Function of several complex variables

From Wikipedia, the free encyclopedia
(Redirected from Several complex variables)

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.

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 locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.[1][2][3] For one complex variable, 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.[4][5] 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.[4] 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 ()[6] and has a different flavour to complex analytic geometry in or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.

Historical perspective[edit]

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem.[7] 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, Pierre Cousin [fr], E. E. Levi, 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, Karl Stein, Wilhelm Wirtinger and Francesco Severi. 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[8] 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[edit]

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, and more generalized Stein space. is also considered to be a complex projective variety, a Kähler manifold,[9] 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 .

Connected space[edit]

Every product of a family of an connected (resp. path-connected) spaces is connected (resp. path-connected).


From the Tychonoff's theorem, the space mapped by the cartesian product consisting of any combination of compact spaces is a compact space.

Holomorphic functions[edit]


When a function f defined on the domain D is complex-differentiable at each point on D, f is said to be holomorphic on D. When the function f defined on the domain D satisfies the following conditions, it is complex-differentiable at the point on D;

Let ,
, since such are uniquely determined, they are called the partial differential coefficients of f, and each are written as

Therefore, when a function f is holomorphic on the domain , then f satisfies the following two conditions.

  1. f is continuous on D[note 4]
  2. f is holomorphic in each variable separately, that is f is separate holomorphicity, namely,
    On the converse, when these conditions are satisfied, the function f is holomorphic (as described later), and this condition is called Osgood's lemma. However, note that condition (B) depends on the properties of the domain (as described later).

Cauchy–Riemann equations[edit]

For each index ν let

and[clarification needed]

(Wirtinger derivative)

Then as expected,

through, let be the Kronecker delta, that is , and if .




This satisfies the Cauchy–Riemann equation of one variable to each index ν, then f is a separate holomorphic.

Cauchy's integral formula I (Polydisc version)[edit]

Prove the sufficiency of two conditions (A) and (B). Let 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 closed polydisc so that it becomes . ( and let be the center of each disk.) Using the Cauchy's integral formula of one variable repeatedly, [note 5]

Because is a rectifiable Jordanian closed curve[note 6] 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,






Cauchy's evaluation formula[edit]

Because the order of products and sums is interchangeable, from (1) we get






f is class -function.

From (2), if f is holomorphic, on polydisc and , the following evaluation equation is obtained.

Therefore, Liouville's theorem hold.

Power series expansion of holomorphic functions on polydisc[edit]

If function f is holomorphic, on polydisc , from the 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.

Radius of convergence of power series[edit]

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 7] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

Laurent series expansion[edit]

Let be holomorphic in the annulus and continuous on their circumference, then there exists the following expansion ;

The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus , where and , and so it is possible to integrate term.[11]

Bochner–Martinelli formula (Cauchy's integral formula II)[edit]

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 the Bochner–Martinelli formula.

Suppose that f is a continuously differentiable function on the closure of a domain D on with piecewise smooth boundary , and let the symbol denotes the exterior or wedge product of differential forms. 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

Identity theorem[edit]

When the function f,g is analytic in the domain D,[note 8] even for several complex variables, the identity theorem[note 9] 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. For a generalized version of the implicit function theorem to complex variables, see the Weierstrass preparation theorem


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[edit]

When , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two.[13] This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.[5][14] However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable. [15]

Analytic continuation[edit]

Let U, V be domain on , such that and , ( is the set/ring of holomorphic functions on U.) 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. When n > 2, the following phenomenon occurs depending on the shape of the boundary : there exists domain U, V, such that all holomorphic functions over the domain U, have an analytic continuation . In other words, there may be not exist a function such that as the natural boundary. 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. In addition, when , it would be that the above V has an intersection part with U other than W. This contributed to advancement of the notion of sheaf cohomology.

Reinhardt domain[edit]

In polydisks, the Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but polydisks and open unit balls are not biholomorphic mapping because the Riemann mapping theorem does not hold, and also, polydisks was possible to separation of variables, but it doesn't always hold for any domain. Therefore, in order to study of the domain of convergence of the power series, it was necessary to make additional restriction on the domain, this was the Reinhardt domain. Early knowledge into the properties of field of study of several complex variables, such as Logarithmically-convex, Hartogs's extension theorem, etc. , were given in the Reinhardt domain.

Let () to be a domain, with centre at a point , such that, together with each point , the domain also contains the set

A domain D is called a Reinhardt domain if it satisfies the following conditions:[16][17]

Let is a arbitrary real numbers, a domain D is invariant under the rotation: .

The Reinhardt domains (subclass of the Hartogs domains[18]) which are defined by the following condition; Together with all points of , the domain contains the set

A Reinhardt domain D is called a complete Reinhardt domain with centre at a point a if together with all point it also contains the polydisc

A complete Reinhardt domain D is star-like with regard 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 the 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 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 10] But, there is an example of a complete Reinhardt domain D which is not logarithmically convex. [19]

Some results[edit]

Hartogs's extension theorem and Hartogs's phenomenon[edit]

When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the were all connected to larger domain.[20]

On the polydisk consisting of two disks when .
Internal domain of
Hartogs's extension theorem (1906);[21] Let f be a holomorphic function on a set G \ K, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain[note 11] on (n ≥ 2) and K is a compact subset of G. If the complement G \ 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.[23][22]
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.[24][25]

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 .[26]

Thullen's classic results[edit]

Thullen's[27] 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:

  1. (polydisc);
  2. (unit ball);
  3. (Thullen domain).

Sunada's results[edit]

Toshikazu Sunada (1978)[28] 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 .

Natural domain of the holomorphic function (domain of holomorphy)[edit]

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.[29] Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for ,[30] later extended to .[31][32])[33] Kiyoshi Oka's[36][37] notion of idéal de domaines indéterminés is interpreted theory of sheaf cohomology by H. Cartan and more development Serre.[note 13][38][39][40][41][42][43][6] In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[44] The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.[4]

Domain of holomorphy[edit]

The sets in the definition. Note: On this section, replace in the figure with D

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.[45]

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[edit]

  • 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).[46]
  • If and are domains of holomorphy, then is a domain of holomorphy.
  • The first Cousin problem is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for .[47] this is also true, with additional topological assumptions, for the second Cousin problem.

