View source for Function of several complex variables

{{Short description|Type of mathematical functions}}
{{Use American English|date = March 2019}}

The theory of '''functions of several complex variables''' is the branch of [[mathematics]] dealing with functions defined on [[#The complex coordinate space|the complex coordinate space]] <math>\mathbb C^n</math>, that is, {{mvar|n}}-tuples of [[complex number]]s. The name of the field dealing with the properties of these functions is called '''several complex variables''' (and [[analytic space]]), which the [[Mathematics Subject Classification]] has as a top-level heading.

As in [[complex analysis|complex analysis of functions of one variable]], which is the case {{math|1=''n'' = 1}}, the functions studied are ''[[holomorphic function|holomorphic]]'' or ''complex analytic'' so that, locally, they are [[power series]] in the variables {{mvar|z<sub>i</sub>}}. Equivalently, they are locally [[uniform convergence|uniform limits]] of [[polynomial]]s; or locally [[Lp space|square-integrable]] solutions to the {{mvar|n}}-dimensional [[#Holomorphic functions|Cauchy–Riemann equations]].<ref name="Hörmander1965">{{cite journal |doi=10.1007/BF02391775|title=L<sup>2</sup> estimates and existence theorems for the <math>\bar \partial</math> operator |year=1965 |last1=Hörmander |first1=Lars |journal=Acta Mathematica |volume=113 |pages=89–152 |s2cid=120051843 |doi-access=free }}</ref><ref>{{cite book |isbn=978-1-4704-4636-9|title=Analysis of Several Complex Variables|last1=Ohsawa|first1=Takeo|year=2002|url={{Google books|IXhoWbo1oCkC|title=Analysis of Several Complex Variables|page=18-IA8|plainurl=yes}}}}</ref><ref name =Błocki2014>{{cite journal |doi=10.1007/s13373-014-0058-2|title=Cauchy–Riemann meet Monge–Ampère|year=2014|last1=Błocki|first1=Zbigniew|journal=Bulletin of Mathematical Sciences|volume=4|issue=3|pages=433–480|s2cid=53582451|doi-access=free}}</ref> For one complex variable, every [[Domain (mathematical analysis)|domain]]<ref name="domain" group="note">That is an [[open set|open]] [[connected space|connected]] [[subset]].
</ref>(<math>D \subset \mathbb C</math>), is the [[domain of holomorphy]] of some function, in other words every domain has a function for which it is the domain of holomorphy.<ref name=Siu1978>{{cite journal |doi=10.1090/S0002-9904-1978-14483-8|title=Pseudoconvexity and the problem of Levi|year=1978|last1=Siu|first1=Yum-Tong|journal=Bulletin of the American Mathematical Society|volume=84|issue=4|pages=481–513|mr=0477104|doi-access=free}}</ref><ref name="Chen2000">{{cite journal |last1=Chen |first1=So-Chin |title=Complex analysis in one and several variables |journal=Taiwanese Journal of Mathematics |year=2000 |volume=4 |issue=4 |pages=531–568 |doi=10.11650/twjm/1500407292|doi-access=free|zbl=0974.32001|jstor=43833225|mr=1799753}}</ref> For several complex variables, this is not the case; there exist domains (<math>D \subset \mathbb C^n,\ n \geq 2</math>) 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.{{R|Siu1978}} Patching the local data of [[meromorphic function]]s, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and [[Projective variety#Complex projective varieties|complex projective varieties]] (<math>\mathbb{CP}^n</math>)<ref name="IWS">{{cite journal |last1=Chong |first1=C.T. |last2=Leong |first2=Y.K. |title=An interview with Jean-Pierre Serre |journal=The Mathematical Intelligencer |year=1986 |volume=8 |issue=4 |pages=8–13|doi=10.1007/BF03026112|s2cid=121138963 }}</ref> and has a different flavour to complex analytic geometry in <math>\mathbb{C}^n</math> or on [[Stein manifold]]s, these are much similar to study of algebraic varieties that is study of the [[algebraic geometry]] than complex analytic geometry.

