{"history":[],"article":{"id":"2c288c04-2254-4522-8f5f-e569148f8443","title":"Function of several complex variables","slug":"function-of-several-complex-variables","content":":::summary\n:::\n\n# \nThe theory of **functions of several complex variables** is the branch of [mathematics](https://bliptext.com/articles/mathematics) dealing with functions defined on [the complex coordinate space](https://bliptext.com/articles/) \\mathbb C^n, that is, n-tuples of [complex numbers](https://bliptext.com/articles/complex-number).\nThe name of the field dealing with the properties of these functions is called **several complex variables** (and [analytic space](https://bliptext.com/articles/analytic-space)), which the [Mathematics Subject Classification](https://bliptext.com/articles/mathematics-subject-classification) has as a top-level heading.\nAs in [complex analysis of functions of one variable](https://bliptext.com/articles/complex-analysis), which is the case\nn = 1\n, the functions studied are [holomorphic](https://bliptext.com/articles/holomorphic-function) or complex analytic so that, locally, they are [power series](https://bliptext.com/articles/power-series) in the variables zi.\nEquivalently, they are locally [uniform limits](https://bliptext.com/articles/uniform-convergence) of [polynomials](https://bliptext.com/articles/polynomial); or locally [square-integrable](https://bliptext.com/articles/lp-space) solutions to the n-dimensional [Cauchy–Riemann equations](https://bliptext.com/articles/).\nFor one complex variable, every [domain](https://bliptext.com/articles/domain-mathematical-analysis), is the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of holomorphy of some function, in other words every [domain](https://bliptext.com/articles/domain-mathematical-analysis) has a function for which it is the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of holomorphy.\nFor 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.\nPatching the local data of [meromorphic functions](https://bliptext.com/articles/meromorphic-function), i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem.\nAlso, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and [complex projective varieties](https://bliptext.com/articles/projective-variety) and has a different flavour to complex analytic geometry in or on [Stein manifolds](https://bliptext.com/articles/stein-manifold), these are much similar to study of algebraic varieties that is study of the [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) than complex analytic geometry.\n\n# Historical perspective\nMany examples of such functions were familiar in nineteenth-century mathematics; [abelian functions](https://bliptext.com/articles/abelian-variety), [theta functions](https://bliptext.com/articles/theta-function), and some [hypergeometric series](https://bliptext.com/articles/hypergeometric-series), and also, as an example of an inverse problem; the [Jacobi inversion problem](https://bliptext.com/articles/jacobi-inversion-problem).\nNaturally also same function of one variable that depends on some complex [parameter](https://bliptext.com/articles/parameter) is a candidate.\nThe theory, however, for many years didn't become a full-fledged field in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), since its characteristic phenomena weren't uncovered.\nThe [Weierstrass preparation theorem](https://bliptext.com/articles/weierstrass-preparation-theorem) would now be classed as [commutative algebra](https://bliptext.com/articles/commutative-algebra); it did justify the local picture, [ramification](https://bliptext.com/articles/ramification-mathematics), that addresses the generalization of the [branch points](https://bliptext.com/articles/branch-point) of [Riemann surface](https://bliptext.com/articles/riemann-surface) theory.\nWith work of [Friedrich Hartogs](https://bliptext.com/articles/friedrich-hartogs), Pierre Cousin (mathematician), [E. E. Levi](https://bliptext.com/articles/e-e-levi), and of [Kiyoshi Oka](https://bliptext.com/articles/kiyoshi-oka) in the 1930s, a general theory began to emerge; others working in the area at the time were [Heinrich Behnke](https://bliptext.com/articles/heinrich-behnke), [Peter Thullen](https://bliptext.com/articles/peter-thullen), [Karl Stein](https://bliptext.com/articles/karl-stein-mathematician), [Wilhelm Wirtinger](https://bliptext.com/articles/wilhelm-wirtinger) and [Francesco Severi](https://bliptext.com/articles/francesco-severi).\nHartogs proved some basic results, such as every [isolated singularity](https://bliptext.com/articles/isolated-singularity) is [removable](https://bliptext.com/articles/removable-singularity), for every analytic function\nwhenever\nn > 1\n. Naturally the analogues of [contour integrals](https://bliptext.com/articles/line-integral) will be harder to handle; when\nn = 2\nan integral surrounding a point should be over a three-dimensional [manifold](https://bliptext.com/articles/manifold) (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a [double integral](https://bliptext.com/articles/double-integral) over a two-dimensional surface.\nThis means that the [residue calculus](https://bliptext.com/articles/residue-calculus) will have to take a very different character.\nAfter 1945 important work in France, in the seminar of [Henri Cartan](https://bliptext.com/articles/henri-cartan), and Germany with [Hans Grauert](https://bliptext.com/articles/hans-grauert) and [Reinhold Remmert](https://bliptext.com/articles/reinhold-remmert), quickly changed the picture of the theory.\nA number of issues were clarified, in particular that of [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nHere a major difference is evident from the one-variable theory; while for every open connected set D in \\Complex we can find a function that will nowhere continue analytically over the boundary, that cannot be said for\nn > 1\n. In fact the D of that kind are rather special in nature (especially in complex coordinate spaces \\mathbb C^n and Stein manifolds, satisfying a condition called [pseudoconvexity](https://bliptext.com/articles/)).\nThe natural domains of definition of functions, continued to the limit, are called [Stein manifolds](https://bliptext.com/articles/stein-manifold) and their nature was to make [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) groups vanish, on the other hand, the [Grauert–Riemenschneider vanishing theorem](https://bliptext.com/articles/grauert-riemenschneider-vanishing-theorem) is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan.\nIn 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](https://bliptext.com/articles/algebraic-geometry), in particular from Grauert's work).\nFrom this point onwards there was a foundational theory, which could be applied to [analytic geometry](https://bliptext.com/articles/algebraic-geometry), [automorphic forms](https://bliptext.com/articles/automorphic-form) of several variables, and [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nThe [deformation theory of complex structures](https://bliptext.com/articles/complex-structure-deformation) and [complex manifolds](https://bliptext.com/articles/complex-manifold) was described in general terms by [Kunihiko Kodaira](https://bliptext.com/articles/kunihiko-kodaira) and [D. C. Spencer](https://bliptext.com/articles/d-c-spencer).\nThe celebrated paper [GAGA](https://bliptext.com/articles/gaga) of [Serre](https://bliptext.com/articles/jean-pierre-serre) pinned down the crossover point from géometrie analytique to géometrie algébrique.\n[C. L. Siegel](https://bliptext.com/articles/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](https://bliptext.com/articles/special-function) side of the theory was subordinated to sheaves.\nThe interest for [number theory](https://bliptext.com/articles/number-theory), certainly, is in specific generalizations of [modular forms](https://bliptext.com/articles/modular-form).\nThe classical candidates are the [Hilbert modular forms](https://bliptext.com/articles/hilbert-modular-form) and [Siegel modular forms](https://bliptext.com/articles/siegel-modular-form).\nThese days these are associated to [algebraic groups](https://bliptext.com/articles/algebraic-group) (respectively the [Weil restriction](https://bliptext.com/articles/weil-restriction) from a [totally real number field](https://bliptext.com/articles/totally-real-number-field) of\nGL(2)\n, and the [symplectic group](https://bliptext.com/articles/symplectic-group)), for which it happens that [automorphic representations](https://bliptext.com/articles/automorphic-representation) can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.\nSubsequent developments included the [hyperfunction](https://bliptext.com/articles/hyperfunction) theory, and the [edge-of-the-wedge theorem](https://bliptext.com/articles/edge-of-the-wedge-theorem), both of which had some inspiration from [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nThere are a number of other fields, such as [Banach algebra](https://bliptext.com/articles/banach-algebra) theory, that draw on several complex variables.\n\n# The complex coordinate space\nThe [complex coordinate space](https://bliptext.com/articles/complex-coordinate-space) \\mathbb C^n is the [Cartesian product](https://bliptext.com/articles/cartesian-product) of n copies of \\mathbb C, and when \\mathbb C^n is a domain of holomorphy, \\mathbb C^n can be regarded as a [Stein manifold](https://bliptext.com/articles/stein-manifold), and more generalized Stein space.\n\\mathbb C^n is also considered to be a [complex projective variety](https://bliptext.com/articles/complex-projective-variety), a [Kähler manifold](https://bliptext.com/articles/k-hler-manifold), etc. It is also an [n-dimensional vector space](https://bliptext.com/articles/dimension-vector-space) over the [complex numbers](https://bliptext.com/articles/complex-number), which gives its dimension\n2n\nover \\mathbb R. Hence, as a set and as a [topological space](https://bliptext.com/articles/topological-space), \\mathbb C^n may be identified to the [real coordinate space](https://bliptext.com/articles/real-coordinate-space) and its [topological dimension](https://bliptext.com/articles/topological-dimension) is thus\n2n\n.\nIn 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](https://bliptext.com/articles/almost-complex-manifold) is specified by a [linear operator](https://bliptext.com/articles/linear-operator) J (such that\nJ2 = −I\n) which defines [multiplication](https://bliptext.com/articles/scalar-multiplication) by the [imaginary unit](https://bliptext.com/articles/imaginary-unit) i.\nAny such space, as a real space, is [oriented](https://bliptext.com/articles/orientation-vector-space).\nOn the [complex plane](https://bliptext.com/articles/complex-plane) thought of as a [Cartesian plane](https://bliptext.com/articles/cartesian-plane), [multiplication](https://bliptext.com/articles/multiplication) by a complex number\nw = u + iv\nmay be represented by the real [matrix](https://bliptext.com/articles/square-matrix)\nwith [determinant](https://bliptext.com/articles/determinant)\nLikewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be [composed from 2 × 2 blocks](https://bliptext.com/articles/block-matrix) of the aforementioned form), then its determinant equals to the [square of absolute value](https://bliptext.com/articles/square-algebra) of the corresponding complex determinant.\nIt is a non-negative number, which implies that [the (real) orientation of the space is never reversed](https://bliptext.com/articles/real-coordinate-space) by a complex operator.\nThe same applies to [Jacobians](https://bliptext.com/articles/jacobian-matrix-and-determinant) of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) from \\mathbb C^n to \\mathbb C^n.\n\n# Holomorphic functions\n\n## Definition\nA function f defined on a domain and with values in \\mathbb{C} is said to be holomorphic at a point z\\in D if it is complex-differentiable at this point, in the sense that there exists a complex linear map such that\nThe function f is said to be holomorphic if it is holomorphic at all points of its domain of definition D.\nIf f is holomorphic, then all the partial maps :\nare holomorphic as functions of one complex variable : we say that f is holomorphic in each variable separately.\nConversely, if f is holomorphic in each variable separately, then f is in fact holomorphic : this is known as [Hartog's theorem](https://bliptext.com/articles/hartogs-s-theorem-on-separate-holomorphicity), or as [Osgood's lemma](https://bliptext.com/articles/osgood-s-lemma) under the additional hypothesis that f is [continuous](https://bliptext.com/articles/continuous-function).\n\n## Cauchy–Riemann equations\nIn one complex variable, a function defined on the plane is holomorphic at a point if and only if its real part u and its imaginary part v satisfy the so-called [Cauchy-Riemann equations](https://bliptext.com/articles/cauchy-riemann-equations) at p :\nIn several variables, a function is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part u and the imaginary part v of f satisfiy the Cauchy Riemann equations :\nUsing the formalism of [Wirtinger derivatives](https://bliptext.com/articles/wirtinger-derivatives), this can be reformulated as :\nor even more compactly using the formalism of [complex differential forms](https://bliptext.com/articles/complex-differential-form), as :\n\n## Cauchy's integral formula I (Polydisc version)\nProve the sufficiency of two conditions (A) and (B).\nLet f meets the conditions of being continuous and separately homorphic on domain D.\nEach disk has a [rectifiable curve](https://bliptext.com/articles/rectifiable-curve) \\gamma, \\gamma_\\nu is piecewise [smoothness](https://bliptext.com/articles/smoothness), class Jordan closed curve.\nLet D_\\nu be the domain surrounded by each \\gamma_\\nu.\nCartesian product closure is.\nAlso, take the closed [polydisc](https://bliptext.com/articles/polydisc) so that it becomes.\n( and let be the center of each disk.) Using the [Cauchy's integral formula](https://bliptext.com/articles/cauchy-s-integral-formula) of one variable repeatedly,\nBecause \\partial D is a rectifiable Jordanian closed curve and f is continuous, so the order of products and sums can be exchanged so the [iterated integral](https://bliptext.com/articles/iterated-integral) can be calculated as a [multiple integral](https://bliptext.com/articles/multiple-integral).\nTherefore,\n\n### Cauchy's evaluation formula\nBecause the order of products and sums is interchangeable, from we get\nf is class -function.\nFrom (2), if f is holomorphic, on polydisc and, the following evaluation equation is obtained.\nTherefore, [Liouville's theorem](https://bliptext.com/articles/liouville-s-theorem-complex-analysis) hold.\n\n### Power series expansion of holomorphic functions on polydisc\nIf 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.\nIn addition, f that satisfies the following conditions is called an analytic function.\nFor each point, f(z) is expressed as a power series expansion that is convergent on D :\nWe have already explained that holomorphic functions on polydisc are analytic.\nAlso, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.\n\n### Radius of convergence of power series\nIt is possible to define a combination of positive real numbers such that the power series converges uniformly at and does not converge uniformly at.\nIn this way it is possible to have a similar, combination of radius of convergence for a one complex variable.\nThis combination is generally not unique and there are an infinite number of combinations.