Holomorphically convex hull[edit]

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 K with the relatively compact components of .

When , if f satisfies the above holomorphic convexity on D it has the following properties. for every compact subset K in D, where denotes the distance between K and . Also, at this time, D is a domain of holomorphy. Therefore, every convex domain is domain of holomorphy.[5]


Hartogs showed that

Hartogs (1906):[21] Let D be a Hartogs's domain on and R be a positive function on D such that the set in defined by and is a domain of holomorphy. Then is a subharmonic function on D.[4]

If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.[note 14] The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.[48]

Definition of plurisubharmonic function[edit]

A function
with domain

is called plurisubharmonic if it is upper semi-continuous, and for every complex line

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.

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.[49]: 39–40 

Strictly plurisubharmonic function[edit]

When the hermitian matrix of u is positive-definite and class , we call u a strict plurisubharmonic function.

(Weakly) pseudoconvex (p-pseudoconvex)[edit]

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 15] 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 .[49]: 49 

Strongly (Strictly) pseudoconvex[edit]

Let X be a complex n-dimensional manifold. Strongly (or Strictly) 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.[49]: 49  Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete[50]) are often used interchangeably,[51] see Lempert[52] for the technical difference.

Levi form[edit]

(Weakly) Levi(–Krzoska) pseudoconvexity[edit]

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

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 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 (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex)[edit]

When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.[5]

Levi total pseudoconvex[edit]

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.[54]

Oka pseudoconvex[edit]

Family of Oka's disk[edit]

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.[54][55]


When holds on any family of Oka's disk, D is called Oka pseudoconvex.[54] Oka's proof of Levi's problem was that when the unramified Riemann domain over [56] 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.[31][55]

Locally pseudoconvex (a.k.a. locally Stein, Cartan pseudoconvex, local Levi property)[edit]

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., let be a holomorphic map, if every point has a neighborhood U such that admits a -plurisubharmonic exhaustion function (weakly 1-complete[57]), in this situation, we call that X is locally pseudoconvex (or locally Stein) over Y. As an old name, it is also called Cartan pseudoconvex. In the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.[58][54] For example, Diederich–Fornæss[59] found local pseudoconvex bounded domains with smooth boundary on non-Kähler manifolds such that is not weakly 1-complete.[60][note 16]

Conditions equivalent to domain of holomorphy[edit]

For a domain the following conditions are equivalent:[note 17]

  1. D is a domain of holomorphy.
  2. D is holomorphically convex.
  3. D is the union of an increasing sequence of analytic polyhedrons in D.
  4. D is pseudoconvex.
  5. 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 solved the this ploblem for unramified Riemann domains over by Kiyoshi Oka,[note 20] but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,[68] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem(equation) with a L2 methods).[1][45][3][69]


Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))[edit]

Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains".[36][37] Specifically, it is a set of pairs , holomorphic on a non-empty open set , such that

  1. If and is arbitrary, then .
  2. For each , then

The origin of indeterminate domains comes from the fact that domains change depending on the pair . Cartan[38][39] translated this notion into the notion of the coherent (sheaf) (Especially, coherent analytic sheaf) in sheaf cohomology.[69][70] This name comes from H. Cartan.[71] Also, Serre (1955)[72] introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf. The notion of coherent (coherent sheaf cohomology) helped solve the problems in several complex variables.[41]

Coherent sheaf[edit]


The definition of the coherent sheaf is as follows.[72][73][74][75] [49]: 83–89  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 .

A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:

  1. 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 ;
  2. for each open set , integer , 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.

Also, Jean-Pierre Serre (1955)[72] 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.

(Oka–Cartan) coherent theorem[edit]

(Oka–Cartan) coherent theorem[36] says that each sheaf that meets the following conditions is a coherent.[76]

  1. the sheaf of germs of holomorphic functions on , or the structure sheaf of complex submanifold or every complex analytic space [77]
  2. the ideal sheaf of an analytic subset A of an open subset of . (Cartan 1950[38])[78][79]
  3. the normalization of the structure sheaf of a complex analytic space[80]

From the above Serre(1955) theorem, is a coherent sheaf, also, (i) is used to prove Cartan's theorems A and B.

Cousin problem[edit]

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and Weierstrass factorization theorem was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). 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.[81] It was Oka who showed[82][83][84][note 21] the conditions for solving first Cousin problem for the domain of holomorphy[note 22] on the complex coordinate space, and also solving the second Cousin problem with additional topological assumptions, the Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic propertie a pure topological,[84][41][33] and Serre[86] called this the Oka principle. They are now posed, and solved, for arbitrary complex manifold M, in terms of conditions on M. M, which satisfies these conditions, is one way to define a Stein manifold. The study of the cousin's problem made us realize that in the study of several complex variables, it is possible to study of global properties from the patching of local data,[38] that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.[44])[41]

First Cousin problem[edit]

Definition without sheaf cohomology words[edit]

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[edit]

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[edit]

Definition without Sheaf cohomology words[edit]

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[edit]

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 (This condition called Oka principle.)

Manifolds and analytic varieties with several complex variables[edit]

Stein manifold (non-compact complex manifold)[edit]

Since a non-compact (open) Riemann surface[87] always has a non-constant single-valued holomorphic function,[88] and satisfies the second axiom of countability, the open Riemann surface is in fact a 1-dimensional complex manifold possessing a holomorphic mapping into the complex plane . (In fact, Gunning and Narasimhan have shown (1967)[89] that every non-compact Riemann surface actually has a holomorphic immersion into the complex plane. In other words, there is a holomorphic mapping into the complex plane whose derivative never vanishes.)[90] 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 . For example, for an arbitrary compact connected complex manifold X, every holomorphic function on it is constant by Liouville's theorem, and so it cannot have any embedding into complex n-space. That is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants. So, consider the conditions under which a complex manifold has a holomorphic function that is not a constant. 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.[91]

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).[92] 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:[93]

  1. X is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
    is also a compact subset of X.
  2. X is holomorphically separable,[note 23] i.e. if are two points in X, then there exists such that
  3. The open neighborhood of every point on the manifold has a holomorphic chart to the .