== Historical perspective ==

Many examples of such functions were familiar in nineteenth-century mathematics; [[abelian variety|abelian functions]], [[theta function]]s, and some [[hypergeometric series]], and also, as an example of an inverse problem; the [[Jacobi inversion problem]].<ref name=":0">{{cite book |doi=10.1007/978-3-642-20554-5_5|chapter=Analytic Functions of Several Complex Variables |title=Complex Analysis 2 |series=Universitext |year=2011 |last1=Freitag |first1=Eberhard |pages=300–346 |isbn=978-3-642-20553-8 }}</ref> Naturally also same function of one variable that depends on some complex [[parameter]] is a candidate. The theory, however, for many years didn't become a full-fledged field in [[mathematical analysis]], since its characteristic phenomena weren't uncovered. The [[Weierstrass preparation theorem]] would now be classed as [[commutative algebra]]; it did justify the local picture, [[Ramification (mathematics)|ramification]], that addresses the generalization of the [[branch point]]s of [[Riemann surface]] theory.

With work of [[Friedrich Hartogs]], {{ill|Pierre Cousin (mathematician)|lt=Pierre Cousin|fr|Pierre Cousin (mathématicien)}}, [[E. E. Levi]], and of [[Kiyoshi Oka]] in the 1930s, a general theory began to emerge; others working in the area at the time were [[Heinrich Behnke]], [[Peter Thullen]], [[Karl Stein (mathematician)|Karl Stein]], [[Wilhelm Wirtinger]] and [[Francesco Severi]]. Hartogs proved some basic results, such as every [[isolated singularity]] is [[removable singularity|removable]], for every analytic function 
<math display=block>f : \mathbb C^n \to \Complex</math>
whenever {{math|1=''n'' > 1}}. Naturally the analogues of [[Line integral|contour integrals]] will be harder to handle; when {{math|1=''n'' = 2}} an integral surrounding a point should be over a three-dimensional [[manifold]] (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a [[double integral]] over a two-dimensional surface. This means that the [[residue calculus]] will have to take a very different character.

After 1945 important work in France, in the seminar of [[Henri Cartan]], and Germany with [[Hans Grauert]] and [[Reinhold Remmert]], quickly changed the picture of the theory. A number of issues were clarified, in particular that of [[analytic continuation]]. Here a major difference is evident from the one-variable theory; while for every open connected set ''D'' in <math>\Complex</math> we can find a function that will nowhere continue analytically over the boundary, that cannot be said for {{math|''n'' > 1}}. In fact the ''D'' of that kind are rather special in nature (especially in complex coordinate spaces <math>\mathbb C^n</math> and Stein manifolds, satisfying a condition called ''[[#Pseudoconvexity|pseudoconvexity]]''). The natural domains of definition of functions, continued to the limit, are called ''[[Stein manifold]]s'' and their nature was to make [[sheaf cohomology]] groups vanish, on the other hand, the [[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.{{R|Siu1978}} In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for [[algebraic geometry]], in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied to [[Algebraic geometry#Analytic geometry|analytic geometry]], {{refn|group=note|A name adopted, confusingly, for the geometry of [[Complex analytic variety|zeroes of analytic functions]]; this is not the [[analytic geometry]] learned at school. (In other words, in the sense of GAGA on Serre.)<ref name="GAGA">{{cite journal | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free |language=fr|zbl=0075.30401}}</ref>}} [[automorphic form]]s of several variables, and [[partial differential equation]]s. The [[complex structure deformation|deformation theory of complex structures]] and [[complex manifold]]s was described in general terms by [[Kunihiko Kodaira]] and [[D. C. Spencer]]. The celebrated paper ''[[GAGA]]'' of [[Jean-Pierre Serre|Serre]]{{R|GAGA|}} pinned down the crossover point from ''géometrie analytique'' to ''géometrie algébrique''.

[[C. L. Siegel]] was heard to complain that the new ''theory of functions of several complex variables'' had few ''functions'' in it, meaning that the [[special function]] side of the theory was subordinated to sheaves. The interest for [[number theory]], certainly, is in specific generalizations of [[modular form]]s. The classical candidates are the [[Hilbert modular form]]s and [[Siegel modular form]]s. These days these are associated to [[algebraic group]]s (respectively the [[Weil restriction]] from a [[totally real number field]] of {{math|[[general linear group|GL]](2)}}, and the [[symplectic group]]), for which it happens that [[automorphic representation]]s can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

Subsequent developments included the [[hyperfunction]] theory, and the [[edge-of-the-wedge theorem]], both of which had some inspiration from [[quantum field theory]]. There are a number of other fields, such as [[Banach algebra]] theory, that draw on several complex variables.

