a subclass of the circular domains, which are defined by the following condition: Together with any , the domain contains the set
i.e. all points of the circle with center and radius that lie on the complex line through and .
A Reinhardt domain D is called a complete Reinhardt domain if together with any point it also contains the polydisc
A complete Reinhardt domain is star-like with respect to its centre a. Therefore, the complete Reinhardt domain is simply connected, also when the complete Reinhardt domain is the boundary line, there is a way to prove Cauchy's integral theorem without using the Jordan curve theorem.
Logarithmically-convex
A Reinhardt domain D is called logarithmically convex if the image of the set
under the mapping
is a convex set in the real space . An important property of logarithmically-convex Reinhardt domains is the following: Every such domain in is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in , and conversely: The domain of convergence of any power series in is a logarithmically-convex Reinhardt domain with centre . [note 11]
Some results
Thullen's classic results
Thullen's[ref 6] classical result says that a 2-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:
- (polydisc);
- (unit ball);
- (Thullen domain).
Hartogs's extension theorem and Hartogs's phenomenon
Look at the example on the Hartogs's phenomenon in terms of the Reinhardt domain.
- On the polydisk consisting of two disks when .
- Internal domain of
- Hartogs's extension theorem (1906); [ref 7] Let f be a holomorphic function on a setG \ K, where G is a domain on ( n ≥ 2) and K is a compact subset of G. If the complementG \ K is connected, then arbitrary holomorphic function f regardless of how it is chosen can be each extended to a unique holomorphic function on G. It is also called Osgood–Brown theorem is that for several complex variables, singular points cannot be isolated points but are accumulation points. This means that the various properties that hold for one-variable complex variables do not hold for several complex variables. The nature of these singularities is also derived from Weierstrass preparation theorem.
From Hartogs's extension theorem the convergence domain extends from to . Looking at this from the perspective of the Reinhardt domain, is the Reinhardt domain containing the center z = 0, and the convergence domain of has been extended to the smallest complete Reinhardt domain containing .[ref 8]
Sunada's results
Toshikazu Sunada (1978)[ref 9] established a generalization of Thullen's result:
- Two n-dimensional bounded Reinhardt domains and are mutually biholomorphic if and only if there exists a transformation given by , being a permutation of the indices), such that .
Область голоморфности
When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of the domain are natural boundaries (i.e. domain of holomorphy), the first result in the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen. Levi's problem shows that the pseudoconvex domain was a domain of holomorphy.[ref 10][ref 11][ref 12][ref 13] Also Kiyoshi Oka's idéal de domaines indéterminés[ref 14] is interpreted by Cartan.[ref 15][note 12] In sheaf[ref 16] theory, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[ref 17]
Definition
When a function f is holomorpic on the domain and cannot directly connect to the domain outside D, including the point of the domain boundary , the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f. In other words, the domain of holomorphy D is the supremum of the domain where the holomorphic function f is holomorphic, and the domain D, which is holomorphic, cannot be extended any more. For Several complex variables, i.e. domain , the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.
Formally, a domain D in the n-dimensional complex coordinate space is called a domain of holomorphy if there do not exist non-empty domain and , and such that for every holomorphic function f on D there exists a holomorphic function g on V with on U.
In the case, the arbitrary domain () is always the domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal.
Holomorphically convex hull
The first result on the properties of the domain of holomorphy is the holomorphic convexity of Henri Cartan and Peter Thullen (1932).[ref 18]
The holomorphically convex hull of a given compact set in the n-dimensional complex space is defined as follows.
Let be a domain , or alternatively for a more general definition, let be an dimensional complex analytic manifold. Further let stand for the set of holomorphic functions on G. For a compact set , the holomorphically convex hull of K is
One obtains a narrower concept of polynomially convex hull by taking instead to be the set of complex-valued polynomial functions on G. The polynomially convex hull contains the holomorphically convex hull.
The domain is called holomorphically convex if for every compact subset is also compact in G. Sometimes this is just abbreviated as holomorph-convex.
When , any domain is holomorphically convex since then is the union of with the relatively compact components of .
If satisfies the above holomorphically convexity it has the following properties. The radius of the polydisc satisfies condition also the compact set satisfies and is the domain. In the time that, any holomorphic function on the domain can be direct analytic continuated up to .