Note that condition (3) can be derived from conditions (1) and (2).[94]

Every non-compact (open) Riemann surface is a Stein manifold[edit]

Let X be a connected, non-compact (open) Riemann surface. A deep theorem of Behnke and Stein (1948)[88] 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 problems[edit]

Cartan extended Levi's problem to Stein manifolds.[95]

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.[96]

This was proved by Bremermann[97] by embedding it in a sufficiently high dimensional , and reducing it to the result of Oka.[31]

Also, Grauert proved for arbitrary complex manifolds M.[note 24][100][33][98]

If the relative compact subset of a arbitrary complex manifold M is a strongly pseudoconvex on M, then M is a holomorphically convex (i.e. Stein manifold). Also, D is itself a Stein manifold.

And Narasimhan[101][102] extended Levi's problem to complex analytic space, a generalized in the singular case of complex manifolds.

A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space.[4]

Levi's problem remains unresolved in the following cases;

Suppose that X is a singular Stein space,[note 25] . Suppose that for all there is an open neighborhood so that is Stein space. Is D itself Stein?[4][104][103]

more generalized

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 ?[103][105]: 109 

and also,

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. [103]


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 .[100] This concept also applies to complex analytic space.[106]

Properties and examples of Stein manifolds[edit]

  • The standard[note 26] complex space is a Stein manifold.
  • Every domain of holomorphy in is a Stein manifold.[13]
  • 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.[107][108][109]

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.[110]
  • 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[111] for Riemann surfaces,[note 27] due to Behnke and Stein.[88]
  • 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,[100] i.e. a smooth real function on X (which can be assumed to be a Morse function) with ,[100] such that the subsets are compact in X for every real number c. This is a solution to the so-called Levi problem,[112] 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.[113] 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.[114]

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)[edit]

Meromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the Riemann-Roch theorem (Riemann's inequality) holds for compact Riemann surfaces (Therefore the theory of compact Riemann surface can be regarded as the theory of (smooth (non-singular) projective) algebraic curve over [115][116]). In fact, compact Riemann surface had a non-constant single-valued meromorphic function[87], and also a compact Riemann surface had enough meromorphic functions. A compact one-dimensional complex manifold was a Riemann sphere . However, the abstract notion of a compact Riemann surface is always algebraizable (The Riemann's existence theorem, Kodaira embedding theorem.),[note 28] but it is not easy to verify which compact complex analytic spaces are algebraizable.[117] In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.[58] However, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic. [118] The generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira,[119] Kodaira also extended the theorem to three-dimensional,[120] and n-dimensional Kähler varieties.[121] Serre formulated the Riemann-Roch theorem as a problem of dimension of coherent sheaf cohomology,[6] and also Serre proved Serre duality.[122] Cartan–Serre proved the following property:[123] the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M.[124] Riemann–Roch on a Riemann surface for a vector bundle was proved by Weil in 1938.[125] Hirzebruch generalized the theorem to compact complex manifolds in 1994[126] and Grothendieck generalized it to a relative version (relative statements about morphisms.).[127] [128] Next, we generalize the result that the compact Riemann surfaces are projective, to the high-dimensional case, specifically, consider the conditions that when embedding of compact complex submanifold X into the complex projective space . [note 29] i.e., gives the conditions when a compact complex manifold is projective. The Kodaira vanishing theorem (1954) and its generalization Nakano vanishing theorem etc. gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of positivity. As an example given by this theorem, Kodaira embedding theorem[129] says that a compact Kähler manifold M, with a Hodge metric, there is a complex-analytic embedding of M into complex projective space of enough high-dimension N. Chow's theorem[130] shows that the complex analytic subspace (subvariety) of a closed complex projective space to be an algebraic that is, so it is the common zero of some homogeneous polynomials, such a relationship is one example of what is called Serre's GAGA principle.[8] The complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties. Then combined with Kodaira's result, a compact Kähler manifold M embeds as an algebraic variety. This gives an example of a complex manifold with enough meromorphic functions. Similarities in the Levi problems on the complex projective space , have been proved in some patterns, for example by Takeuchi.[4][131][132][133] Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory. Also, the deformation theory of compact complex manifolds has developed as Kodaira–Spencer theory. However, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic.[134]

See also[edit]