== The complex coordinate space ==
The [[complex coordinate space]] <math>\mathbb C^n</math> is the [[Cartesian product]] of {{mvar|''n''}} copies of <math>\mathbb C</math>, and when <math>\mathbb C^n</math> is a domain of holomorphy, <math>\mathbb C^n</math> can be regarded as a [[Stein manifold]], and more generalized Stein space. <math>\mathbb C^n</math> is also considered to be a [[complex projective variety]], a [[Kähler manifold]],<ref>{{cite journal |doi=10.2977/prims/1195181825|title=Vanishing theorems on complete Kähler manifolds|year=1984|last1=Ohsawa|first1=Takeo|journal=Publications of the Research Institute for Mathematical Sciences|volume=20|pages=21–38|doi-access=free}}</ref> etc. It is also an [[dimension (vector space)|{{mvar|n}}-dimensional vector space]] over the [[complex number]]s, which gives its dimension {{math|2''n''}} over <math>\mathbb R</math>.<ref group=note>The field of complex numbers is a 2-dimensional vector space over real numbers.</ref> Hence, as a set and as a [[topological space]], <math>\mathbb C^n</math> may be identified to the [[real coordinate space]] <math>\mathbb R^{2n}</math> and its [[topological dimension]] is thus {{math|2''n''}}.

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where [[almost complex manifold|a complex structure]] is specified by a [[linear operator]] {{mvar|J}} (such that {{math|1=''J''<sup> 2</sup> = [[reflection through the origin|−''I'']]}}) which defines [[scalar multiplication|multiplication]] by the [[imaginary unit]] {{mvar|i}}.

Any such space, as a real space, is [[orientation (vector space)|oriented]]. On the [[complex plane]] thought of as a [[Cartesian plane]], [[multiplication]] by a complex number {{math|1=''w'' = ''u'' + ''iv''}} may be represented by the real [[square matrix|matrix]]
:<math>\begin{pmatrix}
u & -v \\
v & u
\end{pmatrix},</math>
with [[determinant]]
:<math>u^2 + v^2 = |w|^2.</math>

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be [[block matrix|composed from 2&nbsp;×&nbsp;2 blocks]] of the aforementioned form), then its determinant equals to the [[square (algebra)#In complex analysis|square of absolute value]] of the corresponding complex determinant. It is a non-negative number, which implies that [[real coordinate space#Orientation|the (real) orientation of the space is never reversed]] by a complex operator. The same applies to [[Jacobian matrix and determinant|Jacobians]] of [[holomorphic function]]s from <math>\mathbb C^n</math> to <math>\mathbb C^n</math>.

== Holomorphic functions ==
=== Definition ===

A function ''f'' defined on a domain <math>D\subset \mathbb{C}^n</math> and with values in <math>\mathbb{C}</math> is said to be holomorphic at a point <math>z\in D</math> if it is complex-differentiable at this point, in the sense that there exists a complex linear map <math>L:\mathbb{C}^n \to \mathbb{C}</math> such that

<math>
f(z+h) = f(z) + L(h) + o(\lVert h\rVert)
</math>

The function ''f'' is said to be holomorphic if it is holomorphic at all points of its domain of definition ''D''.

If ''f'' is holomorphic, then all the partial maps :

<math>z \mapsto f(z_1,\dots,z_{i-1},z,z_{i+1},\dots,z_n)
</math>

are holomorphic as functions of one complex variable : we say that ''f'' is holomorphic in each variable separately. Conversely, if ''f'' is holomorphic in each variable separately, then ''f'' is in fact holomorphic : this is known as [[Hartogs's theorem on separate holomorphicity|Hartog's theorem]], or as [[Osgood's lemma]] under the additional hypothesis that ''f'' is [[continuous function|continuous]].

=== Cauchy–Riemann equations ===

In one complex variable, a function <math>f:\mathbb{C}\to \mathbb{C}</math> defined on the plane is holomorphic at a point <math>p\in \mathbb{C}</math> if and only if its real part <math>u</math> and its imaginary part <math>v</math> satisfy the so-called [[Cauchy–Riemann equations|Cauchy-Riemann equations]] at <math>p</math> :
<math display="block">\frac{\partial u}{\partial x}(p) = \frac{\partial v}{\partial y}(p) \quad \text{ and } \quad\frac{\partial u}{\partial y} (p)=-\frac{\partial v}{\partial x}(p)</math>

In several variables, a function <math>f:\mathbb{C}^n\to \mathbb{C}</math> is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part <math>u</math> and the imaginary part <math>v</math> of <math>f</math> satisfiy the Cauchy Riemann equations :
<math display="block">\forall i\in \{1,\dots,n\},\quad\frac{\partial u}{\partial x_i} = \frac{\partial v}{\partial y_i} \quad \text{ and } \quad\frac{\partial u}{\partial y_i} = -\frac{\partial v}{\partial x_i}</math>