Levi convex (approximate from the inside on the analytic polyhedron domain)
is union of increasing sequence of analytic compact surfaces with paracompact and Holomorphically convex properties such that . i.e. Approximate from the inside by analytic polyhedron. [note 13]
Pseudoconvex
Pseudoconvex Hartogs showed that is subharmonic for the radius of convergence in the Hartogs series when the Hartogs series is a one-variable . If such a relationship holds in the domain of holomorphy of Several complex variables, it looks like a more manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain. Pseudoconvex domain are important, as they allow for classification of domains of holomorphy.
Definition of plurisubharmonic function
- A function
- with domain
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
- with
- the function is a subharmonic function on the set
- In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space as follows. An upper semi-continuous function
- is said to be plurisubharmonic if and only if for any holomorphic map
the function
is subharmonic, where denotes the unit disk.
Strictly plurisubharmonic function
Necessary and sufficient condition that the real-valued function , that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is . When the Hermitian matrix of u is positive-definite and class , we call u a strict plural subharmonic function.
(Weakly) pseudoconvex (p-pseudoconvex)
Weak pseudoconvex[ref 19] is defined as : Let be a domain. One says that X is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function on X such that the set is a relatively compact subset of X for all real numbers x. [note 14] i.e there exists a smooth plurisubharmonic exhaustion function .
Strongly pseudoconvex
Strongly pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function ,i.e. is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.[ref 19]
Levi–Krzoska pseudoconvexity
If boundary (i.e. When D is a strongly pseudoconvex domain.), it can be shown that D has a defining function; i.e., that there exists which is so that , and . Now, D is pseudoconvex iff for every and in the complex tangent space at p, that is,
- , we have
If D does not have a boundary, the following approximation result can be useful.
Proposition 1 If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with (smooth) boundary which are relatively compact in D, such that
This is because once we have a as in the definition we can actually find a exhaustion function.
Levi strongly pseudoconvex (Levi total pseudoconvex)
If for every boundary point of D, there exists an analytic variety passing which lies entirely outside D in some neighborhood around , except the point itself. Domain D that satisfies these conditions is called Levi strongly pseudoconvex or Levi total pseudoconvex.[ref 20]
Oka pseudoconvex
Family of Oka's disk
Let n-functions be continuous on , holomorphic in when the parameter t is fixed in [0, 1], and assume that are not all zero at any point on . Then the set is called an analytic disc de-pending on a parameter t, and is called its shell. If and , Q(t) is called Family of Oka's disk.[ref 20]
Definition
When holds on any Family of Oka's disk, D is called Oka pseudoconvex.[ref 20] Oka's proof of Levi's problem was that when the unramified Riemannian domain[ref 21] was a domain of holomorphy (holomorphically convex), it was proved that it was necessary and sufficient that each boundary point of the domain of holomorphy is an Oka pseudoconvex.[ref 11]
Cartan pseudoconvex (Local Levi property)
For every point there exist a neighbourhood U of x and f holomorphic on ( i.e. be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. Such a property is called local Levi property, and the domain that satisfies this property is called the Cartan pseudoconvex domain. The Cartan pseudoconvex domain is itself a pseudoconvex domain and is a domain of holomorphy.[ref 20]
Equivalent conditions (In connection with Levi problem)
For a domain the following conditions are equivalent.[note 15]:
- D is a domain of holomorphy.
- D is holomorphically convex.
- D is Levi convex.
- D is pseudoconvex.
- D is Cartan pseudoconvex.
The implications ,[note 16] ,[note 17] and are standard results. Proving , i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka,[note 18] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).
Properties of the domain of holomorphy
- If are domains of holomorphy, then their intersection is also a domain of holomorphy.
- If is an increasing sequence of domains of holomorphy, then their union is also a domain of holomorphy (see Behnke–Stein theorem).
- If and are domains of holomorphy, then is a domain of holomorphy.
- The first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem.
Пучок
Coherent sheaf
Definition
The definition of the coherent sheaf is as follows.[ref 27]
A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:
- is of finite type over , that is, every point in has an open neighborhood in such that there is a surjective morphism for some natural number ;
- for any open set , any natural number , and any morphism of -modules, the kernel of is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.
Also, Jean-Pierre Serre (1955)[ref 27] proves that
- If in an exact sequence of sheaves of -modules two of the three sheaves are coherent, then the third is coherent as well.