  1. ^ That is an open connected subset.
  2. ^ A name adopted, confusingly, for the geometry of zeroes of analytic functions; this is not the analytic geometry learned at school. (In other words, in the sense of GAGA on Serre.)[8]
  3. ^ The field of complex numbers is a 2-dimensional vector space over real numbers.
  4. ^ Using Hartogs's theorem on separate holomorphicity, If condition (B) 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.
  5. ^ Note that this formula only holds for polydisc. See §Bochner–Martinelli formula for the Cauchy's integral formula on the more general domain.
  6. ^ According to the Jordan curve theorem, domain D is bounded closed set, that is, each domain is compact.
  7. ^ 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.
  8. ^ 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.
  9. ^ Note that from Hartogs' extension theorem or Weierstrass preparation theorem , the zeros of analytic functions of several variables cannot have isolated singularities.[12] Therefore, for several variables it is not enough that is satisfied at the accumulation point.
  10. ^ When described using the domain of holomorphy, which is a generalization of the convergence domain, a Reinhardt domain is a domain of holomorphy if and only if logarithmically convex.
  11. ^ This theorem holds even if the condition is not restricted to the bounded. i.e. The theorem holds even if this condition is replaced with an open set.[22]
  12. ^ Oka says that[34] the contents of these two papers are different.[35]
  13. ^ The idea of the sheaf itself is by Jean Leray.
  14. ^ In fact, this was proved by Kiyoshi Oka[30] with respect to domain.See Oka's lemma.
  15. ^ 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.
  16. ^ Definition of weakly 1-complete.[61]
  17. ^ 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[62][63] on the complex analytic space (when n = 2, the result by Hirzebruch,[64] when n = 3 the result by Zariski[65] 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: [66]
  18. ^ This relation is called the Cartan–Thullen theorem.[67]
  19. ^ See Oka's lemma
  20. ^ Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.
  21. ^ This is called the classic Cousin problem.[41]
  22. ^ There are some counterexample in the domain of holomorphicity regarding second Cousin problem.[84][85]
  23. ^ From this condition, we can see that the Stein manifold is not compact.
  24. ^ Levi problem is not true for domains in arbitrary manifolds.[33][98][99]
  25. ^ In the case of Stein space with isolated singularities, it has already been positively solved by Narasimhan.[4][103]
  26. ^ ( is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.
  27. ^ The proof method uses an approximation by the polyhedral domain, as in Oka-Weil theorem.
  28. ^ Note that the Riemann extension theorem and its references explained in the linked article includes a generalized version of the Riemann extension theorem by Grothendieck that was proved using the GAGA principle, also every one-dimensional compact complex manifold is a Hodge manifold.
  29. ^ This is the standard method for compactification of , but not the only method like the Riemann sphere that was compactification of .


Inline citations[edit]