Using the formalism of [[Wirtinger derivatives]], this can be reformulated as :
<math display="block">\forall i\in \{1,\dots,n\},\quad \frac{\partial f}{\partial \overline{z_i}} = 0,</math>
or even more compactly using the formalism of [[complex differential form]]s, as :
<math display="block">\bar\partial f=0.</math>

=== Cauchy's integral formula I (Polydisc version) ===
Prove the sufficiency of two conditions (A) and (B). Let ''f'' meets the conditions of being continuous and separately homorphic on domain ''D''. Each disk has a [[rectifiable curve]] <math>\gamma</math>, <math>\gamma_\nu</math> is piecewise [[smoothness]], class <math>\mathcal{C}^1</math> Jordan closed curve. (<math>\nu=1,2,\ldots,n</math>) Let <math>D_\nu</math> be the domain surrounded by each <math>\gamma_\nu</math>. Cartesian product closure <math>\overline{D_1\times D_2\times\cdots\times D_n}</math> is <math>\overline{D_1}\times \overline{D_2}\times\cdots\times \overline{D_n} \in D </math>. Also, take the closed [[polydisc]] <math>\overline{\Delta}</math> so that it becomes <math>\overline{\Delta}\subset{D_1 \times D_2 \times \cdots \times D_n}</math>. <math>\overline{\Delta}(z,r) = \left\{\zeta=(\zeta_1, \zeta_2, \dots, \zeta_n)\in \Complex^n ; \left|\zeta_\nu - z_\nu\right| \leq r_\nu \text{ for all } \nu = 1,\dots,n\right\}</math> and let <math> \{z_\nu \}^n_{\nu=1} </math> be the center of each disk.) Using the [[Cauchy's integral formula]] of one variable repeatedly, <ref group=note>Note that this formula only holds for polydisc. See [[#Bochner–Martinelli formula (Cauchy's integral formula II)|§Bochner–Martinelli formula]] for the Cauchy's integral formula on the more general domain.</ref>

: <math>
\begin{align}
f(z_1,\ldots,z_n) & =\frac{1}{2 \pi i}\int_{\partial D_1}\frac{f(\zeta_1,z_2,\ldots,z_n)}{\zeta_{1}-z_1} \, d\zeta_1 \\[6pt]
& = \frac{1}{(2 \pi i)^{2}} \int_{\partial D_2} \, d\zeta_2\int_{\partial D_1}\frac{f(\zeta_1,\zeta_2,z_3,\ldots,z_n)}{(\zeta_1 - z_1)(\zeta_2 - z_2)} \, d\zeta_1 \\[6pt]
& = \frac{1}{(2 \pi i)^n} \int_{\partial D_n} \, d\zeta_n \cdots \int_{\partial D_2} \, d\zeta_2 \int_{\partial D_1} \frac{f(\zeta_1,\zeta_2,\ldots,\zeta_n)}{(\zeta_1-z_1)(\zeta_2-z_2)\cdots(\zeta_n - z_n)} \, d\zeta_1
\end{align}
</math>

Because <math>\partial D</math> is a rectifiable Jordanian closed curve<ref group=note>According to the Jordan curve theorem, domain ''D'' is bounded closed set, that is, each domain <math>D_\nu</math> is compact.</ref> and ''f'' is continuous, so the order of products and sums can be exchanged so the [[iterated integral]] can be calculated as a [[multiple integral]]. Therefore,

{{NumBlk|::|<math>f(z_1,\dots,z_n)=\frac{1}{(2\pi i)^n}\int_{\partial D_1}\cdots\int_{\partial D_n}\frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - z_1) \cdots (\zeta_n - z_n)} \, d\zeta_1\cdots d\zeta_n</math>|{{EquationRef|1}}}}

==== Cauchy's evaluation formula ====
Because the order of products and sums is interchangeable, from ({{EquationNote|1}}) we get

{{NumBlk|::|<math>\frac{\partial^{k_1 + \cdots + k_n}f(z_1,z_2,\ldots,z_n)}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n} } = \frac{k_1! \cdots k_n!}{(2\pi i)^n} \int_{\partial D_1} \cdots \int_{\partial D_n} \frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - z_1)^{k_1+1} \cdots (\zeta_n - z_n)^{k_n + 1} } \, d\zeta_1\cdots d\zeta_n.</math>|{{EquationRef|2}}}}

''f'' is class <math>\mathcal{C}^{\infty}</math>-function.