\n\n### Laurent series expansion\nLet \\omega(z) be holomorphic in the [annulus](https://bliptext.com/articles/annulus-mathematics) and continuous on their circumference, then there exists the following expansion ;\nThe 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.\n\n## Bochner–Martinelli formula (Cauchy's integral formula II)\nThe Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many possible domains, so we introduce the [Bochner–Martinelli formula](https://bliptext.com/articles/bochner-martinelli-formula).\nSuppose that f is a continuously differentiable function on the closure of a domain D on \\mathbb C^n with piecewise smooth boundary \\partial D, and let the symbol \\land denotes the exterior or [wedge product](https://bliptext.com/articles/wedge-product) of differential forms.\nThen the Bochner–Martinelli formula states that if z is in the domain D then, for \\zeta, z in \\mathbb C^n the Bochner–Martinelli kernel is a [differential form](https://bliptext.com/articles/differential-form) in \\zeta of bidegree (n,n-1), defined by\nIn particular if f is holomorphic the second term vanishes, so\n\n## Identity theorem\nHolomorphic functions of several complex variables satisfy an [identity theorem](https://bliptext.com/articles/identity-theorem), as in one variable : two holomorphic functions defined on the same connected open set and which coincide on an open subset N of D, are equal on the whole open set D.\nThis result can be proven from the fact that holomorphics functions have power series extensions, and it can also be deduced from the one variable case.\nContrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps and coincide on the whole complex line of defined by the equation z_1=0.\nThe [maximal principle](https://bliptext.com/articles/maximal-principle), [inverse function theorem](https://bliptext.com/articles/inverse-function-theorem), and implicit function theorems also hold.\nFor a generalized version of the implicit function theorem to complex variables, see the [Weierstrass preparation theorem](https://bliptext.com/articles/weierstrass-preparation-theorem).\n\n## Biholomorphism\nFrom the establishment of the inverse function theorem, the following mapping can be defined.\nFor the domain U, V of the n-dimensional complex space \\Complex^n, the bijective holomorphic function \\phi:U\\to V and the inverse mapping is also holomorphic.\nAt this time, \\phi is called a U, V biholomorphism also, we say that U and V are biholomorphically equivalent or that they are biholomorphic.\n\n### The Riemann mapping theorem does not hold\nWhen n > 1, open balls and open polydiscs are not biholomorphically equivalent, that is, there is no [biholomorphic mapping](https://bliptext.com/articles/biholomorphic-mapping) between the two.\nThis was proven by [Poincaré](https://bliptext.com/articles/henri-poincar) in 1907 by showing that their [automorphism groups](https://bliptext.com/articles/automorphism-group) have different dimensions as [Lie groups](https://bliptext.com/articles/lie-group).\nHowever, 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.\n\n## Analytic continuation\nLet U, V be domain on, such that and , ( is the set/ring of holomorphic functions on U.) assume that and W is a [connected component](https://bliptext.com/articles/connected-component-topology) of U \\cap V. If f|_W =g|_W then f is said to be connected to V, and g is said to be analytic continuation of f.\nFrom the identity theorem, if g exists, for each way of choosing W it is unique.\nWhen n > 2, the following phenomenon occurs depending on the shape of the boundary \\partial U: there exists domain U, V, such that all holomorphic functions f over the domain U, have an analytic continuation.\nIn other words, there may be not exist a function such that \\partial U as the natural boundary.\nThere is called the Hartogs's phenomenon.\nTherefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables.\nIn addition, when n \\geq 2, it would be that the above V has an intersection part with U other than W.\nThis contributed to advancement of the notion of sheaf cohomology.\n\n# Reinhardt domain\nIn 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.\nTherefore, 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.\nEarly 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.\nLet (n \\geq 1) to be a domain, with centre at a point, such that, together with each point , the domain also contains the set\nA domain D is called a Reinhardt domain if it satisfies the following conditions:\nLet is a arbitrary real numbers, a domain D is invariant under the rotation:.\nThe Reinhardt domains which are defined by the following condition; Together with all points of z^0 \\in D, the domain contains the set\nA Reinhardt domain D is called a complete Reinhardt domain with centre at a point a if together with all point z^0\\in D it also contains the polydisc\nA complete Reinhardt domain D is [star-like](https://bliptext.com/articles/star-like-domain) with regard to its centre a.\nTherefore, the complete Reinhardt domain is [simply connected](https://bliptext.com/articles/simply-connected), also when the complete Reinhardt domain is the boundary line, there is a way to prove the [Cauchy's integral theorem](https://bliptext.com/articles/cauchy-s-integral-theorem) without using the [Jordan curve theorem](https://bliptext.com/articles/jordan-curve-theorem).\n\n## Logarithmically-convex\nWhen a some complete Reinhardt domain to be the domain of convergence of a power series, an additional condition is required, which is called logarithmically-convex.\nA Reinhardt domain D is called [logarithmically convex](https://bliptext.com/articles/logarithmically-convex) if the image of the set\nunder the mapping\nis a [convex set](https://bliptext.com/articles/convex-set) in the real coordinate space \\R^n.\nEvery such domain in \\Complex^n 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 a = 0.\nBut, there is an example of a complete Reinhardt domain D which is not logarithmically convex.\n\n## Some results\n\n### Hartogs's extension theorem and Hartogs's phenomenon\nWhen 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.\nG \\ K\n, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain on \\Complex^n (\nn ≥ 2\n) and K is a compact subset of G. If the [complement](https://bliptext.com/articles/complement-set-theory)\nG \\ K\nis connected, then every holomorphic function f regardless of how it is chosen can be each extended to a unique holomorphic function on G.\nFrom Hartogs's extension theorem the domain of convergence extends from to \\Delta^2.\nLooking 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 \\Delta^2 containing.\n\n### Thullen's classic results\n[Thullen](https://bliptext.com/articles/peter-thullen)'s classical result says that a 2-dimensional bounded Reinhard domain containing the origin is [biholomorphic](https://bliptext.com/articles/biholomorphic) to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:\n\n### Sunada's results\n[Toshikazu Sunada](https://bliptext.com/articles/toshikazu-sunada) (1978) established a generalization of Thullen's result:\n\n# Natural domain of the holomorphic function (domain of holomorphy)\nWhen 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.\nConsidering the domain where the boundaries of the domain are natural boundaries (In the complex coordinate space \\Complex^n call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H.\nCartan and Thullen.\nLevi's problem shows that the pseudoconvex domain was a domain of holomorphy.\n(First for \\Complex^2, later extended to \\Complex^n. ) [Kiyoshi Oka](https://bliptext.com/articles/kiyoshi-oka)'s notion of idéal de domaines indéterminés is interpreted theory of [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) by H.\nCartan and more development Serre.\nIn sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.\nThe notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.\n\n## Domain of holomorphy\nWhen a function f is holomorpic on the domain and cannot directly connect to the domain outside D, including the point of the domain boundary \\partial D, the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f.\nIn 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.\nFor several complex variables, i.e. domain, the boundaries may not be natural boundaries.\nHartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.\nFormally, a domain D in the n-dimensional complex coordinate space \\Complex^n is called a domain of holomorphy if there do not exist non-empty domain U \\subset D and, and such that for every holomorphic function f on D there exists a holomorphic function g on V with f = g on U.\nFor the n=1 case, the every domain was the domain of holomorphy; we can define a holomorphic function with zeros [accumulating](https://bliptext.com/articles/accumulation-point) everywhere on the [boundary](https://bliptext.com/articles/boundary-topology) of the domain, which must then be a natural [boundary](https://bliptext.com/articles/boundary-topology) for a domain of definition of its reciprocal.\n# [The sets in the definition. Note: On this section, replace \\Omega in the figure with D | upload.wikimedia.org/wikipedia/commons/f/fc/Domain///of///holomorphy///illustration.svg]\n\n### Properties of the domain of holomorphy\n\n## Holomorphically convex hull\nLet be a domain, or alternatively for a more general definition, let G be an n dimensional [complex analytic manifold](https://bliptext.com/articles/complex-analytic-manifold).\nFurther let stand for the set of holomorphic functions on G.\nFor a compact set K \\subset G, the **holomorphically convex hull** of K is\nOne obtains a narrower concept of **polynomially convex hull** by taking instead to be the set of complex-valued polynomial functions on G.\nThe polynomially convex hull contains the holomorphically convex hull.\nThe domain G is called **holomorphically convex** if for every compact subset is also compact in G.\nSometimes this is just abbreviated as holomorph-convex.\nWhen n=1, every domain G is holomorphically convex since then \\hat{K}_G is the union of K with the relatively compact components of.\nWhen n\\geq 1, if f satisfies the above holomorphic convexity on D it has the following properties.\nfor every compact subset K in D, where\ndenotes the distance between K and.\nAlso, at this time, D is a domain of holomorphy.\nTherefore, every convex domain is domain of holomorphy.\n\n## Pseudoconvexity\nHartogs showed that\n\"Hartogs (1906): Let D be a Hartogs's domain on \\mathbb{C} and R be a positive function on D such that the set \\Omega in defined by z_1 \\in D and is a domain of holomorphy. Then is a subharmonic function on D.\"\nIf such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.\nThe [subharmonic function](https://bliptext.com/articles/subharmonic-function) looks like a kind of [convex function](https://bliptext.com/articles/convex-function), so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity).\nPseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy.\nA domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.\n\n### Definition of plurisubharmonic function\nis called **plurisubharmonic** if it is [upper semi-continuous](https://bliptext.com/articles/semi-continuous-function), and for every complex line\nthe function\nis subharmonic, where denotes the unit disk.\nIn one-complex variable, necessary and sufficient condition that the real-valued function u=u(z), that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is.\nTherefore, if u is of class, then u is plurisubharmonic if and only if the [hermitian matrix](https://bliptext.com/articles/hermitian-matrix) is positive semidefinite.\nEquivalently, a -function u is plurisubharmonic if and only if is a [positive (1,1)-form](https://bliptext.com/articles/positive-form).\n\n#### Strictly plurisubharmonic function\nWhen the hermitian matrix of u is positive-definite and class, we call u a strict plurisubharmonic function.\n\n### (Weakly) pseudoconvex (p-pseudoconvex)\nWeak pseudoconvex is defined as : Let be a domain.\nOne says that X is pseudoconvex if there exists a [continuous](https://bliptext.com/articles/continuous-function) [plurisubharmonic function](https://bliptext.com/articles/plurisuperharmonic-function) \\varphi on X such that the set is a [relatively compact](https://bliptext.com/articles/relatively-compact) subset of X for all real numbers x.\ni.e. there exists a smooth plurisubharmonic exhaustion function.\nOften, the definition of pseudoconvex is used here and is written as; Let X be a complex n-dimensional manifold.\nThen is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function.\n\n### Strongly (Strictly) pseudoconvex\nLet X be a complex n-dimensional manifold.\n**Strongly (or Strictly) pseudoconvex** if there exists a smooth **strictly** plurisubharmonic exhaustion function, i.e., H\\psi is positive definite at every point.\nThe strongly pseudoconvex domain is the pseudoconvex domain.\nStrongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete ) are often used interchangeably, see Lempert for the technical difference.\n\n### Levi form\n\n#### (Weakly) Levi(–Krzoska) pseudoconvexity\nIf boundary, it can be shown that D has a defining function; i.e., that there exists which is so that , and.\nNow, D is pseudoconvex iff for every and w in the complex tangent space at p, that is,\nIf D does not have a boundary, the following approximation result can be useful.\n**Proposition 1** If D is pseudoconvex, then there exist [bounded](https://bliptext.com/articles/bounded-set), strongly Levi pseudoconvex domains with class -boundary which are relatively compact in D, such that\nThis is because once we have a \\varphi as in the definition we can actually find a exhaustion function.\n\n#### Strongly (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex)\nWhen the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.\n\n### Levi total pseudoconvex\nIf for every boundary point \\rho of D, there exists an [analytic variety](https://bliptext.com/articles/analytic-variety) \\mathcal{B} passing \\rho which lies entirely outside D in some neighborhood around \\rho, except the point \\rho itself.\nDomain D that satisfies these conditions is called Levi total pseudoconvex.\n\n### Oka pseudoconvex\n\n#### Family of Oka's disk\nLet n-functions be continuous on, holomorphic in |u|< 1 when the parameter t is fixed in [0, 1], and assume that are not all zero at any point on \\Delta.\nThen the set is called an analytic disc de-pending on a parameter t, and is called its shell.\nIf and, Q(t) is called Family of Oka's disk.\n\n#### Definition\nWhen holds on any family of Oka's disk, D is called Oka pseudoconvex.\nOka's proof of Levi's problem was that when the [unramified](https://bliptext.com/articles/unramified) Riemann domain over 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.\n\n### Locally pseudoconvex (a.k.a. locally Stein, Cartan pseudoconvex, local Levi property)\nFor every point there exist a neighbourhood U of x and f holomorphic.\n( i.e. U \\cap D be holomorphically convex.) such that f cannot be extended to any neighbourhood of x.\ni.e., let be a holomorphic map, if every point y\\in Y has a neighborhood U such that admits a -plurisubharmonic exhaustion function (weakly 1-complete ), in this situation, we call that X is locally pseudoconvex (or locally Stein) over Y.\nAs an old name, it is also called Cartan pseudoconvex.\nIn \\Complex^n the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.\nFor example, Diederich–Fornæss found local pseudoconvex bounded domains \\Omega with smooth boundary on non-Kähler manifolds such that \\Omega is not weakly 1-complete.\n\n## Conditions equivalent to domain of holomorphy\nFor a domain the following conditions are equivalent:\n