  1. ^ a b Hörmander, Lars (1965). "L2 estimates and existence theorems for the operator". Acta Mathematica. 113: 89–152. doi:10.1007/BF02391775. S2CID 120051843.
  2. ^ Ohsawa, Takeo (2002). Analysis of Several Complex Variables. ISBN 978-1-4704-4636-9.
  3. ^ a b Błocki, Zbigniew (2014). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences. 4 (3): 433–480. doi:10.1007/s13373-014-0058-2. S2CID 53582451.
  4. ^ a b c d e f g h Siu, Yum-Tong (1978). "Pseudoconvexity and the problem of Levi". Bulletin of the American Mathematical Society. 84 (4): 481–513. doi:10.1090/S0002-9904-1978-14483-8. MR 0477104.
  5. ^ a b c d e Chen, So-Chin (2000). "Complex analysis in one and several variables". Taiwanese Journal of Mathematics. 4 (4): 531–568. doi:10.11650/twjm/1500407292. JSTOR 43833225. MR 1799753. Zbl 0974.32001.
  6. ^ a b c Chong, C.T.; Leong, Y.K. (1986). "An interview with Jean-Pierre Serre". The Mathematical Intelligencer. 8 (4): 8–13. doi:10.1007/BF03026112. S2CID 121138963.
  7. ^ Freitag, Eberhard (2011). "Analytic Functions of Several Complex Variables". Complex Analysis 2. Universitext. pp. 300–346. doi:10.1007/978-3-642-20554-5_5. ISBN 978-3-642-20553-8.
  8. ^ a b c Serre, Jean-Pierre (1956). "Géométrie algébrique et géométrie analytique". Annales de l'Institut Fourier (in French). 6: 1–42. doi:10.5802/aif.59. ISSN 0373-0956. MR 0082175. Zbl 0075.30401.
  9. ^ Ohsawa, Takeo (1984). "Vanishing theorems on complete Kähler manifolds". Publications of the Research Institute for Mathematical Sciences. 20: 21–38. doi:10.2977/prims/1195181825.
  10. ^ Solomentsev, E.D. (2001) [1994], "Weierstrass theorem", Encyclopedia of Mathematics, EMS Press
  11. ^ Ozaki, Shigeo; Onô, Isao (February 1, 1953). "Analytic Functions of Several Complex Variables". Science Reports of the Tokyo Bunrika Daigaku, Section A. 4 (98/103): 262–270. JSTOR 43700400.
  12. ^ Martin, W. T. (1945). "Functions of Several Complex Variables". The American Mathematical Monthly. 52 (1): 17–27. doi:10.1080/00029890.1945.11991493. JSTOR 2304831.
  13. ^ a b Field, M (1982). "Complex Manifolds". Several Complex Variables and Complex Manifolds I. pp. 134–186. doi:10.1017/CBO9781107325562.005. ISBN 9780521283014.
  14. ^ Poincare, M. Henri (1907). "Les fonctions analytiques de deux variables et la représentation conforme". Rendiconti del Circolo Matematico di Palermo. 23: 185–220. doi:10.1007/BF03013518. S2CID 123480258.
  15. ^ Siu, Yum-Tong (1991). "Uniformization in Several Complex Variables". In Wu, Hung-Hsi (ed.). Contemporary Geometry. p. 494. doi:10.1007/978-1-4684-7950-8. ISBN 978-1-4684-7950-8.
  16. ^ Jarnicki, Marek; Pflug, Peter (2008). First Steps in Several Complex Variables: Reinhardt Domains. doi:10.4171/049. ISBN 978-3-03719-049-4.
  17. ^ Sakai, Eiichi (1970). "Meromorphic or Holomorphic Completion of a Reinhardt Domain". Nagoya Mathematical Journal. 38: 1–12. doi:10.1017/S0027763000013465. S2CID 118248529.
  18. ^ Chirka, E.M. (2001) [1994], "Hartogs domain", Encyclopedia of Mathematics, EMS Press
  19. ^ Range, R. Michael (1986). "Domains of Holomorphy and Pseudoconvexity". Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics. Vol. 108. p. 10.1007/978-1-4757-1918-5_2. doi:10.1007/978-1-4757-1918-5_2. ISBN 978-1-4419-3078-1.
  20. ^ Krantz, Steven G. (2008). "The Hartogs extension phenomenon redux". Complex Variables and Elliptic Equations. 53 (4): 343–353. doi:10.1080/17476930701747716. S2CID 121700550.
  21. ^ a b Hartogs, Fritz (1906), "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, JFM 37.0443.01
  22. ^ a b Simonič, Aleksander (2016). "Elementary approach to the Hartogs extension theorem". arXiv:1608.00950 [math.CV].
  23. ^ Laufer, Henry B. (1 June 1966). "Some remarks about a theorem of Hartogs". Proceedings of the American Mathematical Society. 17 (6): 1244–1249. doi:10.1090/S0002-9939-1966-0201675-2. JSTOR 2035718.
  24. ^ Merker, Joël; Porten, Egmont (2007). "A Morse-theoretical proof of the Hartogs extension theorem". Journal of Geometric Analysis. 17 (3): 513–546. doi:10.1007/BF02922095. S2CID 449210.
  25. ^ Boggess, A.; Dwilewicz, R.J.; Slodkowski, Z. (2013). "Hartogs extension for generalized tubes in Cn". Journal of Mathematical Analysis and Applications. 402 (2): 574–578. doi:10.1016/j.jmaa.2013.01.049.
  26. ^ Cartan, Henri (1931). "Les fonctions de deux variables complexes et le problème de la représentation analytique". Journal de Mathématiques Pures et Appliquées. 10: 1–116. Zbl 0001.28501.
  27. ^ Thullen, Peter (1931). "Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veränderlichen die Invarianz des Mittelpunktes von Kreiskörpern". Mathematische Annalen. 104: 244–259. doi:10.1007/bf01457933. S2CID 121072397.
  28. ^ Sunada, Toshikazu (1978). "Holomorphic equivalence problem for bounded Reinhardt domains". Mathematische Annalen. 235 (2): 111–128. doi:10.1007/BF01405009. S2CID 124324696.
  29. ^ Cartan, Henri; Thullen, Peter (1932). "Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen Regularitäts-und Konvergenzbereiche". Mathematische Annalen. 106: 617–647. doi:10.1007/BF01455905.
  30. ^ a b Oka, Kiyoshi (1943), "Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes", Tohoku Mathematical Journal, First Series, 49: 15–52, ISSN 0040-8735, Zbl 0060.24006
  31. ^ a b c Oka, Kiyoshi (1953), "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur", Japanese Journal of Mathematics: Transactions and Abstracts, 23: 97–155, doi:10.4099/jjm1924.23.0_97, ISSN 0075-3432
  32. ^ Hans J. Bremermann (1954), "Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum vonn komplexen Veränderlichen", Mathematische Annalen, 106: 63–91, doi:10.1007/BF01360125, S2CID 119837287
  33. ^ a b c d Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7. S2CID 119685542.
  34. ^ Oka, Kiyoshi (1953). Merker, j.; Noguchi, j. (eds.). "Sur les formes objectives et les contenus subjectifs dans les sciences math'ematiques; Propos post'erieur" (PDF).
