Functions of multiple variables which are complex numbers
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As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi. Equivalently, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, the every domain[note 1](), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy.[ref 1] For several complex variables, this is not the case; there exist domains () that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field.[ref 1] Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and Complex projective varieties () and has a different flavour to complex analytic geometry in or on Stein manifolds.
whenever n > 1. Naturally the analogues of contour integrals will be harder to handle; when n = 2 an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface. This means that the residue calculus will have to take a very different character.
After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory; while for every open connected set D in we can find a function that will nowhere continue analytically over the boundary, that cannot be said for n > 1. In fact the D of that kind are rather special in nature (especially in complex coordinate spaces and Stein manifolds, satisfying a condition called pseudoconvexity). The natural domains of definition of functions, continued to the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish, also, the property that the sheaf cohomology group disappears is also found in other high-dimensional complex manifolds, indicating that the Hodge manifold is projective. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work).
the above equations (1) and (2) turn to be equivalent.
Cauchy's integral formula
f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve , is piecewise smoothness, class Jordan closed curve. () Let be the domain surrounded by each . Cartesian product closure is . Also, take the polydisc so that it becomes . ( and let be the center of each disk.) Using Cauchy's integral formula of one variable repeatedly,
Power series expansion of holomorphic functions on polydisc
If function f is holomorphic, on polydisc , from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.
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 on polydisc are analytic. Also, from the theorem derived by Weierstrass , we can see that the analytic function on polydisc (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.
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.
The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many domain, so we introduce Bochner–Martinelli formula.
Suppose that f is a continuously differentiable function on the closure of a domain D on with piecewise smooth boundary . Then the Bochner–Martinelli formula states that if z is in the domain D then, for , z in the Bochner–Martinelli kernel is a differential form in of bidegree defined by
In particular if f is holomorphic the second term vanishes, so
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.
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 arbitrarily. Several complex variables have restrictions on this domain, and depending on the shape of the domain , all analytic functions g belonging to V are connected to , and there may be not exist function g with as the natural boundary. In other words, V cannot be defined arbitrarily. 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 analytic 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.
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 coordinate space , , with centre at a point , with the following property; Together with each 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 7] and a subclass of the circular domains, which are defined by the following condition; Together with all points of , 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 all point it also contains the polydisc
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 every power series in is a logarithmically-convex Reinhardt domain with centre . [note 11]
Hartogs's extension theorem and Hartogs's phenomenon
Hartogs's extension theorem (1906);[ref 8] Let f be a holomorphic function on a setG\K, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain[note 12] on (n ≥ 2) and K is a compact subset of G. If the complementG\K is connected, then every holomorphic function f regardless of how it is chosen can be each extended to a unique holomorphic function on G. [ref 9]
It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for holomorphic functions of several complex variables. The nature of these singularities is also derived from Weierstrass preparation theorem. A generalization of this theorem using the same method as Hartogs was proved in 2007.[ref 10][ref 11]
From Hartogs's extension theorem the domain of convergence extends from to . Looking at this from the perspective of the Reinhardt domain, is the Reinhardt domain containing the center z = 0, and the domain of convergence of has been extended to the smallest complete Reinhardt domain containing .[ref 12]
Thullen's classic results
Thullen's[ref 13] 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:
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 .
Natural domain of the holomorphic function (Domain of holomorphy)
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 (In the complex coordinate space call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen.[ref 15] Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for ,[ref 16] later extended to .[ref 17][ref 18])[ref 19] Also Kiyoshi Oka's idéal de domaines indéterminés[ref 20][ref 21] is interpreted by H. Cartan.[ref 22][note 13] In Sheaf cohomology,[ref 23] the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[ref 24] The concept of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.[ref 1]
Domain of holomorphy
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.
For the case, the every domain () was 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.
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;[ref 25] this is also true, with additional topological assumptions, for the second Cousin problem.
Holomorphically convex hull
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 , every 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 (), every 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 14]
Pseudoconvex Hartogs showed that is subharmonic for the radius of convergence in the Hartogs series when the Hartogs series is a one-variable .[ref 1] 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.[note 15] 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.