D is a domain of holomorphy.

D is holomorphically convex.

D is the union of an increasing sequence of [analytic polyhedrons](https://bliptext.com/articles/analytic-polyhedron) in D.

D is pseudoconvex.

D is Locally pseudoconvex.

\nThe implications, , and are standard results.\nProving, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally.\nThis is called the [Levi problem](https://bliptext.com/articles/levi-problem) (after [E. E. Levi](https://bliptext.com/articles/eugenio-elia-levi)) and was solved for unramified Riemann domains over by Kiyoshi Oka, but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity, and then by [Lars Hörmander](https://bliptext.com/articles/lars-h-rmander) using methods from functional analysis and partial differential equations (a consequence of -problem(equation) with a [L2 methods](https://bliptext.com/articles/ohsawa-takegoshi-l2-extension-theorem)).\n\n# Sheaves\nThe introduction of [sheaves](https://bliptext.com/articles/sheaf-mathematics) into several complex variables allowed the reformulation of and solution to several important problems in the field.\n\n## Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))\nOka introduced the notion which he termed \"idéal de domaines indéterminés\" or \"ideal of indeterminate domains\".\nSpecifically, it is a set (I) of pairs (f, \\delta), f holomorphic on a non-empty open set \\delta, such that\n

If and is arbitrary, then .

For each, then

\n\nThe origin of indeterminate domains comes from the fact that domains change depending on the pair (f, \\delta).\nCartan translated this notion into the notion of the **[coherent](https://bliptext.com/articles/coherent-[sheaf](https://bliptext.com/articles/sheaf-mathematics))** (**[sheaf](https://bliptext.com/articles/sheaf-mathematics)**) (Especially, **[coherent](https://bliptext.com/articles/coherent-[sheaf](https://bliptext.com/articles/sheaf-mathematics))** analytic **[sheaf](https://bliptext.com/articles/sheaf-mathematics)**) in **[sheaf](https://bliptext.com/articles/sheaf-mathematics)** cohomology.\nThis name comes from\nH. Cartan.\nAlso, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.\nThe notion of coherent ([coherent sheaf cohomology](https://bliptext.com/articles/coherent-sheaf-cohomology)) helped solve the problems in several complex variables.\n\n## Coherent sheaf\n\n### Definition\nThe definition of the coherent sheaf is as follows.\nA **quasi-coherent sheaf** on a [ringed space](https://bliptext.com/articles/ringed-space) is a sheaf \\mathcal F of -[modules](https://bliptext.com/articles/sheaf-of-modules) which has a local presentation, that is, every point in X has an open neighborhood U in which there is an [exact sequence](https://bliptext.com/articles/exact-sequence)\nfor some (possibly infinite) sets I and J.\nA **coherent sheaf** on a ringed space is a sheaf \\mathcal F satisfying the following two properties:\n

\\mathcal F is of finite type over, that is, every point in X has an [open neighborhood](https://bliptext.com/articles/open-neighborhood) U in X such that there is a surjective morphism for some natural number n;

for each open set, integer n > 0, and arbitrary morphism of -modules, the kernel of \\varphi is of finite type.

\n\nMorphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.\nAlso, [Jean-Pierre Serre](https://bliptext.com/articles/jean-pierre-serre) (1955) proves that\n\n### (Oka–Cartan) coherent theorem\n(Oka–Cartan) coherent theorem says that each sheaf that meets the following conditions is a coherent.\n

the sheaf of [germs](https://bliptext.com/articles/germ-mathematics) of holomorphic functions on, or the structure sheaf of complex submanifold or every complex analytic space

the ideal sheaf of an analytic subset A of an open subset of.\n(Cartan 1950)