  35. ^ Noguchi, J. "Related to Works of Dr. Kiyoshi OKA".
  36. ^ a b c Oka, Kiyoshi (1950). "Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques". Bulletin de la Société Mathématique de France. 2: 1–27. doi:10.24033/bsmf.1408., Oka, Kiyoshi (1961). "Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques" (PDF). Iwanami Shoten, Tokyo (Oka's Original Version).[note 12]
  37. ^ a b Oka, Kiyoshi (1951), "Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental", Journal of the Mathematical Society of Japan, 3 (1): 204–214, doi:10.2969/jmsj/00310204, Oka, Kiyoshi (1951), "Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental (Suite)", Journal of the Mathematical Society of Japan, 3 (2): 259–278, doi:10.2969/jmsj/00320259
  38. ^ a b c d Cartan, Henri (1950). "Idéaux et modules de fonctions analytiques de variables complexes". Bulletin de la Société Mathématique de France. 2: 29–64. doi:10.24033/bsmf.1409.
  39. ^ a b Cartan, Henri (1953). "Variétés analytiques complexes et cohomologie". Colloque sur les fonctions de plusieurs variables, Bruxelles: 41–55. MR 0064154. Zbl 0053.05301.
  40. ^ Cartan, H.; Eilenberg, Samuel; Serre, J-P. "Séminaire Henri Cartan, Tome 3 (1950-1951)".
  41. ^ a b c d e Chorlay, Renaud (January 2010). "From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept". Archive for History of Exact Sciences. 64 (1): 1–73. doi:10.1007/s00407-009-0052-3. JSTOR 41342411. S2CID 73633995.
  42. ^ Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften. Vol. 136. 1990. doi:10.1007/978-3-662-02661-8. ISBN 978-3-642-08082-1.
  43. ^ Serre, Jean-Pierre (1953). "Quelques problèmes globaux rélatifs aux variétés de Stein". Centre Belge Rech. Math., Colloque Fonctions Plusieurs Variables, Bruxelles du 11 Au 14 Mars: 67–58. Zbl 0053.05302.
  44. ^ a b Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P. "Séminaire Henri Cartan, Tome 4 (1951-1952)".
  45. ^ a b Forstnerič, Franc (2011). "Stein Manifolds". Stein Manifolds and Holomorphic Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 56. doi:10.1007/978-3-642-22250-4. ISBN 978-3-642-22249-8.
  46. ^ Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen. 116: 204–216. doi:10.1007/BF01597355. S2CID 123982856.
  47. ^ Kajiwara, Joji (1 January 1965). "Relations between domains of holomorphy and multiple Cousin's problems". Kodai Mathematical Journal. 17 (4). doi:10.2996/kmj/1138845123.
  48. ^ Range, R. Michael (2012). "WHAT IS...a Pseudoconvex Domain?". Notices of the American Mathematical Society. 59 (2): 1. doi:10.1090/noti798.
  49. ^ a b c d Complex Analytic and Differential Geometry
  50. ^ Fritzsche, Klaus; Grauert, Hans (6 December 2012). From Holomorphic Functions to Complex Manifolds. ISBN 9781468492736.
  51. ^ Krantz, Steven George (2001). Function Theory of Several Complex Variables. ISBN 9780821827246.
  52. ^ Lempert, Laszlo (1981). "La métrique de Kobayashi et la représentation des domaines sur la boule". Bulletin de la Société Mathématique de France. 109: 427–474. doi:10.24033/bsmf.1948.
  53. ^ Shon, Kwang Ho (1987). "Stein Neighborhood Bases for Product Sets of Polydiscs and Open Intervals". Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics. 41: 45–80. doi:10.2206/kyushumfs.41.45.
  54. ^ a b c d Sin Hitomatsu (1958), "On some conjectures concerning pseudo-convex domains", Journal of the Mathematical Society of Japan, 6 (2): 177–195, doi:10.2969/jmsj/00620177, Zbl 0057.31503
  55. ^ a b Kajiwara, Joji (1959). "Some Results on the Equivalence of Complex-Analytic Fibre Bundles". Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics. 13: 37–48. doi:10.2206/kyushumfs.13.37.
  56. ^ Solomentsev, E.D. (2001) [1994], "Riemannian domain", Encyclopedia of Mathematics, EMS Press
  57. ^ Ohsawa, Takeo (2018). "On the local pseudoconvexity of certain analytic families of ". Annales de l'Institut Fourier. 68 (7): 2811–2818. doi:10.5802/aif.3226.
  58. ^ a b Ohsawa, Takeo (February 2021). "NISHIno's Rigidity, Locally pseudoconvex maps, and holomorphic motions (Topology of pseudoconvex domains and analysis of reproducing kernels)". RIMS Kôkyûroku. 2175: 27–46. hdl:2433/263965.
  59. ^ Diederich, Klas; Fornæss, John Erik (1982). "A smooth pseudoconvex domain without pseudoconvex exhaustion". Manuscripta Mathematica. 39: 119–123. doi:10.1007/BF01312449.
  60. ^ Ohsawa, Takeo (2012). "Hartogs type extension theorems on some domains in Kähler manifolds". Annales Polonici Mathematici. 106: 243–254. doi:10.4064/ap106-0-19. S2CID 123827662.
  61. ^ Ohsawa, Takeo (1981). "Weakly 1-Complete Manifold and Levi Problem". Publications of the Research Institute for Mathematical Sciences. 17: 153–164. doi:10.2977/prims/1195186709.
  62. ^ Heinrich Behnke & Karl Stein (1951), "Modifikationen komplexer Mannigfaltigkeiten und Riernannscher Gebiete", Mathematische Annalen, 124: 1–16, doi:10.1007/BF01343548, S2CID 120455177, Zbl 0043.30301
  63. ^ Onishchik, A.L. (2001) [1994], "Modification", Encyclopedia of Mathematics, EMS Press
  64. ^ Friedrich Hirzebruch (1953), "Über vierdimensionaleRIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen", Mathematische Annalen, 126: 1–22, doi:10.1007/BF01343146, hdl:21.11116/0000-0004-3A47-C, S2CID 122862268
  65. ^ Oscar Zariski (1944), "Reduction of the Singularities of Algebraic Three Dimensional Varieties", Annals of Mathematics, Second Series, 45 (3): 472–542, doi:10.2307/1969189, JSTOR 1969189
  66. ^ Hans Grauert & Reinhold Remmert (1956), "Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie", Commentarii Mathematici Helvetici, 31: 152–183, doi:10.1007/BF02564357, S2CID 117913713, Zbl 0073.30301
  67. ^ Tsurumi, Kazuyuki; Jimbo, Toshiya (1969). "Some properties of holomorphic convexity in general function algebras". Science Reports of the Tokyo Kyoiku Daigaku, Section A. 10 (249/262): 178–183. JSTOR 43698735.