In one-complex variable, 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 . There fore, if is of class , then is plurisubharmonic if and only if the hermitian matrix is positive semidefinite.
When the hermitian matrix of u is positive-definite and class , we call u a strict plurisubharmonic function.
(Weakly) pseudoconvex (p-pseudoconvex)
Weak pseudoconvex is defined as: Let be a domain. One says that X is pseudoconvex if there exists a continuousplurisubharmonic function on X such that the set is a relatively compact subset of X for all real numbers x. [note 16] i.e. there exists a smooth plurisubharmonic exhaustion function . Often, the definition of pseudoconvex is used here and is written as; Let X be a complex n-dimensional manifold. Then is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function .[ref 26](p49)
Strongly (Strictly) pseudoconvex
Let X be a complex n-dimensional manifold. 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 26](p49) The strong Levi pseudoconvex domain is simply called strong pseudoconvex and is often called strictly pseudoconvex to make it clear that it has a strictly plurisubharmonic exhaustion function in relation to the fact that it may not have a strictly plurisubharmonic exhaustion function.[ref 27]
(Weakly) Levi(–Krzoska) pseudoconvexity
If boundary , 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
For arbitrary complex manifold, Levi (–Krzoska) pseudoconvexity does not always have an plurisubharmonic exhaustion function, i.e. it does not necessarily have a (p-)pseudoconvex domain.[ref 27]
If D does not have a boundary, the following approximation result can be useful.
Proposition 1If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with class -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.
Strongly Levi (–Krzoska) pseudoconvexity (Strongly pseudoconvex)
When the Levi (–Krzoska) form is positive-definite, it is called Strongly Levi (–Krzoska) pseudoconvex or often called simply 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 total pseudoconvex.[ref 28]
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 28][ref 29]
When holds on any Family of Oka's disk, D is called Oka pseudoconvex.[ref 28] Oka's proof of Levi's problem was that when the unramified Riemannian domain[ref 30] 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 17][ref 29]
Locally pseudoconvex (locally Stein,Cartan pseudoconvex,Local Levi property)
For every point there exist a neighbourhood U of x and f holomorphic. ( i.e. be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. i.e. A holomorphic map will be said to be locally pseudoconvex if every point has a neighborhood U such that is Stein (weakly 1-complete). In this situation, we shall also say that X is locally pseudoconvex over Y. This was also called locally Stein and was classically called Cartan Pseudoconvex. In the Clocally pseudoconvexdomain is itself a pseudoconvex domain and is a domain of holomorphy.[ref 31][ref 28]
Conditions equivalent to domain of holomorphy
For a domain the following conditions are equivalent.[note 17]:
D is a domain of holomorphy.
D is holomorphically convex.
D is Levi convex.
D is pseudoconvex.
D is Locally pseudoconvex.
The implications ,[note 18],[note 19] 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 20] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).
The definition of the coherent sheaf is as follows.[ref 37]
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 arbitrary open set , arbitrary natural number , and arbitrary morphism of -modules, the kernel of is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.
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.
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 cohomology 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.
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 cohomology 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 cohomology 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
Manifolds with several complex variables
Stein manifold (Non-compact manifold)
Since an open Riemann surface[ref 43] always has a non-constant single-valued holomorphic function,[ref 44] and satisfies the second axiom of countability, the open Riemann surface can be thought of complex manifold to have a holomorphic embedding into a one-dimensional complex plane . 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 arbitrary compact connected complex manifold X: every 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 holomorphic embedded into are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.[note 21]
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 45] 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.