A quasi-coherent sheaf on a ringed space is a sheaf of -modules which has a local presentation, that is, every point in has an open neighborhood in which there is an exact sequence
for some (possibly infinite) sets and .
Oka's coherent theorem for sheaf of holomorphic function germ
Kiyoshi Oka (1950)[ref 14][ref 28] proved the following
- Sheaf of holomorphic function germ on the complex manifold is the coherent sheaf. Therefore, from the above Serre(1955) theorem, is also a coherent sheaf. This theorem is also used to prove Cartan's theorems A and B.
Ideal sheaf
If is a closed subscheme of a locally Noetherian scheme , the sheaf of all regular functions vanishing on is coherent. Likewise, if is a closed analytic subspace of a complex analytic space , the ideal sheaf is coherent.
Cousin problem
In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given pole, and Weierstrass factorization theorem was able to create a global meromorphic function from a given zero. The theory of Riemann's surface suggests that in multivariate complex functions, the similar theorem that holds for one-variable complex functions does not hold unless Several restrictions are added to the open Complex manifold. This problem is called the Cousin problem and is formulated in Sheaf cohomology terms. They were introduced in special cases by Pierre Cousin in 1895.[ref 29] It was Kiyoshi Oka who gave the complete answer to this question.[ref 30][ref 31][ref 32]
First Cousin problem
Definition without Sheaf words
Each difference 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.
Definition using Sheaf words
Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. If the next map is surjective, Cousin first problem can be solved.
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.
Second Cousin problem
Definition without Sheaf words
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.
Definition using Sheaf words
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 groups, and the quotient sheaf is well-defined. If the next map is surjective, then Second Cousin problem can be solved.
The long exact sheaf cohomology sequence associated to the quotient is
so the second Cousin problem is solvable in all cases provided that
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 H1 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
Многообразия с несколькими комплексными переменными
Stein manifold (Non-compact complex manifold)
Since an open Riemann surface[ref 33] always has a non-constant single-valued holomorphic function[ref 34] and satisfies the second axiom of countability, the Riemann surface was considered for embedding the one-dimensional complex plane into a complex manifolds. In fact, taking one point at infinity on the one-dimensional complex plane extended it to the Riemann sphere. The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of , whereas it is "rare" for a complex manifold to have a holomorphic embedding into . Consider for example any compact connected complex manifold X: any holomorphic function on it is constant by Liouville's theorem. Now that we know that for Several complex variables, complex manifolds do not always have holomorphic functions that are not constants, consider the conditions that have holomorphic functions. Now if we had a holomorphic embedding of X into , then the coordinate functions of would restrict to nonconstant holomorphic functions on X, contradicting compactness, except in the case that X is just a point. Complex manifolds that can be embedded into are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.
A Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951).[ref 35] A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. If the univalent domain on is connection to a manifold, can be regarded as a complex manifold and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.
Definition
Suppose X is a paracompact complex manifolds of complex dimension and let denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold:
- X is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
- is also a compact subset of X.
- X is holomorphically separable, i.e. if are two points in X, then there exists such that
- The open neighborhood of any point on the manifold has a holomorphic Chart to the .
Non-compact (open) Riemann surfaces are Stein
Let X be a connected, non-compact (open) Riemann surface. A deep theorem (1939)[ref 36] of Heinrich Behnke and Stein (1948)[ref 34] asserts that X is a Stein manifold.
Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so . The exponential sheaf sequence leads to the following exact sequence:
Now Cartan's theorem B shows that , therefore .
This is related to the solution of the second (multiplicative) Cousin problem.
Levi problem
Cartan extended Levi's problem to Stein manifolds.[ref 37]
- If the relative compact open subset of the Stein manifold X is a Cartan pseudoconvex, then D is a Stein manifold, and conversely, if D is a Cartan pseudoconvex, then X is a Stein manifold. i.e. Then X is a Stein manifold if and only if D is locally the Stein manifold. [ref 38]
This was proved by Bremermann[ref 39] by embedding it in a sufficiently high dimensional , and reducing it to the result of Oka.[ref 11]
Also, Grauert proved for arbitrary complex manifolds.[ref 40][ref 13]
- If the relative compact subset of a arbitrary complex manifold X is a strongly pseudoconvex[note 19] on X, then X is a holomorphically convex (i.e. Stein manifold). Also, D is itself a Stein manifold.