  68. ^ Fornæss, John Erik (1978). "A counterexample for the Levi problem for branched Riemann domains over ". Mathematische Annalen. 234 (3): 275–277. doi:10.1007/BF01420649.
  69. ^ a b Noguchi, Junjiro (2019). "A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem". Notices of the International Congress of Chinese Mathematicians. 7 (2): 19–24. arXiv:1807.08246. doi:10.4310/ICCM.2019.V7.N2.A2. S2CID 119619733.
  70. ^ Noguchi, Junjiro (2016). Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.x). p. XVIII, 397. doi:10.1007/978-981-10-0291-5. ISBN 978-981-10-0289-2. S2CID 125752012.
  71. ^ Noguchi, Junjiro (2016). Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.33). p. XVIII, 397. doi:10.1007/978-981-10-0291-5. ISBN 978-981-10-0289-2. S2CID 125752012.
  72. ^ a b c Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents" (PDF), Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874
  73. ^ Grothendiec, Alexander; Dieudonn, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas (ch.0 § 5. FAISCEAUX QUASI-COHÉRENTS ET FAISCEAUX COHÉRENTS (0.5.1–0.5.3))". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083. S2CID 121855488.
  74. ^ Remmert, R. (1994). "Local Theory of Complex Spaces". Several Complex Variables VII §6. Calculs of Coherent sheaves. Encyclopaedia of Mathematical Sciences. Vol. 74. pp. 7–96. doi:10.1007/978-3-662-09873-8_2. ISBN 978-3-642-08150-7.
  75. ^ Ohsawa, Takeo (10 December 2018). L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds. Springer Monographs in Mathematics. doi:10.1007/978-4-431-55747-0. ISBN 9784431568513.
  76. ^ Noguchi, Junjiro (2019), "A Weak Coherence Theorem and Remarks to the Oka Theory" (PDF), Kodai Math. J., 42 (3): 566–586, arXiv:1704.07726, doi:10.2996/kmj/1572487232, S2CID 119697608
  77. ^ Grauert, H.; Remmert, R. (6 December 2012). Coherent Analytic Sheaves. p. 60. ISBN 978-3-642-69582-7.
  78. ^ Grauert, H.; Remmert, R. (6 December 2012). Coherent Analytic Sheaves. p. 84. ISBN 978-3-642-69582-7.
  79. ^ Demailly, Jean-Pierre. "Basic results on Sheaves and Analytic Sets" (PDF). Institut Fourier.
  80. ^ Grauert, Hans; Remmert, Reinhold (1984). "Normalization of Complex Spaces". Coherent Analytic Sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 265. pp. 152–166. doi:10.1007/978-3-642-69582-7_8. ISBN 978-3-642-69584-1.
  81. ^ Cousin, Pierre (1895). "Sur les fonctions de n variables complexes". Acta Mathematica. 19: 1–61. doi:10.1007/BF02402869.
  82. ^ Oka, Kiyoshi (1936). "Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles". Journal of Science of the Hiroshima University. 6: 245–255. doi:10.32917/hmj/1558749869.
  83. ^ Oka, Kiyoshi (1937). "Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie". Journal of Science of the Hiroshima University. 7: 115–130. doi:10.32917/hmj/1558576819.
  84. ^ a b c Oka, Kiyoshi (1939). "Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin". Journal of Science of the Hiroshima University. 9: 7–19. doi:10.32917/hmj/1558490525.
  85. ^ Serre, Jean-Pierre (2003). "Quelques problèmes globaux rélatifs aux variétés de Stein". Oeuvres - Collected Papers I (in French). Springer Berlin Heidelberg. p. XXIII, 598. ISBN 978-3-642-39815-5.
  86. ^ Serre, J. -P. "Applications de la théorie générale à divers problèmes globaux". Séminaire Henri Cartan. 4: 1–26.
  87. ^ a b Weyl, Hermann (2009) [1913], The concept of a Riemann surface (3rd ed.), New York: Dover Publications, ISBN 978-0-486-47004-7, MR 0069903
  88. ^ a b c Heinrich Behnke & Karl Stein (1948), "Entwicklung analytischer Funktionen auf Riemannschen Flächen", Mathematische Annalen, 120: 430–461, doi:10.1007/BF01447838, S2CID 122535410, Zbl 0038.23502
  89. ^ Gunning, R. C.; Narasimhan, Raghavan (1967). "Immersion of open Riemann surfaces". Mathematische Annalen. 174 (2): 103–108. doi:10.1007/BF01360812. S2CID 122162708.
  90. ^ Fornaess, J.E.; Forstneric, F; Wold, E.F (2020). "The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan". In Breaz, Daniel; Rassias, Michael Th. (eds.). Advancements in Complex Analysis – Holomorphic Approximation. Springer Nature. pp. 133–192. arXiv:1802.03924. doi:10.1007/978-3-030-40120-7. ISBN 978-3-030-40119-1. S2CID 220266044.
  91. ^ Patyi, Imre (2011). "On complex Banach manifolds similar to Stein manifolds". Comptes Rendus Mathematique. 349 (1–2): 43–45. arXiv:1010.3738. doi:10.1016/j.crma.2010.11.020. S2CID 119631664.
  92. ^ 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, MR 0043219, S2CID 122647212
  93. ^ Noguchi, Junjiro (2011). "Another Direct Proof of Oka's Theorem (Oka IX)" (PDF). J. Math. Sci. Univ. Tokyo. 19 (4). arXiv:1108.2078. MR 3086750.
  94. ^ Grauert, Hans (1955). "Charakterisierung der holomorph vollständigen komplexen Räume". Mathematische Annalen. 129: 233–259. doi:10.1007/BF01362369. S2CID 122840967.
  95. ^ Cartan, Henri (1957). "Variétés analytiques réelles et variétés analytiques complexes". Bulletin de la Société Mathématique de France. 85: 77–99. doi:10.24033/bsmf.1481.
  96. ^ Barth, Theodore J. (1968). "Families of nonnegative divisors". Trans. Amer. Math. Soc. 131: 223–245. doi:10.1090/S0002-9947-1968-0219751-3.
  97. ^ Bremermann, Hans J. (1957). "On Oka's theorem for Stein manifolds". Seminars on Analytic Functions. Institute for Advanced Study (Princeton, N.J.). 1: 29–35. Zbl 0192.18304.
  98. ^ a b Sibony, Nessim (2018). "Levi problem in complex manifolds". Mathematische Annalen. 371 (3–4): 1047–1067. arXiv:1610.07768. doi:10.1007/s00208-017-1539-x. S2CID 119670805.
  99. ^ Grauert, Hans (1963). "Bemerkenswerte pseudokonvexe Mannigfaltigkeiten". Mathematische Zeitschrift. 81 (5): 377–391. doi:10.1007/BF01111528. S2CID 122214512.
  100. ^ a b c d Hans Grauert (1958), "On Levi's Problem and the Imbedding of Real-Analytic Manifolds", Annals of Mathematics, Second Series, 68 (2): 460–472, doi:10.2307/1970257, JSTOR 1970257, Zbl 0108.07804