the normalization of the structure sheaf of a complex analytic space

\n\nFrom the above Serre(1955) theorem, is a coherent sheaf, also, (i) is used to prove [Cartan's theorems A and B](https://bliptext.com/articles/cartan-s-theorems-a-and-b).\n\n## Cousin problem\nIn the case of one variable complex functions, [Mittag-Leffler's theorem](https://bliptext.com/articles/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](https://bliptext.com/articles/weierstrass-factorization-theorem) was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem).\nHowever, these theorems do not hold in several complex variables because the singularities of [analytic function in several complex variables](https://bliptext.com/articles/analytic-function) are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology.\nThey were first introduced in special cases by Pierre Cousin in 1895.\nIt was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy on the complex coordinate space, also solving the second Cousin problem with additional topological assumptions.\nThe Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological; Serre called this the [Oka principle](https://bliptext.com/articles/).\nThey are now posed, and solved, for arbitrary complex manifold M, in terms of conditions on M.\nM, which satisfies these conditions, is one way to define a Stein manifold.\nThe 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, that is it has developed the theory of sheaf cohomology.\n(e.g.Cartan seminar.)\n\n### First Cousin problem\nWithout the language of sheaves, the problem can be formulated as follows.\nOn a complex manifold M, one is given several meromorphic functions f_i along with domains U_i where they are defined, and where each difference f_i-f_j is holomorphic (wherever the difference is defined).\nThe first Cousin problem then asks for a meromorphic function f on M such that f-f_i is holomorphic on U_i; in other words, that f shares the [singular](https://bliptext.com/articles/mathematical-singularity) behaviour of the given local function.\nNow, let **K** be the sheaf of meromorphic functions and **O** the sheaf of holomorphic functions on M.\nThe first Cousin problem can always be solved if the following map is surjective:\nBy the [long exact cohomology sequence](https://bliptext.com/articles/long-exact-sequence-in-homology),\nis exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,**O**) vanishes.\nIn particular, by [Cartan's theorem B](https://bliptext.com/articles/cartan-s-theorems-a-and-b), the Cousin problem is always solvable if M is a Stein manifold.\n\n### Second Cousin problem\nThe second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio f_i/f_j is a non-vanishing holomorphic function (where said difference is defined).\nIt asks for a meromorphic function f on M such that f/f_i is holomorphic and non-vanishing.\nLet be the sheaf of holomorphic functions that vanish nowhere, and the sheaf of meromorphic functions that are not identically zero.\nThese are both then sheaves of [abelian groups](https://bliptext.com/articles/abelian-group), and the quotient sheaf is well-defined.\nIf the following map \\phi is surjective, then Second Cousin problem can be solved:\nThe long exact sheaf cohomology sequence associated to the quotient is\nso the second Cousin problem is solvable in all cases provided that\nThe cohomology group for the multiplicative structure on can be compared with the cohomology group with its additive structure by taking a logarithm.\nThat is, there is an exact sequence of sheaves\nwhere the leftmost sheaf is the locally constant sheaf with fiber 2\\pi i\\Z.\nThe obstruction to defining a logarithm at the level of H1 is in H^2(M,\\Z), from the long exact cohomology sequence\nWhen M is a Stein manifold, the middle arrow is an isomorphism because for q > 0 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.)\n\n# Manifolds and analytic varieties with several complex variables\n\n## Stein manifold (non-compact Kähler manifold)\nSince a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, and satisfies the [second axiom of countability](https://bliptext.com/articles/second-axiom-of-countability), the open Riemann surface is in fact a 1-dimensional complex manifold possessing a holomorphic mapping into the complex plane \\mathbb C. (In fact, Gunning and Narasimhan have shown (1967) 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.) The [Whitney embedding theorem](https://bliptext.com/articles/whitney-embedding-theorem) tells us that every smooth n-dimensional manifold can be [embedded](https://bliptext.com/articles/embedding) as a smooth submanifold of, whereas it is \"rare\" for a complex manifold to have a holomorphic embedding into \\mathbb C^n.\nFor 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.\nThat is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants.\nSo, consider the conditions under which a complex manifold has a holomorphic function that is not a constant.\nNow if we had a holomorphic embedding of X into \\mathbb C^n, then the coordinate functions of \\mathbb C^n would restrict to nonconstant holomorphic functions on X, contradicting compactness, except in the case that X is just a point.\nComplex manifolds that can be holomorphic embedded into \\mathbb C^n are called Stein manifolds.\nAlso Stein manifolds satisfy the second axiom of countability.\nA **Stein manifold** is a complex [submanifold](https://bliptext.com/articles/submanifold) of the [vector space](https://bliptext.com/articles/vector-space) of n complex dimensions.\nThey were introduced by and named after Karl Stein (1951).\nA **Stein space** is similar to a Stein manifold but is allowed to have singularities.\nStein spaces are the analogues of [affine varieties](https://bliptext.com/articles/affine-variety) or [affine schemes](https://bliptext.com/articles/affine-scheme) in algebraic geometry.\nIf the univalent domain on \\mathbb C^n is connection to a manifold, can be regarded as a [complex manifold](https://bliptext.com/articles/complex-manifold) and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity.\nTherefore, the Stein manifold is the properties of the domain of definition of the (maximal) [analytic continuation](https://bliptext.com/articles/analytic-continuation) of an analytic function.\n\n### Definition\nSuppose X is a [paracompact](https://bliptext.com/articles/paracompact) [complex manifolds](https://bliptext.com/articles/complex-manifold) of complex dimension n and let denote the ring of holomorphic functions on X.\nWe call X a **Stein manifold** if the following conditions hold:\n

X is holomorphically convex, i.e. for every compact subset K \\subset X, the so-called holomorphically convex hull,\nis also a compact subset of X.

X is [holomorphically separable](https://bliptext.com/articles/holomorphically-separable), i.e. if x \\neq y are two points in X, then there exists such that

The open neighborhood of every point on the manifold has a holomorphic [chart](https://bliptext.com/articles/atlas-topology) to the .