Properties and examples of Stein manifolds
- The standard[note 20] complex space is a Stein manifold.
- Every domain of holomorphy in is a Stein manifold.
- It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
- The embedding theorem for Stein manifolds states the following: Every Stein manifold X of complex dimension n can be embedded into by a biholomorphic proper map.[ref 42][ref 43]
These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).
- Every Stein manifold of (complex) dimension n has the homotopy type of an n-dimensional CW-Complex.
- In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem [ref 44] for Riemann surfaces,[note 21] due to Behnke and Stein.[ref 34]
- Every Stein manifold X is holomorphically spreadable, i.e. for every point , there are n holomorphic functions defined on all of X which form a local coordinate system when restricted to some open neighborhood of x.
- The first Cousin problem can always be solved on a Stein manifold.
- Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold.[ref 40] The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function,[ref 45][ref 46] i.e. a smooth real function on X (which can be assumed to be a Morse function) with , such that the subsets are compact in X for every real number c. This is a solution to the so-called Levi problem,[ref 47] named after E. E. Levi (1911). The function invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains. A Stein domain is the preimage . Some authors call such manifolds therefore strictly pseudoconvex manifolds.
- Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface X with a real-valued Morse function f on X such that, away from the critical points of f, the field of complex tangencies to the preimage is a contact structure that induces an orientation on Xc agreeing with the usual orientation as the boundary of That is, is a Stein filling of Xc.
Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers. See for example Cartan's theorems A and B, relating to sheaf cohomology.
In the GAGA set of analogies, Stein manifolds correspond to affine varieties.
Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".
Complex projective varieties (Compact complex manifold)
Considered in the case of one-variable complex function, the compact Riemann surface had a non-constant single-valued meromorphic function.[ref 33] The compact one-dimensional complex manifold was the Riemann sphere . However, for high-dimensional (several complex variables) compact complex manifolds, the existence of meromorphic functions cannot be easily indicated because the singularity is not an isolated point. Consider expanding the closed (compact) Riemann surface to a higher dimension, such as, embedding close Complex submanifold M into the . In high-dimensional complex manifolds, the phenomenon that the sheaf cohomology group disappears occurs, and it is Kodaira vanishing theorem and its generalization Nakano vanishing theorem etc. that gives the condition for this phenomenon to occur. Kodaira embedding theorem[ref 48] gives complex Kähler manifold M, with a Hodge metric large enough dimension N into complex projective space, and also embedding from Chow's theorem[ref 49] into Algebraic manifold. These give an example embeddings in manifolds with meromorphic functions.
Смотрите также
- Complex geometry
- CR manifold
- Harmonic maps
- Harmonic morphisms
- Infinite-dimensional holomorphy
- Oka–Weil theorem
Аннотации
- ^ That is an open connected subset.
- ^ a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the analytic geometry learned at school
- ^ The field of complex numbers is a 2-dimensional vector space over real numbers.
- ^ This may seem nontrivial, but it's known as Osgood's lemma. Osgood's lemma can be proved from the establishment of Cauchy's integral formula, also Cauchy's integral formula can be proved by assuming separate holomorphicity and continuity, so it is appropriate to define it in this way.
- ^ It is not separate continuous.
- ^ Using Hartogs's theorem on separate holomorphicity, If condition (ii) is met, it will be derived to be continuous. But, there is no theorem similar to several real variables, and there is no theorem that indicates the continuity of the function, assuming differentiability.
- ^ According to the Jordan curve theorem, domain D is bounded closed set.
- ^ But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge.
- ^ For several variables, the boundary of any domain is not always the natural boundary, so depending on how the domain is taken, there may not be a holomorphic function that makes that domain the natural boundary. See domain of holomorphy for an example of a condition where the boundary of a domain is a natural boundary.
- ^ Note that from Hartogs' extension theorem or Weierstrass preparation theorem , the zeros of holomorphic functions of several variables are not isolated points. Therefore, for several variables it is not enough that is satisfied at the accumulation point.
- ^ The final paragraph reduces to: A Reinhardt domain is a domain of holomorphy if and only if it is logarithmically convex.
- ^ The idea of the sheaf itself is by Jean Leray.
- ^ cannot be "touched from inside" by a sequence of analytic surfaces
- ^ This is a hullomorphically convex hull condition expressed by a plurisubharmonic function. For this reason, it is also called p-pseudoconvex or simply p-convex.
