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The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables, that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function is n-tuples of complex numbers, classically studied on the complex coordinate space .
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 z_{i}. 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.
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series. Naturally also same function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged field in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory.
With work of Friedrich Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function
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).
From this point onwards there was a foundational theory, which could be applied to analytic geometry, ^{ [note 2] } automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper GAGA of Serre ^{ [ref 2] } pinned down the crossover point from géometrie analytique to géometrie algébrique.
C. L. Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it, meaning that the special function side of the theory was subordinated to sheaves. The interest for number theory, certainly, is in specific generalizations of modular forms. The classical candidates are the Hilbert modular forms and Siegel modular forms. These days these are associated to algebraic groups (respectively the Weil restriction from a totally real number field of GL(2), and the symplectic group), for which it happens that automorphic representations can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.
Subsequent developments included the hyperfunction theory, and the edge-of-the-wedge theorem, both of which had some inspiration from quantum field theory. There are a number of other fields, such as Banach algebra theory, that draw on several complex variables.
The complex coordinate space is the Cartesian product of n copies of , and when is a domain of holomorphy, can be regarded as a Stein manifold. is also considered to be a complex projective variety, a Kähler manifold,^{ [ref 3] } etc. It is also an n-dimensional vector space over the complex numbers, which gives its dimension 2n over .^{ [note 3] } Hence, as a set and as a topological space, may be identified to the real coordinate space and its topological dimension is thus 2n.
In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J^{ 2} = −I ) which defines multiplication by the imaginary unit i.
Any such space, as a real space, is oriented. On the complex plane thought of as a Cartesian plane, multiplication by a complex number w = u + iv may be represented by the real matrix
with determinant
Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be composed from 2 × 2 blocks of the aforementioned form), then its determinant equals to the square of absolute value of the corresponding complex determinant. It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator. The same applies to Jacobians of holomorphic functions from to .
Every product of a family of an connected (resp. path-connected) spaces is connected (resp. path-connected).
From Tychonoff's theorem, the space mapped by the cartesian product consisting of any combination of compact spaces is a compact space.
A function f defined on a domain is called holomorphic if f satisfies the following two conditions.^{ [note 4] }^{ [ref 4] }
| (1) |
which is a generalization of the Cauchy–Riemann equations.
For each index λ let
and generalize the usual Cauchy–Riemann equation for one variable for each index λ, then we obtain
| (2) |
Let
through
the above equations (1) and (2) turn to be equivalent.
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,
Because is a rectifiable Jordanian closed curve^{ [note 7] } and f is continuous, so the order of products and sums can be exchanged so the iterated integral can be calculated as a multiple integral. Therefore,
| (3) |
Because the order of products and sums is interchangeable, from ( 3 ) we get
| (4) |
f is class -function.
From (4), if f is holomorphic, on polydisc and , the following evaluation equation is obtained.
Therefore, Liouville's theorem hold.
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.
| (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 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.
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
When the function f,g is analytic 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 point of analytic. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold.
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.
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 6] }
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
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.
A Reinhardt domain D is called logarithmically convex if the image of the set
under the mapping
is a convex set in the real coordinate space .
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] }
Look at the example on the Hartogs's phenomenon in terms of the Reinhardt domain.
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^{ [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:
Toshikazu Sunada (1978)^{ [ref 14] } established a generalization of Thullen's result:
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] }
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.
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 .
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.
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
the function
is subharmonic, where denotes the unit disk.
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.
Equivalently, a -function u is plurisubharmonic if and only if is a positive (1,1)-form.^{ [ref 26] }^{(pp39–40)}
When the hermitian matrix of u is positive-definite and class , we call u a strict plurisubharmonic function.
Weak pseudoconvex is defined as : Let be a domain. One says that X is 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 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)}
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] }
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,
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.
When the Levi (–Krzoska) form is positive-definite, it is called Strongly Levi (–Krzoska) pseudoconvex or often called simply Strongly 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] }
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] }
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] }
For a domain the following conditions are equivalent.^{ [note 17] }:
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:
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.
Also, Jean-Pierre Serre (1955)^{ [ref 37] } proves that
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^{ [ref 20] } says that each sheaf that meets the following conditions is a coherent.^{ [ref 38] }
From the above Serre(1955) theorem, is a coherent sheaf, also, (i) is used to prove Cartan's theorems A and B.
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.
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. However, these theorems do not hold because the singularities of analytic function in several complex variables is not isolated points, 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 39] } It was Kiyoshi Oka who gave the complete answer to this question.^{ [ref 40] }^{ [ref 41] }^{ [ref 42] }
Each difference is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that is holomorphic on U_{i}; in other words, that f shares the singular behaviour of the given local function.
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 H^{1}(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold.
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.
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 H^{1} 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
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 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:^{ [ref 46] }
Let X be a connected, non-compact (open) Riemann surface. A deep theorem (1939)^{ [ref 47] } of Heinrich Behnke and Stein (1948)^{ [ref 44] } 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.
Cartan extended Levi's problem to Stein manifolds.^{ [ref 48] }
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] }
Also, Grauert proved for arbitrary complex manifolds M.^{ [note 23] }^{ [ref 53] }^{ [ref 19] }^{ [ref 51] }
And Narasimhan^{ [ref 54] }^{ [ref 55] } extended Levi's problem to Complex analytic space, a generalized in the singular case of complex manifolds.
Levi's problem remains unresolved in the following cases;
more generalized
and also,
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] }
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).
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".
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 manifold M, 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] }
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about x_{0} converges to the function in some neighborhood for every x_{0} in its domain.
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
In mathematics, complex geometry is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.
In the theory of several complex variables and complex manifolds in mathematics, 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). 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.
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.
In mathematics, plurisubharmonic functions form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions plurisubharmonic functions can be defined in full generality on complex analytic spaces.
In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space A^{p}(D) is the space of all holomorphic functions in D for which the p-norm is finite:
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space C^{n}. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set.
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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