\nNote that condition (3) can be derived from conditions (1) and (2).\n\n### Every non-compact (open) Riemann surface is a Stein manifold\nLet X be a connected, non-compact (open) [Riemann surface](https://bliptext.com/articles/riemann-surface).\nA [deep theorem](https://bliptext.com/articles/behnke-stein-theorem-on-stein-manifolds) of Behnke and Stein (1948) asserts that X is a Stein manifold.\nAnother result, attributed to [Hans Grauert](https://bliptext.com/articles/hans-grauert) and [Helmut Röhrl](https://bliptext.com/articles/helmut-r-hrl) (1956), states moreover that every [holomorphic vector bundle](https://bliptext.com/articles/holomorphic-vector-bundle) on X is trivial.\nIn particular, every line bundle is trivial, so.\nThe [exponential sheaf sequence](https://bliptext.com/articles/exponential-sheaf-sequence) leads to the following exact sequence:\nNow [Cartan's theorem B](https://bliptext.com/articles/cartan-s-theorems-a-and-b) shows that, therefore.\nThis is related to the solution of the [second (multiplicative) Cousin problem](https://bliptext.com/articles/cousin-problems).\n\n### Levi problems\nCartan extended Levi's problem to Stein manifolds.\nThis was proved by Bremermann by embedding it in a sufficiently high dimensional, and reducing it to the result of Oka.\nAlso, Grauert proved for arbitrary **complex** manifolds M.\nAnd Narasimhan extended Levi's problem to [complex analytic space](https://bliptext.com/articles/complex-analytic-space), a generalized in the singular case of complex manifolds.\nLevi's problem remains unresolved in the following cases;\nmore generalized\nand also,\nThis means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space.\n\n#### K-complete\nGrauert introduced the concept of K-complete in the proof of Levi's problem.\nLet X is complex manifold, X is K-complete if, to each point x_0\\in X, there exist finitely many holomorphic map of X into \\Complex^p, p = p(x_0), such that x_0 is an isolated point of the set.\nThis concept also applies to complex analytic space.\n\n### Properties and examples of Stein manifolds\nThese facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the [ambient space](https://bliptext.com/articles/ambient-space) (because the embedding is biholomorphic).\nNumerous further characterizations of such manifolds exist, in particular capturing the property of their having \"many\" [holomorphic functions](https://bliptext.com/articles/holomorphic-function) taking values in the complex numbers.\nSee for example [Cartan's theorems A and B](https://bliptext.com/articles/cartan-s-theorems-a-and-b), relating to [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology).\nIn the [GAGA](https://bliptext.com/articles/gaga) set of analogies, Stein manifolds correspond to [affine varieties](https://bliptext.com/articles/affine-variety).\nStein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit \"many\" holomorphic functions from the complex numbers into themselves.\nIt is known that a Stein manifold is elliptic if and only if it is [fibrant](https://bliptext.com/articles/fibrant-object) in the sense of so-called \"holomorphic homotopy theory\".\n\n## Complex projective varieties (compact complex manifold)\nMeromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the [Riemann-Roch theorem](https://bliptext.com/articles/riemann-roch-theorem) ([Riemann's inequality](https://bliptext.com/articles/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](https://bliptext.com/articles/algebraic-curve) over \\mathbb{C} ).\nIn fact, compact Riemann surface had a non-constant single-valued meromorphic function, and also a compact Riemann surface had enough meromorphic functions.\nA compact one-dimensional complex manifold was a Riemann sphere.\nHowever, the abstract notion of a compact Riemann surface is always algebraizable (The [Riemann's existence theorem](https://bliptext.com/articles/riemann-s-existence-theorem), [Kodaira embedding theorem](https://bliptext.com/articles/kodaira-embedding-theorem).), but it is not easy to verify which compact complex analytic spaces are algebraizable.\nIn fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.\nHowever, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic.\nThe generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira, Kodaira also extended the theorem to three-dimensional, and n-dimensional Kähler varieties.\nSerre formulated the Riemann–Roch theorem as a problem of dimension of [coherent sheaf cohomology](https://bliptext.com/articles/coherent-sheaf-cohomology), and also Serre proved [Serre duality](https://bliptext.com/articles/serre-duality).\nCartan and Serre proved the following property: the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M. Riemann–Roch on a Riemann surface for a vector bundle was proved by [Weil](https://bliptext.com/articles/vector-bundles-on-algebraic-curves) in 1938.\n[Hirzebruch](https://bliptext.com/articles/hirzebruch-riemann-roch-theorem) generalized the theorem to compact complex manifolds in 1994 and [Grothendieck](https://bliptext.com/articles/grothendieck-hirzebruch-riemann-roch-theorem) generalized it to a relative version (relative statements about [morphisms](https://bliptext.com/articles/morphism).).\nNext, the generalization of the result that \"the compact Riemann surfaces are projective\" to the high-dimension.\nIn particular, consider the conditions that when embedding of compact complex submanifold X into the complex projective space.\nThe [vanishing theorem](https://bliptext.com/articles/vanishing-theorem) (was first introduced by [Kodaira](https://bliptext.com/articles/kodaira-vanishing-theorem) in 1953) gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of [positivity](https://bliptext.com/articles/positive-form).\nAs an application of this theorem, the [Kodaira embedding theorem](https://bliptext.com/articles/kodaira-embedding-theorem) says that a compact [Kähler manifold](https://bliptext.com/articles/k-hler-manifold) M, with a Hodge metric, there is a complex-analytic embedding of M into [complex projective space](https://bliptext.com/articles/complex-projective-space) of enough high-dimension N.\nIn addition the [Chow's theorem](https://bliptext.com/articles/gaga) 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](https://bliptext.com/articles/algebraic-geometry-and-analytic-geometry).\nThe complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties.\nThen combined with Kodaira's result, a compact Kähler manifold M embeds as an algebraic variety.\nThis result gives an example of a complex manifold with enough meromorphic functions.\nBroadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties.\nThe combination of analytic and algebraic methods for complex projective varieties lead to areas such as [Hodge theory](https://bliptext.com/articles/hodge-theory).\nAlso, the [deformation theory](https://bliptext.com/articles/deformation-theory) of compact complex manifolds has developed as Kodaira–Spencer theory.\nHowever, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic.\nAnalogy of the Levi problems on the complex projective space by Takeuchi.\n\n# Annotation\n\n# Inline citations\n\n## Textbooks\n\n## Encyclopedia of Mathematics","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:24:34.040Z","updated_at":"2025-01-25T19:24:34.040Z","search_vector":"'complex':4 'function':1 'sever':3 'variabl':5","search_content":" \n \n\n# \nThe theory of **functions of several complex variables** is the branch of [mathematics](https://bliptext.com/articles/mathematics) dealing with functions defined on [the complex coordinate space](https://bliptext.com/articles/) \\mathbb C^n, that is, n-tuples of [complex numbers](https://bliptext.com/articles/complex-number).\nThe name of the field dealing with the properties of these functions is called **several complex variables** (and [analytic space](https://bliptext.com/articles/analytic-space)), which the [Mathematics Subject Classification](https://bliptext.com/articles/mathematics-subject-classification) has as a top-level heading.\nAs in [complex analysis of functions of one variable](https://bliptext.com/articles/complex-analysis), which is the case\nn = 1\n, the functions studied are [holomorphic](https://bliptext.com/articles/holomorphic-function) or complex analytic so that, locally, they are [power series](https://bliptext.com/articles/power-series) in the variables zi.\nEquivalently, they are locally [uniform limits](https://bliptext.com/articles/uniform-convergence) of [polynomials](https://bliptext.com/articles/polynomial); or locally [square-integrable](https://bliptext.com/articles/lp-space) solutions to the n-dimensional [Cauchy–Riemann equations](https://bliptext.com/articles/).\nFor one complex variable, every [domain](https://bliptext.com/articles/domain-mathematical-analysis), is the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of holomorphy of some function, in other words every [domain](https://bliptext.com/articles/domain-mathematical-analysis) has a function for which it is the [domain](https://bliptext.com/articles/domain-mathematical-analysis) of holomorphy.\nFor 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.\nPatching the local data of [meromorphic functions](https://bliptext.com/articles/meromorphic-function), i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem.\nAlso, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and [complex projective varieties](https://bliptext.com/articles/projective-variety) and has a different flavour to complex analytic geometry in or on [Stein manifolds](https://bliptext.com/articles/stein-manifold), these are much similar to study of algebraic varieties that is study of the [algebraic geometry](https://bliptext.com/articles/algebraic-geometry) than complex analytic geometry.\n\n# Historical perspective\nMany examples of such functions were familiar in nineteenth-century mathematics; [abelian functions](https://bliptext.com/articles/abelian-variety), [theta functions](https://bliptext.com/articles/theta-function), and some [hypergeometric series](https://bliptext.com/articles/hypergeometric-series), and also, as an example of an inverse problem; the [Jacobi inversion problem](https://bliptext.com/articles/jacobi-inversion-problem).\nNaturally also same function of one variable that depends on some complex [parameter](https://bliptext.com/articles/parameter) is a candidate.\nThe theory, however, for many years didn't become a full-fledged field in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), since its characteristic phenomena weren't uncovered.\nThe [Weierstrass preparation theorem](https://bliptext.com/articles/weierstrass-preparation-theorem) would now be classed as [commutative algebra](https://bliptext.com/articles/commutative-algebra); it did justify the local picture, [ramification](https://bliptext.com/articles/ramification-mathematics), that addresses the generalization of the [branch points](https://bliptext.com/articles/branch-point) of [Riemann surface](https://bliptext.com/articles/riemann-surface) theory.\nWith work of [Friedrich Hartogs](https://bliptext.com/articles/friedrich-hartogs), Pierre Cousin (mathematician), [E. E. Levi](https://bliptext.com/articles/e-e-levi), and of [Kiyoshi Oka](https://bliptext.com/articles/kiyoshi-oka) in the 1930s, a general theory began to emerge; others working in the area at the time were [Heinrich Behnke](https://bliptext.com/articles/heinrich-behnke), [Peter Thullen](https://bliptext.com/articles/peter-thullen), [Karl Stein](https://bliptext.com/articles/karl-stein-mathematician), [Wilhelm Wirtinger](https://bliptext.com/articles/wilhelm-wirtinger) and [Francesco Severi](https://bliptext.com/articles/francesco-severi).\nHartogs proved some basic results, such as every [isolated singularity](https://bliptext.com/articles/isolated-singularity) is [removable](https://bliptext.com/articles/removable-singularity), for every analytic function\nwhenever\nn > 1\n. Naturally the analogues of [contour integrals](https://bliptext.com/articles/line-integral) will be harder to handle; when\nn = 2\nan integral surrounding a point should be over a three-dimensional [manifold](https://bliptext.com/articles/manifold) (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a [double integral](https://bliptext.com/articles/double-integral) over a two-dimensional surface.\nThis means that the [residue calculus](https://bliptext.com/articles/residue-calculus) will have to take a very different character.\nAfter 1945 important work in France, in the seminar of [Henri Cartan](https://bliptext.com/articles/henri-cartan), and Germany with [Hans Grauert](https://bliptext.com/articles/hans-grauert) and [Reinhold Remmert](https://bliptext.com/articles/reinhold-remmert), quickly changed the picture of the theory.\nA number of issues were clarified, in particular that of [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nHere a major difference is evident from the one-variable theory; while for every open connected set D in \\Complex we can find a function that will nowhere continue analytically over the boundary, that cannot be said for\nn > 1\n. In fact the D of that kind are rather special in nature (especially in complex coordinate spaces \\mathbb C^n and Stein manifolds, satisfying a condition called [pseudoconvexity](https://bliptext.com/articles/)).\nThe natural domains of definition of functions, continued to the limit, are called [Stein manifolds](https://bliptext.com/articles/stein-manifold) and their nature was to make [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) groups vanish, on the other hand, the [Grauert–Riemenschneider vanishing theorem](https://bliptext.com/articles/grauert-riemenschneider-vanishing-theorem) is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan.\nIn 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](https://bliptext.com/articles/algebraic-geometry), in particular from Grauert's work).\nFrom this point onwards there was a foundational theory, which could be applied to [analytic geometry](https://bliptext.com/articles/algebraic-geometry), [automorphic forms](https://bliptext.com/articles/automorphic-form) of several variables, and [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nThe [deformation theory of complex structures](https://bliptext.com/articles/complex-structure-deformation) and [complex manifolds](https://bliptext.com/articles/complex-manifold) was described in general terms by [Kunihiko Kodaira](https://bliptext.com/articles/kunihiko-kodaira) and [D. C. Spencer](https://bliptext.com/articles/d-c-spencer).\nThe celebrated paper [GAGA](https://bliptext.com/articles/gaga) of [Serre](https://bliptext.com/articles/jean-pierre-serre) pinned down the crossover point from géometrie analytique to géometrie algébrique.\n[C. L. Siegel](https://bliptext.com/articles/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](https://bliptext.com/articles/special-function) side of the theory was subordinated to sheaves.\nThe interest for [number theory](https://bliptext.com/articles/number-theory), certainly, is in specific generalizations of [modular forms](https://bliptext.com/articles/modular-form).\nThe classical candidates are the [Hilbert modular forms](https://bliptext.com/articles/hilbert-modular-form) and [Siegel modular forms](https://bliptext.com/articles/siegel-modular-form).\nThese days these are associated to [algebraic groups](https://bliptext.com/articles/algebraic-group) (respectively the [Weil restriction](https://bliptext.com/articles/weil-restriction) from a [totally real number field](https://bliptext.com/articles/totally-real-number-field) of\nGL(2)\n, and the [symplectic group](https://bliptext.com/articles/symplectic-group)), for which it happens that [automorphic representations](https://bliptext.com/articles/automorphic-representation) can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.\nSubsequent developments included the [hyperfunction](https://bliptext.com/articles/hyperfunction) theory, and the [edge-of-the-wedge theorem](https://bliptext.com/articles/edge-of-the-wedge-theorem), both of which had some inspiration from [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nThere are a number of other fields, such as [Banach algebra](https://bliptext.com/articles/banach-algebra) theory, that draw on several complex variables.\n\n# The complex coordinate space\nThe [complex coordinate space](https://bliptext.com/articles/complex-coordinate-space) \\mathbb C^n is the [Cartesian product](https://bliptext.com/articles/cartesian-product) of n copies of \\mathbb C, and when \\mathbb C^n is a domain of holomorphy, \\mathbb C^n can be regarded as a [Stein manifold](https://bliptext.com/articles/stein-manifold), and more generalized Stein space.\n\\mathbb C^n is also considered to be a [complex projective variety](https://bliptext.com/articles/complex-projective-variety), a [Kähler manifold](https://bliptext.com/articles/k-hler-manifold), etc. It is also an [n-dimensional vector space](https://bliptext.com/articles/dimension-vector-space) over the [complex numbers](https://bliptext.com/articles/complex-number), which gives its dimension\n2n\nover \\mathbb R. Hence, as a set and as a [topological space](https://bliptext.com/articles/topological-space), \\mathbb C^n may be identified to the [real coordinate space](https://bliptext.com/articles/real-coordinate-space) and its [topological dimension](https://bliptext.com/articles/topological-dimension) is thus\n2n\n.\nIn 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](https://bliptext.com/articles/almost-complex-manifold) is specified by a [linear operator](https://bliptext.com/articles/linear-operator) J (such that\nJ2 = −I\n) which defines [multiplication](https://bliptext.com/articles/scalar-multiplication) by the [imaginary unit](https://bliptext.com/articles/imaginary-unit) i.\nAny such space, as a real space, is [oriented](https://bliptext.com/articles/orientation-vector-space).\nOn the [complex plane](https://bliptext.com/articles/complex-plane) thought of as a [Cartesian plane](https://bliptext.com/articles/cartesian-plane), [multiplication](https://bliptext.com/articles/multiplication) by a complex number\nw = u + iv\nmay be represented by the real [matrix](https://bliptext.com/articles/square-matrix)\nwith [determinant](https://bliptext.com/articles/determinant)\nLikewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be [composed from 2 × 2 blocks](https://bliptext.com/articles/block-matrix) of the aforementioned form), then its determinant equals to the [square of absolute value](https://bliptext.com/articles/square-algebra) of the corresponding complex determinant.\nIt is a non-negative number, which implies that [the (real) orientation of the space is never reversed](https://bliptext.com/articles/real-coordinate-space) by a complex operator.\nThe same applies to [Jacobians](https://bliptext.com/articles/jacobian-matrix-and-determinant) of [holomorphic functions](https://bliptext.com/articles/holomorphic-function) from \\mathbb C^n to \\mathbb C^n.\n\n# Holomorphic functions\n\n## Definition\nA function f defined on a domain and with values in \\mathbb{C} is said to be holomorphic at a point z\\in D if it is complex-differentiable at this point, in the sense that there exists a complex linear map such that\nThe function f is said to be holomorphic if it is holomorphic at all points of its domain of definition D.\nIf f is holomorphic, then all the partial maps :\nare holomorphic as functions of one complex variable : we say that f is holomorphic in each variable separately.\nConversely, if f is holomorphic in each variable separately, then f is in fact holomorphic : this is known as [Hartog's theorem](https://bliptext.com/articles/hartogs-s-theorem-on-separate-holomorphicity), or as [Osgood's lemma](https://bliptext.com/articles/osgood-s-lemma) under the additional hypothesis that f is [continuous](https://bliptext.com/articles/continuous-function).\n\n## Cauchy–Riemann equations\nIn one complex variable, a function defined on the plane is holomorphic at a point if and only if its real part u and its imaginary part v satisfy the so-called [Cauchy-Riemann equations](https://bliptext.com/articles/cauchy-riemann-equations) at p :\nIn several variables, a function is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part u and the imaginary part v of f satisfiy the Cauchy Riemann equations :\nUsing the formalism of [Wirtinger derivatives](https://bliptext.com/articles/wirtinger-derivatives), this can be reformulated as :\nor even more compactly using the formalism of [complex differential forms](https://bliptext.com/articles/complex-differential-form), as :\n\n## Cauchy's integral formula I (Polydisc version)\nProve the sufficiency of two conditions (A) and (B).\nLet f meets the conditions of being continuous and separately homorphic on domain D.\nEach disk has a [rectifiable curve](https://bliptext.com/articles/rectifiable-curve) \\gamma, \\gamma_\\nu is piecewise [smoothness](https://bliptext.com/articles/smoothness), class Jordan closed curve.\nLet D_\\nu be the domain surrounded by each \\gamma_\\nu.\nCartesian product closure is.\nAlso, take the closed [polydisc](https://bliptext.com/articles/polydisc) so that it becomes.\n( and let be the center of each disk.) Using the [Cauchy's integral formula](https://bliptext.com/articles/cauchy-s-integral-formula) of one variable repeatedly,\nBecause \\partial D is a rectifiable Jordanian closed curve and f is continuous, so the order of products and sums can be exchanged so the [iterated integral](https://bliptext.com/articles/iterated-integral) can be calculated as a [multiple integral](https://bliptext.com/articles/multiple-integral).\nTherefore,\n\n### Cauchy's evaluation formula\nBecause the order of products and sums is interchangeable, from we get\nf is class -function.\nFrom (2), if f is holomorphic, on polydisc and, the following evaluation equation is obtained.\nTherefore, [Liouville's theorem](https://bliptext.com/articles/liouville-s-theorem-complex-analysis) hold.\n\n### Power series expansion of holomorphic functions on polydisc\nIf 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.\nIn addition, f that satisfies the following conditions is called an analytic function.\nFor each point, f(z) is expressed as a power series expansion that is convergent on D :\nWe have already explained that holomorphic functions on polydisc are analytic.\nAlso, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.\n\n### Radius of convergence of power series\nIt is possible to define a combination of positive real numbers such that the power series converges uniformly at and does not converge uniformly at.\nIn this way it is possible to have a similar, combination of radius of convergence for a one complex variable.\nThis combination is generally not unique and there are an infinite number of combinations.\n\n### Laurent series expansion\nLet \\omega(z) be holomorphic in the [annulus](https://bliptext.com/articles/annulus-mathematics) and continuous on their circumference, then there exists the following expansion ;\nThe 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.\n\n## Bochner–Martinelli formula (Cauchy's integral formula II)\nThe Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many possible domains, so we introduce the [Bochner–Martinelli formula](https://bliptext.com/articles/bochner-martinelli-formula).\nSuppose that f is a continuously differentiable function on the closure of a domain D on \\mathbb C^n with piecewise smooth boundary \\partial D, and let the symbol \\land denotes the exterior or [wedge product](https://bliptext.com/articles/wedge-product) of differential forms.\nThen the Bochner–Martinelli formula states that if z is in the domain D then, for \\zeta, z in \\mathbb C^n the Bochner–Martinelli kernel is a [differential form](https://bliptext.com/articles/differential-form) in \\zeta of bidegree (n,n-1), defined by\nIn particular if f is holomorphic the second term vanishes, so\n\n## Identity theorem\nHolomorphic functions of several complex variables satisfy an [identity theorem](https://bliptext.com/articles/identity-theorem), as in one variable : two holomorphic functions defined on the same connected open set and which coincide on an open subset N of D, are equal on the whole open set D.\nThis result can be proven from the fact that holomorphics functions have power series extensions, and it can also be deduced from the one variable case.\nContrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps and coincide on the whole complex line of defined by the equation z_1=0.\nThe [maximal principle](https://bliptext.com/articles/maximal-principle), [inverse function theorem](https://bliptext.com/articles/inverse-function-theorem), and implicit function theorems also hold.\nFor a generalized version of the implicit function theorem to complex variables, see the [Weierstrass preparation theorem](https://bliptext.com/articles/weierstrass-preparation-theorem).\n\n## Biholomorphism\nFrom the establishment of the inverse function theorem, the following mapping can be defined.\nFor the domain U, V of the n-dimensional complex space \\Complex^n, the bijective holomorphic function \\phi:U\\to V and the inverse mapping is also holomorphic.\nAt this time, \\phi is called a U, V biholomorphism also, we say that U and V are biholomorphically equivalent or that they are biholomorphic.\n\n### The Riemann mapping theorem does not hold\nWhen n > 1, open balls and open polydiscs are not biholomorphically equivalent, that is, there is no [biholomorphic mapping](https://bliptext.com/articles/biholomorphic-mapping) between the two.\nThis was proven by [Poincaré](https://bliptext.com/articles/henri-poincar) in 1907 by showing that their [automorphism groups](https://bliptext.com/articles/automorphism-group) have different dimensions as [Lie groups](https://bliptext.com/articles/lie-group).\nHowever, 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.\n\n## Analytic continuation\nLet U, V be domain on, such that and , ( is the set/ring of holomorphic functions on U.) assume that and W is a [connected component](https://bliptext.com/articles/connected-component-topology) of U \\cap V. If f|_W =g|_W then f is said to be connected to V, and g is said to be analytic continuation of f.\nFrom the identity theorem, if g exists, for each way of choosing W it is unique.\nWhen n > 2, the following phenomenon occurs depending on the shape of the boundary \\partial U: there exists domain U, V, such that all holomorphic functions f over the domain U, have an analytic continuation.\nIn other words, there may be not exist a function such that \\partial U as the natural boundary.\nThere is called the Hartogs's phenomenon.\nTherefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables.\nIn addition, when n \\geq 2, it would be that the above V has an intersection part with U other than W.\nThis contributed to advancement of the notion of sheaf cohomology.\n\n# Reinhardt domain\nIn 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.\nTherefore, 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.\nEarly 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.\nLet (n \\geq 1) to be a domain, with centre at a point, such that, together with each point , the domain also contains the set\nA domain D is called a Reinhardt domain if it satisfies the following conditions:\nLet is a arbitrary real numbers, a domain D is invariant under the rotation:.