From (2), if ''f'' is holomorphic, on polydisc <math>\left\{ \zeta=(\zeta_1, \zeta_2, \dots, \zeta_n) \in \Complex^n ; | \zeta_\nu - z_\nu | \leq r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math> and <math>|f| \leq {M}</math>, the following evaluation equation is obtained.

: <math>\left|\frac{\partial^{k_1 + \cdots + k_n} f(\zeta_1,\zeta_2,\ldots,\zeta_n)}{{\partial z_1}^{k_1} \cdots \partial {z_n}^{k_n}} \right| \leq \frac{Mk_1! \cdots k_n!}{{r_1}^{k_1} \cdots {r_n}^{k_n}}</math>

Therefore, [[Liouville's theorem (complex analysis)|Liouville's theorem]] hold.

==== Power series expansion of holomorphic functions on polydisc ====
If function ''f'' is holomorphic, on polydisc <math>\{ z=(z_1, z_2, \dots, z_n) \in \mathbb C^n ; | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}</math>, from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

: <math>\begin{align}
& f(z)=\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n} (z_1 - a_1)^{k_1} \cdots (z_n - a_n)^{k_n}\ , \\
& c_{k_1 \cdots k_n}=\frac{1}{(2\pi i)^n}\int_{\partial D_1}\cdots\int_{\partial D_n}\frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - a_1)^{k_1 + 1} \cdots(\zeta_n - a_n)^{k_n + 1} } \, d\zeta_1\cdots d\zeta_n
\end{align} </math>

In addition, ''f'' that satisfies the following conditions is called an analytic function.

For each point <math>a=(a_1,\dots,a_n)\in D \subset \mathbb C^n</math>, <math>f(z)</math> is expressed as a power series expansion that is convergent on ''D'' :

: <math>f(z)=\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1 - a_1)^{k_1}\cdots(z_n - a_n)^{k_n}\ ,</math>

We have already explained that holomorphic functions on polydisc are analytic. Also, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.

:If a sequence of functions <math>f_1,\ldots,f_n</math> which converges uniformly on compacta inside a domain ''D'', the limit function ''f'' of <math>f_v</math> also uniformly on compacta inside a domain ''D''. Also, respective partial derivative of <math>f_v</math> also compactly converges on domain ''D'' to the corresponding derivative of ''f''.

:<math>\frac{\partial^{k_1 + \cdots + k_n}f}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n}} = \sum_{v=1}^\infty \frac{\partial^{k_1 + \cdots + k_n} f_v}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n}} </math><ref>{{Eom| title = Weierstrass theorem | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 49192}}</ref>

==== Radius of convergence of power series ====
It is possible to define a combination of positive real numbers <math>\{r_\nu \ (\nu = 1,\dots,n) \}</math> such that the power series <math display="inline">\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}\ </math> converges uniformly at <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math> and does not converge uniformly at <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math>.

In this way it is possible to have a similar, combination of radius of convergence<ref group=note>But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge.</ref> for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

==== Laurent series expansion ====
Let <math>\omega(z)</math> be holomorphic in the [[Annulus (mathematics)|annulus]] <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; r_\nu < |z| <R_\nu, \text{ for all } \nu + 1,\dots, n\right\}</math> and continuous on their circumference, then there exists the following expansion ;

: <math>
\begin{align}\omega(z) & = \sum_{k=0}^{\infty}\frac{1}{k!}\frac{1}{(2\pi i)^n} \int_{|\zeta_\nu|=R_\nu}\cdots\int\omega(\zeta)\times\left[\frac{d^k}{dz^k}\frac{1}{\zeta-z}\right]_{z=0}df_{\zeta}\cdot z^k \\[6pt]
&+\sum_{k=1}^\infty \frac{1}{k!}\frac{1}{2\pi i}\int_{|\zeta_\nu| = r_\nu}\cdots\int\omega(\zeta) \times \left(0,\cdots,\sqrt{\frac{k!}{\alpha_{1}!\cdots\alpha_{n}!}}\cdot\zeta_{n}^{\alpha_1-1}\cdots\zeta_{n}^{\alpha_n-1},\cdots 0\right)df_{\zeta}\cdot\frac{1}{z^k}\ (\alpha_1 + \cdots + \alpha_n = k)
\end{align}
</math>

The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus <math>r'_\nu < |z| < R'_\nu</math>, where <math>r'_\nu > r_\nu</math> and <math>R'_\nu < R_\nu</math>, and so it is possible to integrate term.<ref>{{cite journal | last1=Ozaki | first1=Shigeo | last2=Onô | first2=Isao | title=Analytic Functions of Several Complex Variables | journal=Science Reports of the Tokyo Bunrika Daigaku, Section A | volume=4 | issue=98/103 | date=February 1, 1953 | pages=262–270|jstor=43700400}}</ref>

=== Bochner–Martinelli formula (Cauchy's integral formula II) ===

The 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]].