Suppose X is a paracompactcomplex 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:[ref 46]
X is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
Cartan extended Levi's problem to Stein manifolds.[ref 48]
If the relative compact open subset of the Stein manifold X is a Locally pseudoconvex, then D is a Stein manifold, and conversely, if D is a Locally 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 49]
This was proved by Bremermann[ref 50] by embedding it in a sufficiently high dimensional , and reducing it to the result of Oka.[ref 17]
A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space.[ref 1][ref 56]
Levi's problem remains unresolved in the following cases;
Suppose that X is a singular Stein space, . Suppose that for all there is an open neighborhood so that is Stein space. Is D itself Stein?[ref 1][ref 57][ref 58]
Suppose that N be a Stein space and f an injective, and also a Riemann unbranched domain, such that map f is a locally pseudoconvex map (i.e. Stein morphism). Then M is itself Stein?[ref 58][ref 59](p109)
Suppose that X be a Stein space and an increasing union of Stein open sets. Then D is itself Stein?
This means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space. [ref 58]
Grauert introduced the concept of K-complete in the proof of Levi's problem.
Let X is complex manifold, X is K-complete if, to each point , there exist finitely many holomorphic map of X into ,, such that is an isolated point of the set .[ref 53] This concept also applies to complex space.[ref 60]
Properties and examples of Stein manifolds
The standard[note 24] 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.
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. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function,[ref 53] 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 64] 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 domain.[ref 65] 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.
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 manifold)
Meromorphic function in one-variable complex function were studied in the closed Riemann surface, because the compact Riemann surface had a non-constant single-valued meromorphic function.[ref 43] 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. In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.[ref 31] Consider expanding the closed (compact) Riemann surface to a higher dimension, such as, embedding close Complex submanifold M into the complex projective space , in particular, Complex projective varieties 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 66] gives complex Kähler manifoldM, with a Hodge metric large enough dimension N into complex projective space, and also Chow's theorem[ref 67] shows that the analytic subspace of a closed complex projective space is an algebraic manifold, and when combined with Kodaira's result, M embeds as an algebraic manifold. These give an example embeddings in manifolds with meromorphic functions. Similarities in Levi problems on the complex projective space , have been proved in some patterns, for example by Takeuchi.[ref 1][ref 68][ref 69][ref 70]
↑ 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.
↑ 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 each domain is not always the natural boundary, so depending on how the domain is taken, there may not be a analytic 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.
↑ 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 32][ref 33] on the complex analytic space (when n = 2, the result by Hirzebruch,[ref 34] when n = 3 the result by Zariski[ref 35] 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 36]
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In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence of domains of holomorphy is again a domain of holomorphy. It was proved by was proved by Heinrich Behnke and Karl Stein in 1938.
In mathematics, a tube domain is a generalization of the notion of a vertical strip in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained.
Bloch's Principle is a philosophical principle in mathematics stated by André Bloch.
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↑ Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem", Math. Ann. (in German), 123: 201–222, doi:10.1007/bf02054949, MR0043219, S2CID122647212
↑ Noguchi, Junjiro (2011). "Another Direct Proof of Oka's Theorem (Oka IX)". arXiv:1108.2078.Cite journal requires |journal= (help)
↑ Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen. 116: 204–216. doi:10.1007/BF01597355. S2CID123982856.
↑ Andreotti, Aldo; Narasimhan, Raghavan (1964). "Oka's Heftungslemma and the Levi Problem for Complex Spaces". Transactions of the American Mathematical Society. 111 (2): 345–366. doi:10.2307/1994247. JSTOR1994247.
↑ Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics. 82 (4): 917–934. doi:10.2307/2372949. JSTOR2372949.
↑ Eliashberg, Yakov; Gromov, Mikhael (1992). "Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 +1". Annals of Mathematics. Second Series. 136 (1): 123–135. doi:10.2307/2946547. JSTOR2946547.
↑ Kodaira, K. (1954). "On Kahler Varieties of Restricted Type (An Intrinsic Characterization of Algebraic Varieties)". Annals of Mathematics. Second Series. 60 (1): 28–48. doi:10.2307/1969701. JSTOR1969701.
↑ Takeuchi, Akira (1964). "Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projecti". Journal of the Mathematical Society of Japan. 10.2969/jmsj/01620159 (2). doi:10.2969/jmsj/01620159.
Cartan, Henri; Takahashi, Reiji (1992). Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes (in French) (6é. ed., nouv. tired.). Paris: Hermann. p.231. ISBN9782705652159.