- ^ In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification[ref 22][ref 23] on the complex analytic space (when n = 2, the result by Hirzebruch,[ref 24] when n = 3 the result by Zariski[ref 25] for algebraic varietie.), but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy. [ref 26]
- ^ The Cartan–Thullen theorem
- ^ See Oka's lemma
- ^ Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.
- ^ This condition cannot be replaced with a Cartan pseudoconvex as in the case of a Stein manifold, that is, "If the closed complex submanifold in the non-branched domain on is a Cartan pseudoconvex, it is a Stein manifold." is currently unresolved.[ref 41]
- ^ ( is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.
- ^ The proof method uses an approximation by the polyhedral domain, as in Oka-Weil theorem.
Рекомендации
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Textbooks
- H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen (1934)
- Salomon Bochner and W. T. Martin Several Complex Variables (1948)
- Forster, Otto (1981), Lectures on Riemann surfaces, Graduate Text in Mathematics, 81, New-York: Springer Verlag, ISBN 0-387-90617-7
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- B.V. Shabat, Introduction of complex analysis, 1–2, Moscow (1985) (In Russian)
- V.S. Vladimirov, Methods of the theory of functions of many complex variables, M.I.T. (1966) (Translated from Russian)
- Boris Vladimirovich Shabat, Introduction to Complex Analysis, AMS, 1992
- Lars Hörmander (1990) [1966], An Introduction to Complex Analysis in Several Variables (3rd ed.), North Holland, ISBN 978-1-493-30273-4
- Henri Cartan,Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Paris, Hermann, 1975.
- Elementary theory of analytic functions of one or several complex variables, Dover 1995 (English translation edition)
- Krantz, Steven G. (1992), Function Theory of Several Complex Variables (Second ed.), AMS Chelsea Publishing, p. 340, doi:10.1090/chel/340, ISBN 978-0-8218-2724-6
- R. Michael Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer 1986, 1998
- Volker Scheidemann, Introduction to complex analysis in several variables, Birkhäuser, 2005, ISBN 3-7643-7490-X
- Noguchi, Junjiro (2016), Analytic Function Theory of Several Variables Elements of Oka's Coherence, p. XVIII, 397, doi:10.1007/978-981-10-0291-5, ISBN 978-981-10-0289-2
Encyclopedia of Mathematics
- Solomentsev, E.D. (2001) [1994], "Power series", Encyclopedia of Mathematics, EMS Press
- Solomentsev, E.D. (2001) [1994], "Biholomorphic mapping", Encyclopedia of Mathematics, EMS Press
- Solomentsev, E.D. (2001) [1994], "Reinhardt domain", Encyclopedia of Mathematics, EMS Press
- Chirka, E.M. (2001) [1994], "Hartogs theorem", Encyclopedia of Mathematics, EMS Press
- Onishchik, A.L. (2001) [1994], "Pseudo-convex and pseudo-concave", Encyclopedia of Mathematics, EMS Press
- Solomentsev, E.D. (2001) [1994], "Plurisubharmonic function", Encyclopedia of Mathematics, EMS Press
- Onishchik, A.L. (2001) [1994], "Coherent sheaf", Encyclopedia of Mathematics, EMS Press
- Chirka, E.M. (2001) [1994], "Oka theorems", Encyclopedia of Mathematics, EMS Press
- Chirka, E.M. (2001) [1994], "Cousin problems", Encyclopedia of Mathematics, EMS Press
- Onishchik, A.L. (2001) [1994], "Stein manifold", Encyclopedia of Mathematics, EMS Press
PlanetMath
- This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: Reinhardt domain, Holomorphically convex, Domain of holomorphy, polydisc, biholomorphically equivalent.
дальнейшее чтение
- Krantz, Steven G. (1987), "What is Several Complex Variables?", The American Mathematical Monthly, 94 (3): 236–256, doi:10.2307/2323391, JSTOR 2323391
- Oka, Kiyoshi; R., Remmert(Ed.) (1984), Collected Papers, Springer-Verlag Berlin Heidelberg, p. XIV, 226, ISBN 978-3-662-43412-3CS1 maint: extra text: authors list (link)
Внешние ссылки
- Tasty Bits of Several Complex Variables open source book by Jiří Lebl
- Complex Analytic and Differential Geometry