Suppose that ''f'' is a continuously differentiable function on the closure of a domain ''D'' on <math>\mathbb C^n</math> with piecewise smooth boundary <math>\partial D</math>, and let the symbol <math>\land</math> denotes the exterior or [[wedge product]] of differential forms. Then the Bochner–Martinelli formula states that if ''z'' is in the domain ''D'' then, for <math>\zeta</math>, ''z'' in <math>\mathbb C^n</math> the Bochner–Martinelli kernel <math>\omega(\zeta,z)</math> is a [[differential form]] in <math>\zeta</math> of bidegree <math>(n,n-1)</math>, defined by
:<math>\omega(\zeta,z) = \frac{(n-1)!}{(2\pi i)^n}\frac{1}{|z-\zeta|^{2n}}
\sum_{1\le j\le n}(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \land d\zeta_1 \land \cdots \land d\zeta_j \land \cdots \land d\overline\zeta_n \land d\zeta_n</math>
:<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\land\omega(\zeta,z).</math>

In particular if ''f'' is holomorphic the second term vanishes, so 
:<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z). </math>

=== Identity theorem ===
Holomorphic functions of several complex variables satisfy an [[identity theorem]], as in one variable : two holomorphic functions defined on the same connected open set <math>D\subset \mathbb{C}^n</math> and which coincide on an open subset ''N'' of ''D'', are equal on the whole open set ''D''. This 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. Contrary 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 <math>f(z_1,z_2)=0</math> and <math>g(z_1,z_2)=z_1 </math>coincide on the whole complex line of <math>\mathbb{C}^2</math> defined by the equation <math>z_1=0</math>.

The [[maximal principle]], [[inverse function theorem]], and implicit function theorems also hold. For a generalized version of the implicit function theorem to complex variables, see the [[Weierstrass preparation theorem]].

=== Biholomorphism ===
From the establishment of the inverse function theorem, the following mapping can be defined.

For the domain ''U'', ''V'' of the ''n''-dimensional complex space <math>\Complex^n</math>, the bijective holomorphic function <math>\phi:U\to V</math> and the inverse mapping <math>\phi^{-1}:V\to U</math> is also holomorphic. At this time, <math>\phi</math> is called a ''U'', ''V'' biholomorphism also, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorphic.

==== The Riemann mapping theorem does not hold ====
When <math>n > 1</math>, open balls and open polydiscs are ''not'' biholomorphically equivalent, that is, there is no [[biholomorphic mapping]] between the two.<ref name=FieldCM1982>{{cite book |doi=10.1017/CBO9781107325562.005|chapter=Complex Manifolds |title=Several Complex Variables and Complex Manifolds I |year=1982 |pages=134–186 |isbn=9780521283014|last1=Field|first1=M}}</ref> This was proven by [[Henri Poincaré|Poincaré]] in 1907 by showing that their [[automorphism group]]s have different dimensions as [[Lie group]]s.{{R|Chen2000}}<ref>{{cite journal |last1=Poincare |first1=M. Henri |title=Les fonctions analytiques de deux variables et la représentation conforme |journal=Rendiconti del Circolo Matematico di Palermo |date=1907 |volume=23 |pages=185–220 |doi=10.1007/BF03013518|doi-access=free|s2cid=123480258 |url=https://zenodo.org/record/2037590 }}</ref> However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable.<ref>{{cite book |last1=Siu |first1=Yum-Tong |editor1-last=Wu |editor1-first=Hung-Hsi |title=Contemporary Geometry |isbn=978-1-4684-7950-8 |url={{Google books|title=Contemporary Geometry|u53VBwAAQBAJ|page=95|plainurl=yes}}|page=494|doi=10.1007/978-1-4684-7950-8|chapter=Uniformization in Several Complex Variables|year=1991 |chapter-url=https://doi.org/10.1007/978-1-4684-7950-8_5}}</ref>