\nThe Reinhardt domains which are defined by the following condition; Together with all points of z^0 \\in D, the domain contains the set\nA Reinhardt domain D is called a complete Reinhardt domain with centre at a point a if together with all point z^0\\in D it also contains the polydisc\nA complete Reinhardt domain D is [star-like](https://bliptext.com/articles/star-like-domain) with regard to its centre a.\nTherefore, the complete Reinhardt domain is [simply connected](https://bliptext.com/articles/simply-connected), also when the complete Reinhardt domain is the boundary line, there is a way to prove the [Cauchy's integral theorem](https://bliptext.com/articles/cauchy-s-integral-theorem) without using the [Jordan curve theorem](https://bliptext.com/articles/jordan-curve-theorem).\n\n## Logarithmically-convex\nWhen a some complete Reinhardt domain to be the domain of convergence of a power series, an additional condition is required, which is called logarithmically-convex.\nA Reinhardt domain D is called [logarithmically convex](https://bliptext.com/articles/logarithmically-convex) if the image of the set\nunder the mapping\nis a [convex set](https://bliptext.com/articles/convex-set) in the real coordinate space \\R^n.\nEvery such domain in \\Complex^n 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 a = 0.\nBut, there is an example of a complete Reinhardt domain D which is not logarithmically convex.\n\n## Some results\n\n### Hartogs's extension theorem and Hartogs's phenomenon\nWhen 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.\nG \\ K\n, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain on \\Complex^n (\nn ≥ 2\n) and K is a compact subset of G. If the [complement](https://bliptext.com/articles/complement-set-theory)\nG \\ K\nis connected, then every holomorphic function f regardless of how it is chosen can be each extended to a unique holomorphic function on G.\nFrom Hartogs's extension theorem the domain of convergence extends from to \\Delta^2.\nLooking 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 \\Delta^2 containing.\n\n### Thullen's classic results\n[Thullen](https://bliptext.com/articles/peter-thullen)'s classical result says that a 2-dimensional bounded Reinhard domain containing the origin is [biholomorphic](https://bliptext.com/articles/biholomorphic) to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:\n\n### Sunada's results\n[Toshikazu Sunada](https://bliptext.com/articles/toshikazu-sunada) (1978) established a generalization of Thullen's result:\n\n# Natural domain of the holomorphic function (domain of holomorphy)\nWhen 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.\nConsidering the domain where the boundaries of the domain are natural boundaries (In the complex coordinate space \\Complex^n call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H.\nCartan and Thullen.\nLevi's problem shows that the pseudoconvex domain was a domain of holomorphy.\n(First for \\Complex^2, later extended to \\Complex^n. ) [Kiyoshi Oka](https://bliptext.com/articles/kiyoshi-oka)'s notion of idéal de domaines indéterminés is interpreted theory of [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) by H.\nCartan and more development Serre.\nIn sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.\nThe notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.\n\n## Domain of holomorphy\nWhen a function f is holomorpic on the domain and cannot directly connect to the domain outside D, including the point of the domain boundary \\partial D, the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f.\nIn 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.\nFor several complex variables, i.e. domain, the boundaries may not be natural boundaries.\nHartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.\nFormally, a domain D in the n-dimensional complex coordinate space \\Complex^n is called a domain of holomorphy if there do not exist non-empty domain U \\subset D and, and such that for every holomorphic function f on D there exists a holomorphic function g on V with f = g on U.\nFor the n=1 case, the every domain was the domain of holomorphy; we can define a holomorphic function with zeros [accumulating](https://bliptext.com/articles/accumulation-point) everywhere on the [boundary](https://bliptext.com/articles/boundary-topology) of the domain, which must then be a natural [boundary](https://bliptext.com/articles/boundary-topology) for a domain of definition of its reciprocal.\n# [The sets in the definition. Note: On this section, replace \\Omega in the figure with D | upload.wikimedia.org/wikipedia/commons/f/fc/Domain///of///holomorphy///illustration.svg]\n\n### Properties of the domain of holomorphy\n\n## Holomorphically convex hull\nLet be a domain, or alternatively for a more general definition, let G be an n dimensional [complex analytic manifold](https://bliptext.com/articles/complex-analytic-manifold).\nFurther let stand for the set of holomorphic functions on G.\nFor a compact set K \\subset G, the **holomorphically convex hull** of K is\nOne obtains a narrower concept of **polynomially convex hull** by taking instead to be the set of complex-valued polynomial functions on G.\nThe polynomially convex hull contains the holomorphically convex hull.\nThe domain G is called **holomorphically convex** if for every compact subset is also compact in G.\nSometimes this is just abbreviated as holomorph-convex.\nWhen n=1, every domain G is holomorphically convex since then \\hat{K}_G is the union of K with the relatively compact components of.\nWhen n\\geq 1, if f satisfies the above holomorphic convexity on D it has the following properties.\nfor every compact subset K in D, where\ndenotes the distance between K and.\nAlso, at this time, D is a domain of holomorphy.\nTherefore, every convex domain is domain of holomorphy.\n\n## Pseudoconvexity\nHartogs showed that\n\"Hartogs (1906): Let D be a Hartogs's domain on \\mathbb{C} and R be a positive function on D such that the set \\Omega in defined by z_1 \\in D and is a domain of holomorphy. Then is a subharmonic function on D.\"\nIf such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.\nThe [subharmonic function](https://bliptext.com/articles/subharmonic-function) looks like a kind of [convex function](https://bliptext.com/articles/convex-function), so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity).\nPseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy.\nA domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.\n\n### Definition of plurisubharmonic function\nis called **plurisubharmonic** if it is [upper semi-continuous](https://bliptext.com/articles/semi-continuous-function), and for every complex line\nthe function\nis subharmonic, where denotes the unit disk.\nIn one-complex variable, necessary and sufficient condition that the real-valued function u=u(z), that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is.\nTherefore, if u is of class, then u is plurisubharmonic if and only if the [hermitian matrix](https://bliptext.com/articles/hermitian-matrix) is positive semidefinite.\nEquivalently, a -function u is plurisubharmonic if and only if is a [positive (1,1)-form](https://bliptext.com/articles/positive-form).\n\n#### Strictly plurisubharmonic function\nWhen the hermitian matrix of u is positive-definite and class, we call u a strict plurisubharmonic function.\n\n### (Weakly) pseudoconvex (p-pseudoconvex)\nWeak pseudoconvex is defined as : Let be a domain.\nOne says that X is pseudoconvex if there exists a [continuous](https://bliptext.com/articles/continuous-function) [plurisubharmonic function](https://bliptext.com/articles/plurisuperharmonic-function) \\varphi on X such that the set is a [relatively compact](https://bliptext.com/articles/relatively-compact) subset of X for all real numbers x.\ni.e. there exists a smooth plurisubharmonic exhaustion function.\nOften, the definition of pseudoconvex is used here and is written as; Let X be a complex n-dimensional manifold.\nThen is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function.\n\n### Strongly (Strictly) pseudoconvex\nLet X be a complex n-dimensional manifold.\n**Strongly (or Strictly) pseudoconvex** if there exists a smooth **strictly** plurisubharmonic exhaustion function, i.e., H\\psi is positive definite at every point.\nThe strongly pseudoconvex domain is the pseudoconvex domain.\nStrongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete ) are often used interchangeably, see Lempert for the technical difference.\n\n### Levi form\n\n#### (Weakly) Levi(–Krzoska) pseudoconvexity\nIf boundary, it can be shown that D has a defining function; i.e., that there exists which is so that , and.\nNow, D is pseudoconvex iff for every and w in the complex tangent space at p, that is,\nIf D does not have a boundary, the following approximation result can be useful.\n**Proposition 1** If D is pseudoconvex, then there exist [bounded](https://bliptext.com/articles/bounded-set), strongly Levi pseudoconvex domains with class -boundary which are relatively compact in D, such that\nThis is because once we have a \\varphi as in the definition we can actually find a exhaustion function.\n\n#### Strongly (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex)\nWhen the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.\n\n### Levi total pseudoconvex\nIf for every boundary point \\rho of D, there exists an [analytic variety](https://bliptext.com/articles/analytic-variety) \\mathcal{B} passing \\rho which lies entirely outside D in some neighborhood around \\rho, except the point \\rho itself.\nDomain D that satisfies these conditions is called Levi total pseudoconvex.\n\n### Oka pseudoconvex\n\n#### Family of Oka's disk\nLet n-functions be continuous on, holomorphic in |u| D is a domain of holomorphy. \n D is holomorphically convex. \n D is the union of an increasing sequence of [analytic polyhedrons](https://bliptext.com/articles/analytic-polyhedron) in D. \n D is pseudoconvex. \n D is Locally pseudoconvex. \nThe implications, , and are standard results.\nProving, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally.\nThis is called the [Levi problem](https://bliptext.com/articles/levi-problem) (after [E. E. Levi](https://bliptext.com/articles/eugenio-elia-levi)) and was solved for unramified Riemann domains over by Kiyoshi Oka, but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity, and then by [Lars Hörmander](https://bliptext.com/articles/lars-h-rmander) using methods from functional analysis and partial differential equations (a consequence of -problem(equation) with a [L2 methods](https://bliptext.com/articles/ohsawa-takegoshi-l2-extension-theorem)).\n\n# Sheaves\nThe introduction of [sheaves](https://bliptext.com/articles/sheaf-mathematics) into several complex variables allowed the reformulation of and solution to several important problems in the field.\n\n## Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))\nOka introduced the notion which he termed \"idéal de domaines indéterminés\" or \"ideal of indeterminate domains\".\nSpecifically, it is a set (I) of pairs (f, \\delta), f holomorphic on a non-empty open set \\delta, such that\n If and is arbitrary, then . \n For each, then \n \nThe origin of indeterminate domains comes from the fact that domains change depending on the pair (f, \\delta).\nCartan translated this notion into the notion of the **[coherent](https://bliptext.com/articles/coherent-[sheaf](https://bliptext.com/articles/sheaf-mathematics))** (**[sheaf](https://bliptext.com/articles/sheaf-mathematics)**) (Especially, **[coherent](https://bliptext.com/articles/coherent-[sheaf](https://bliptext.com/articles/sheaf-mathematics))** analytic **[sheaf](https://bliptext.com/articles/sheaf-mathematics)**) in **[sheaf](https://bliptext.com/articles/sheaf-mathematics)** cohomology.\nThis name comes from\nH. Cartan.\nAlso, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.\nThe notion of coherent ([coherent sheaf cohomology](https://bliptext.com/articles/coherent-sheaf-cohomology)) helped solve the problems in several complex variables.\n\n## Coherent sheaf\n\n### Definition\nThe definition of the coherent sheaf is as follows.\nA **quasi-coherent sheaf** on a [ringed space](https://bliptext.com/articles/ringed-space) is a sheaf \\mathcal F of -[modules](https://bliptext.com/articles/sheaf-of-modules) which has a local presentation, that is, every point in X has an open neighborhood U in which there is an [exact sequence](https://bliptext.com/articles/exact-sequence)\nfor some (possibly infinite) sets I and J.\nA **coherent sheaf** on a ringed space is a sheaf \\mathcal F satisfying the following two properties:\n \\mathcal F is of finite type over, that is, every point in X has an [open neighborhood](https://bliptext.com/articles/open-neighborhood) U in X such that there is a surjective morphism for some natural number n; \n for each open set, integer n > 0, and arbitrary morphism of -modules, the kernel of \\varphi is of finite type. \n \nMorphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.\nAlso, [Jean-Pierre Serre](https://bliptext.com/articles/jean-pierre-serre) (1955) proves that\n\n### (Oka–Cartan) coherent theorem\n(Oka–Cartan) coherent theorem says that each sheaf that meets the following conditions is a coherent.\n the sheaf of [germs](https://bliptext.com/articles/germ-mathematics) of holomorphic functions on, or the structure sheaf of complex submanifold or every complex analytic space \n the ideal sheaf of an analytic subset A of an open subset of.\n(Cartan 1950) \n the normalization of the structure sheaf of a complex analytic space \n \nFrom the above Serre(1955) theorem, is a coherent sheaf, also, (i) is used to prove [Cartan's theorems A and B](https://bliptext.com/articles/cartan-s-theorems-a-and-b).\n\n## Cousin problem\nIn the case of one variable complex functions, [Mittag-Leffler's theorem](https://bliptext.com/articles/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](https://bliptext.com/articles/weierstrass-factorization-theorem) was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem).\nHowever, these theorems do not hold in several complex variables because the singularities of [analytic function in several complex variables](https://bliptext.com/articles/analytic-function) are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology.\nThey were first introduced in special cases by Pierre Cousin in 1895.\nIt was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy on the complex coordinate space, also solving the second Cousin problem with additional topological assumptions.\nThe Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological; Serre called this the [Oka principle](https://bliptext.com/articles/).\nThey are now posed, and solved, for arbitrary complex manifold M, in terms of conditions on M.\nM, which satisfies these conditions, is one way to define a Stein manifold.\nThe 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, that is it has developed the theory of sheaf cohomology.\n(e.g.Cartan seminar.)\n\n### First Cousin problem\nWithout the language of sheaves, the problem can be formulated as follows.\nOn a complex manifold M, one is given several meromorphic functions f_i along with domains U_i where they are defined, and where each difference f_i-f_j is holomorphic (wherever the difference is defined).\nThe first Cousin problem then asks for a meromorphic function f on M such that f-f_i is holomorphic on U_i; in other words, that f shares the [singular](https://bliptext.com/articles/mathematical-singularity) behaviour of the given local function.\nNow, let **K** be the sheaf of meromorphic functions and **O** the sheaf of holomorphic functions on M.\nThe first Cousin problem can always be solved if the following map is surjective:\nBy the [long exact cohomology sequence](https://bliptext.com/articles/long-exact-sequence-in-homology),\nis exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,**O**) vanishes.\nIn particular, by [Cartan's theorem B](https://bliptext.com/articles/cartan-s-theorems-a-and-b), the Cousin problem is always solvable if M is a Stein manifold.\n\n### Second Cousin problem\nThe second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio f_i/f_j is a non-vanishing holomorphic function (where said difference is defined).\nIt asks for a meromorphic function f on M such that f/f_i is holomorphic and non-vanishing.\nLet be the sheaf of holomorphic functions that vanish nowhere, and the sheaf of meromorphic functions that are not identically zero.\nThese are both then sheaves of [abelian groups](https://bliptext.com/articles/abelian-group), and the quotient sheaf is well-defined.\nIf the following map \\phi is surjective, then Second Cousin problem can be solved:\nThe long exact sheaf cohomology sequence associated to the quotient is\nso the second Cousin problem is solvable in all cases provided that\nThe cohomology group for the multiplicative structure on can be compared with the cohomology group with its additive structure by taking a logarithm.\nThat is, there is an exact sequence of sheaves\nwhere the leftmost sheaf is the locally constant sheaf with fiber 2\\pi i\\Z.\nThe obstruction to defining a logarithm at the level of H1 is in H^2(M,\\Z), from the long exact cohomology sequence\nWhen M is a Stein manifold, the middle arrow is an isomorphism because for q > 0 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.)\n\n# Manifolds and analytic varieties with several complex variables\n\n## Stein manifold (non-compact Kähler manifold)\nSince a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, and satisfies the [second axiom of countability](https://bliptext.com/articles/second-axiom-of-countability), the open Riemann surface is in fact a 1-dimensional complex manifold possessing a holomorphic mapping into the complex plane \\mathbb C. (In fact, Gunning and Narasimhan have shown (1967) 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.) The [Whitney embedding theorem](https://bliptext.com/articles/whitney-embedding-theorem) tells us that every smooth n-dimensional manifold can be [embedded](https://bliptext.com/articles/embedding) as a smooth submanifold of, whereas it is \"rare\" for a complex manifold to have a holomorphic embedding into \\mathbb C^n.\nFor 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.\nThat is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants.\nSo, consider the conditions under which a complex manifold has a holomorphic function that is not a constant.\nNow if we had a holomorphic embedding of X into \\mathbb C^n, then the coordinate functions of \\mathbb C^n would restrict to nonconstant holomorphic functions on X, contradicting compactness, except in the case that X is just a point.\nComplex manifolds that can be holomorphic embedded into \\mathbb C^n are called Stein manifolds.\nAlso Stein manifolds satisfy the second axiom of countability.\nA **Stein manifold** is a complex [submanifold](https://bliptext.com/articles/submanifold) of the [vector space](https://bliptext.com/articles/vector-space) of n complex dimensions.\nThey were introduced by and named after Karl Stein (1951).\nA **Stein space** is similar to a Stein manifold but is allowed to have singularities.\nStein spaces are the analogues of [affine varieties](https://bliptext.com/articles/affine-variety) or [affine schemes](https://bliptext.com/articles/affine-scheme) in algebraic geometry.\nIf the univalent domain on \\mathbb C^n is connection to a manifold, can be regarded as a [complex manifold](https://bliptext.com/articles/complex-manifold) and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity.\nTherefore, the Stein manifold is the properties of the domain of definition of the (maximal) [analytic continuation](https://bliptext.com/articles/analytic-continuation) of an analytic function.\n\n### Definition\nSuppose X is a [paracompact](https://bliptext.com/articles/paracompact) [complex manifolds](https://bliptext.com/articles/complex-manifold) of complex dimension n and let denote the ring of holomorphic functions on X.\nWe call X a **Stein manifold** if the following conditions hold:\n X is holomorphically convex, i.e. for every compact subset K \\subset X, the so-called holomorphically convex hull,\nis also a compact subset of X. \n X is [holomorphically separable](https://bliptext.com/articles/holomorphically-separable), i.e. if x \\neq y are two points in X, then there exists such that \n The open neighborhood of every point on the manifold has a holomorphic [chart](https://bliptext.com/articles/atlas-topology) to the . \nNote that condition (3) can be derived from conditions (1) and (2).\n\n### Every non-compact (open) Riemann surface is a Stein manifold\nLet X be a connected, non-compact (open) [Riemann surface](https://bliptext.com/articles/riemann-surface).\nA [deep theorem](https://bliptext.com/articles/behnke-stein-theorem-on-stein-manifolds) of Behnke and Stein (1948) asserts that X is a Stein manifold.\nAnother result, attributed to [Hans Grauert](https://bliptext.com/articles/hans-grauert) and [Helmut Röhrl](https://bliptext.com/articles/helmut-r-hrl) (1956), states moreover that every [holomorphic vector bundle](https://bliptext.com/articles/holomorphic-vector-bundle) on X is trivial.\nIn particular, every line bundle is trivial, so.\nThe [exponential sheaf sequence](https://bliptext.com/articles/exponential-sheaf-sequence) leads to the following exact sequence:\nNow [Cartan's theorem B](https://bliptext.com/articles/cartan-s-theorems-a-and-b) shows that, therefore.\nThis is related to the solution of the [second (multiplicative) Cousin problem](https://bliptext.com/articles/cousin-problems).\n\n### Levi problems\nCartan extended Levi's problem to Stein manifolds.\nThis was proved by Bremermann by embedding it in a sufficiently high dimensional, and reducing it to the result of Oka.\nAlso, Grauert proved for arbitrary **complex** manifolds M.\nAnd Narasimhan extended Levi's problem to [complex analytic space](https://bliptext.com/articles/complex-analytic-space), a generalized in the singular case of complex manifolds.\nLevi's problem remains unresolved in the following cases;\nmore generalized\nand also,\nThis means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space.\n\n#### K-complete\nGrauert introduced the concept of K-complete in the proof of Levi's problem.\nLet X is complex manifold, X is K-complete if, to each point x_0\\in X, there exist finitely many holomorphic map of X into \\Complex^p, p = p(x_0), such that x_0 is an isolated point of the set.\nThis concept also applies to complex analytic space.\n\n### Properties and examples of Stein manifolds\nThese facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the [ambient space](https://bliptext.com/articles/ambient-space) (because the embedding is biholomorphic).\nNumerous further characterizations of such manifolds exist, in particular capturing the property of their having \"many\" [holomorphic functions](https://bliptext.com/articles/holomorphic-function) taking values in the complex numbers.\nSee for example [Cartan's theorems A and B](https://bliptext.com/articles/cartan-s-theorems-a-and-b), relating to [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology).\nIn the [GAGA](https://bliptext.com/articles/gaga) set of analogies, Stein manifolds correspond to [affine varieties](https://bliptext.com/articles/affine-variety).\nStein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit \"many\" holomorphic functions from the complex numbers into themselves.\nIt is known that a Stein manifold is elliptic if and only if it is [fibrant](https://bliptext.com/articles/fibrant-object) in the sense of so-called \"holomorphic homotopy theory\".\n\n## Complex projective varieties (compact complex manifold)\nMeromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the [Riemann-Roch theorem](https://bliptext.com/articles/riemann-roch-theorem) ([Riemann's inequality](https://bliptext.com/articles/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](https://bliptext.com/articles/algebraic-curve) over \\mathbb{C} ).\nIn fact, compact Riemann surface had a non-constant single-valued meromorphic function, and also a compact Riemann surface had enough meromorphic functions.\nA compact one-dimensional complex manifold was a Riemann sphere.\nHowever, the abstract notion of a compact Riemann surface is always algebraizable (The [Riemann's existence theorem](https://bliptext.com/articles/riemann-s-existence-theorem), [Kodaira embedding theorem](https://bliptext.com/articles/kodaira-embedding-theorem).), but it is not easy to verify which compact complex analytic spaces are algebraizable.\nIn fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.\nHowever, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic.\nThe generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira, Kodaira also extended the theorem to three-dimensional, and n-dimensional Kähler varieties.\nSerre formulated the Riemann–Roch theorem as a problem of dimension of [coherent sheaf cohomology](https://bliptext.com/articles/coherent-sheaf-cohomology), and also Serre proved [Serre duality](https://bliptext.com/articles/serre-duality).\nCartan and Serre proved the following property: the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M. Riemann–Roch on a Riemann surface for a vector bundle was proved by [Weil](https://bliptext.com/articles/vector-bundles-on-algebraic-curves) in 1938.\n[Hirzebruch](https://bliptext.com/articles/hirzebruch-riemann-roch-theorem) generalized the theorem to compact complex manifolds in 1994 and [Grothendieck](https://bliptext.com/articles/grothendieck-hirzebruch-riemann-roch-theorem) generalized it to a relative version (relative statements about [morphisms](https://bliptext.com/articles/morphism).).\nNext, the generalization of the result that \"the compact Riemann surfaces are projective\" to the high-dimension.\nIn particular, consider the conditions that when embedding of compact complex submanifold X into the complex projective space.\nThe [vanishing theorem](https://bliptext.com/articles/vanishing-theorem) (was first introduced by [Kodaira](https://bliptext.com/articles/kodaira-vanishing-theorem) in 1953) gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of [positivity](https://bliptext.com/articles/positive-form).\nAs an application of this theorem, the [Kodaira embedding theorem](https://bliptext.com/articles/kodaira-embedding-theorem) says that a compact [Kähler manifold](https://bliptext.com/articles/k-hler-manifold) M, with a Hodge metric, there is a complex-analytic embedding of M into [complex projective space](https://bliptext.com/articles/complex-projective-space) of enough high-dimension N.\nIn addition the [Chow's theorem](https://bliptext.com/articles/gaga) 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](https://bliptext.com/articles/algebraic-geometry-and-analytic-geometry).\nThe complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties.\nThen combined with Kodaira's result, a compact Kähler manifold M embeds as an algebraic variety.\nThis result gives an example of a complex manifold with enough meromorphic functions.\nBroadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties.\nThe combination of analytic and algebraic methods for complex projective varieties lead to areas such as [Hodge theory](https://bliptext.com/articles/hodge-theory).\nAlso, the [deformation theory](https://bliptext.com/articles/deformation-theory) of compact complex manifolds has developed as Kodaira–Spencer theory.\nHowever, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic.\nAnalogy of the Levi problems on the complex projective space by Takeuchi.\n\n# Annotation\n\n# Inline citations\n\n## Textbooks\n\n## Encyclopedia of Mathematics","revisionCount":0},"hasMore":false,"nextCursor":null}

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