  101. ^ Narasimhan, Raghavan (1961). "The Levi problem for complex spaces". Mathematische Annalen. 142 (4): 355–365. doi:10.1007/BF01451029. S2CID 120565581.
  102. ^ Narasimhan, Raghavan (1962). "The Levi problem for complex spaces II". Mathematische Annalen. 146 (3): 195–216. doi:10.1007/BF01470950. S2CID 179177434.
  103. ^ a b c d Coltoiu, Mihnea (2009). "The Levi problem on Stein spaces with singularities. A survey". arXiv:0905.2343 [math.CV].
  104. ^ Fornæss, John Erik; Sibony, Nessim (2001). "Some open problems in higher dimensional complex analysis and complex dynamics". Publicacions Matemàtiques. 45 (2): 529–547. doi:10.5565/PUBLMAT_45201_11. JSTOR 43736735.
  105. ^ Ohsawa, Takeo (10 December 2018). L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds. Springer Monographs in Mathematics. doi:10.1007/978-4-431-55747-0. ISBN 9784431568513.
  106. ^ 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:10.1090/S0002-9947-1964-0159961-3. JSTOR 1994247.
  107. ^ Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics. 82 (4): 917–934. doi:10.2307/2372949. JSTOR 2372949.
  108. ^ 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. JSTOR 2946547.
  109. ^ Remmert, Reinhold (1956). "Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris (in French). 243: 118–121. Zbl 0070.30401.
  110. ^ Forster, Otto (1967). "Some remarks on parallelizable Stein manifolds". Bulletin of the American Mathematical Society. 73 (5): 712–716. doi:10.1090/S0002-9904-1967-11839-1.
  111. ^ Simha, R. R. (1989). "The Behnke-Stein Theorem for Open Riemann Surfaces". Proceedings of the American Mathematical Society. 105 (4): 876–880. doi:10.1090/S0002-9939-1989-0953748-X. JSTOR 2047046.
  112. ^ Onishchik, A.L. (2001) [1994], "Levi problem", Encyclopedia of Mathematics, EMS Press
  113. ^ Ohsawa, Takeo (1982). "A Stein domain with smooth boundary which has a product structure". Publications of the Research Institute for Mathematical Sciences. 18 (3): 1185–1186. doi:10.2977/prims/1195183303.
  114. ^ Neeman, Amnon (1988). "Steins, Affines and Hilbert's Fourteenth Problem". Annals of Mathematics. 127 (2): 229–244. doi:10.2307/2007052. JSTOR 2007052.
  115. ^ Miranda, Rick (1995). Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics. Vol. 5. doi:10.1090/gsm/005. ISBN 9780821802687.
  116. ^ Arapura, Donu (15 February 2012). Algebraic Geometry over the Complex Numbers. ISBN 9781461418092.
  117. ^ Danilov, V. I. (1996). "Cohomology of Algebraic Varieties". Algebraic Geometry II. Encyclopaedia of Mathematical Sciences. Vol. 35. pp. 1–125. doi:10.1007/978-3-642-60925-1_1. ISBN 978-3-642-64607-2.
  118. ^ Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. S2CID 197660097. Zbl 0367.14001.
  119. ^ Kodaira, Kunihiko (1951). "The Theorem of Riemann-Roch on Compact Analytic Surfaces". American Journal of Mathematics. 73 (4): 813–875. doi:10.2307/2372120. JSTOR 2372120.
  120. ^ Kodaira, Kunihiko (1952). "The Theorem of Riemann-Roch for Adjoint Systems on 3-Dimensional Algebraic Varieties". Annals of Mathematics. 56 (2): 298–342. doi:10.2307/1969802. JSTOR 1969802.
  121. ^ Kodaira, Kunihiko (1952). "On the Theorem of Riemann-Roch for Adjoint Systems on Kahlerian Varieties". Proceedings of the National Academy of Sciences of the United States of America. 38 (6): 522–527. Bibcode:1952PNAS...38..522K. doi:10.1073/pnas.38.6.522. JSTOR 88542. PMC 1063603. PMID 16589138.
  122. ^ Serre, Jean-Pierre (1955), "Un théorème de dualité", Commentarii Mathematici Helvetici, 29: 9–26, doi:10.1007/BF02564268, MR 0067489, S2CID 123643759
  123. ^ Cartan, Henri; Serre, Jean-Pierre (1953). "Un théorème de finitude concernant les variétés analytiques compactes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris. 237: 128–130. Zbl 0050.17701.
  124. ^ Brînzănescu, Vasile (1996). "Vector bundles over complex manifolds". Holomorphic Vector Bundles over Compact Complex Surfaces. Lecture Notes in Mathematics. Vol. 1624. pp. 1–27. doi:10.1007/BFb0093697. ISBN 978-3-540-61018-2.
  125. ^ Weil, A. (1938). "Zur algebraischen Theorie der algebraischen Funktionen. (Aus einem Brief an H. Hasse.)". Journal für die reine und angewandte Mathematik. 179: 129–133. doi:10.1515/crll.1938.179.129.
  126. ^ Hirzebruch, Friedrich (1966). Topological Methods in Algebraic Geometry. doi:10.1007/978-3-642-62018-8. ISBN 978-3-540-58663-0.
  127. ^ Berthelot, Pierre (1971). Alexandre Grothendieck; Luc Illusie (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Lecture Notes in Mathematics. Vol. 225. Springer Science+Business Media. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.
  128. ^ Borel, Armand; Serre, Jean-Pierre (1958). "Le théorème de Riemann–Roch". Bulletin de la Société Mathématique de France. 86: 97–136. doi:10.24033/bsmf.1500. MR 0116022.
  129. ^ 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. JSTOR 1969701.
  130. ^ Chow, Wei-Liang (1949). "On Compact Complex Analytic Varieties". American Journal of Mathematics. 71 (2): 893–914. doi:10.2307/2372375. JSTOR 2372375.
  131. ^ Ohsawa, Takeo (2012). "On the complement of effective divisors with semipositive normal bundle". Kyoto Journal of Mathematics. 52 (3). doi:10.1215/21562261-1625181. S2CID 121799985.
  132. ^ Matsumoto, Kazuko (2018). "Takeuchi's equality for the levi form of the Fubini–Study distance to complex submanifolds in complex projective spaces". Kyushu Journal of Mathematics. 72 (1): 107–121. doi:10.2206/kyushujm.72.107.
  133. ^ Takeuchi, Akira (1964). "Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projecti". Journal of the Mathematical Society of Japan. 16 (2). doi:10.2969/jmsj/01620159. S2CID 122894640.
  134. ^ Calabi, Eugenio; Eckmann, Beno (1953). "A Class of Compact, Complex Manifolds Which are not Algebraic". Annals of Mathematics. 58 (3): 494–500. doi:10.2307/1969750. JSTOR 1969750.


Encyclopedia of Mathematics[edit]

Further reading[edit]

External links[edit]