=== Analytic continuation ===
Let ''U, V'' be domain on <math>\mathbb{C}^n</math>, such that <math>f \in \mathcal{O}(U)</math> and <math>g \in \mathcal{O}(V)</math>, (<math>\mathcal{O}(U)</math> is the set/ring of holomorphic functions on ''U''.) assume that <math>U,\ V,\ U \cap V \ne \varnothing</math> and <math>W</math> is a [[connected component (topology)|connected component]] of <math>U \cap V</math>. If <math>f|_W =g|_W</math> then ''f'' is said to be connected to ''V'', and ''g'' is said to be analytic continuation of ''f''. From the identity theorem, if ''g'' exists, for each way of choosing ''W'' it is unique. When n > 2, the following phenomenon occurs depending on the shape of the boundary <math>\partial U</math>: there exists domain ''U'', ''V'', such that all holomorphic functions <math>f</math> over the domain ''U'', have an analytic continuation <math>g \in \mathcal{O}(V)</math>. In other words, there may be not exist a function <math>f \in \mathcal{O}(U)</math> such that <math>\partial U</math> as the natural boundary. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. In addition, when <math>n  \geq 2</math>, it would be that the above ''V'' has an intersection part with ''U'' other than ''W''. This contributed to advancement of the notion of sheaf cohomology.

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

Let <math>D \subset \Complex^n</math> (<math> n \geq 1</math>) to be a domain, with centre at a point <math>a = (a_1,\dots,a_n) \in \Complex^n</math>, such that, together with each point <math>z^0 = (z_1^0,\dots,z_n^0)\in D</math>, the domain also contains the set

: <math> \left\{ z = (z_1, \dots, z_n) ; \left|z_\nu - a_\nu \right| = \left|z_\nu^0 - a_\nu\right|,\ \nu = 1, \dots, n \right\} .
</math>

A domain ''D'' is called a Reinhardt domain if it satisfies the following conditions:<ref>{{cite book |doi=10.4171/049 |title=First Steps in Several Complex Variables: Reinhardt Domains |date=2008 |last1=Jarnicki |first1=Marek |last2=Pflug |first2=Peter |isbn=978-3-03719-049-4|url={{Google books|TZuc66RB-rQC|keywords=reinhardt domains|plainurl=yes}}}}</ref><ref>{{cite journal |doi=10.1017/S0027763000013465|title=Meromorphic or Holomorphic Completion of a Reinhardt Domain|year=1970|last1=Sakai|first1=Eiichi|journal=Nagoya Mathematical Journal|volume=38|pages=1–12|s2cid=118248529 |doi-access=free}}</ref>

Let <math>\theta_\nu \;(\nu = 1,\dots,n)</math> is a arbitrary real numbers, a domain ''D'' is invariant under the rotation: <math>\left\{z^0 - a_\nu \right\} \to \left\{e^{i\theta_\nu} (z_\nu^0 - a_\nu) \right\}</math>.

The Reinhardt domains which are defined by the following condition; Together with all points of <math>z^0 \in D</math>, the domain contains the set

: <math> \left\{ z = ( z_1, \dots, z_n ) ; z = a + \left(z^0 - a\right) e^{i \theta} ,\ 0 \leq \theta < 2 \pi \right\}.</math>

A Reinhardt domain ''D'' is called a complete Reinhardt domain with centre at a point ''a'' if together with all point <math>z^0\in D</math> it also contains the polydisc

: <math>
\left\{ z = ( z_1, \dots, z_n) ; \left|z_\nu - a_\nu \right| \leq \left|z_\nu^0 - a_\nu \right| , \ \nu = 1, \dots, n \right\}.
</math>

A complete Reinhardt domain ''D'' is [[Star-like domain|star-like]] with regard to its centre ''a''. Therefore, the complete Reinhardt domain is [[simply connected]], also when the complete Reinhardt domain is the boundary line, there is a way to prove the [[Cauchy's integral theorem]] without using the [[Jordan curve theorem]].

=== Logarithmically-convex ===
When 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.

A Reinhardt domain ''D'' is called [[logarithmically convex]] if the image <math>\lambda(D^{*})</math> of the set

: <math> 
D ^{*} = \{ z = (z_1, \dots, z_n) \in D ; z_1, \dots, z_n \neq 0 \}
</math>

under the mapping

: <math> 
\lambda ; z \rightarrow \lambda(z) = (\ln|z_1|, \dots, \ln |z_n|)
</math>

is a [[convex set]] in the real coordinate space <math>\R^n</math>.

Every such domain in <math>\Complex^n</math> is the interior of the set of points of absolute convergence of some power series in <math display="inline">\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1 - a_1)^{k_1}\cdots(z_n - a_n)^{k_n}\ </math>, and conversely; The domain of convergence of every power series in <math>z_1,\dots,z_n</math> is a logarithmically-convex Reinhardt domain with centre <math>a = 0</math>. 
<ref group=note>When described using the [[#Domain of holomorphy|domain of holomorphy]], which is a generalization of the convergence domain, a Reinhardt domain is a domain of holomorphy if and only if logarithmically convex.</ref> But, there is an example of a complete Reinhardt domain D which is not logarithmically convex.<ref>{{cite book |page=10.1007/978-1-4757-1918-5_2|doi=10.1007/978-1-4757-1918-5_2 |chapter=Domains of Holomorphy and Pseudoconvexity |title=Holomorphic Functions and Integral Representations in Several Complex Variables |series=Graduate Texts in Mathematics |date=1986 |last1=Range |first1=R. Michael |volume=108 |isbn=978-1-4419-3078-1 |url={{Google books|mv_pBwAAQBAJ|page=80|plainurl=yes}}}}</ref>

=== Some results ===

==== Hartogs's extension theorem and Hartogs's phenomenon ====
When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the <math>\mathbb{C}^n</math> were all connected to larger domain.<ref>{{cite journal |doi=10.1080/17476930701747716 |title=The Hartogs extension phenomenon redux |year=2008 |last1=Krantz |first1=Steven G. |journal=Complex Variables and Elliptic Equations |volume=53 |issue=4 |pages=343–353 |s2cid=121700550 }}</ref>

:On the polydisk consisting of two disks <math>\Delta^2=\{z\in\Complex^2;|z_1|<1,|z_2|<1\}</math> when <math>0 <\varepsilon < 1</math>.

:Internal domain of <math>H_\varepsilon = \{z=(z_1,z_2)\in\Delta^2;|z_1|<\varepsilon\ \cup \ 1-\varepsilon< |z_2|\}\ (0 <\varepsilon < 1) </math>

::Hartogs's extension theorem (1906);<ref name=Hartogs1906>{{Citation|last = Hartogs|first = Fritz|author-link = Friedrich Hartogs|title = Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.| journal = Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse| language = de| volume = 36| pages = 223–242| year = 1906| url = https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up| jfm = 37.0443.01
}}</ref> Let ''f'' be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G''&nbsp;\&nbsp;''K''}}, where {{mvar|G}} is a bounded (surrounded by a rectifiable closed Jordan curve) domain{{refn|group=note|1=This theorem holds even if the condition is not restricted to the bounded. i.e. The theorem holds even if this condition is replaced with an open set.<ref name=Simonič2016>{{cite arXiv|eprint=1608.00950|last1=Simonič|first1=Aleksander|title=Elementary approach to the Hartogs extension theorem|year=2016|class=math.CV }}</ref>}} on <math>\Complex^n</math> ({{math|''n'' ≥ 2}}) and ''K'' is a compact subset of ''G''. If the [[Complement (set theory)|complement]] {{math|''G''&nbsp;\&nbsp;''K''}} is connected, then every holomorphic function ''f'' regardless of how it is chosen can be each extended to a unique holomorphic function on ''G''.<ref>{{cite journal |last1=Laufer |first1=Henry B. |title=Some remarks about a theorem of Hartogs |journal=[[Proceedings of the American Mathematical Society]] |date=1 June 1966 |volume=17 |issue=6 |pages=1244–1249 |doi=10.1090/S0002-9939-1966-0201675-2 |doi-access=free |jstor=2035718}}</ref>{{R|Simonič2016}}
:It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for holomorphic functions of several complex variables. The nature of these singularities is also derived from [[Weierstrass preparation theorem]]. A generalization of this theorem using the same method as Hartogs was proved in 2007.<ref>{{cite journal |doi=10.1007/BF02922095|doi-access=free|title=A Morse-theoretical proof of the Hartogs extension theorem|year=2007|last1=Merker|first1=Joël|last2=Porten|first2=Egmont|journal=Journal of Geometric Analysis|volume=17|issue=3|pages=513–546|s2cid=449210|arxiv=math/0610985}}</ref><ref>{{cite journal |doi=10.1016/j.jmaa.2013.01.049|title=Hartogs extension for generalized tubes in Cn|year=2013|last1=Boggess|first1=A.|last2=Dwilewicz|first2=R.J.|last3=Slodkowski|first3=Z.|journal=Journal of Mathematical Analysis and Applications|volume=402|issue=2|pages=574–578|doi-access=free}}</ref>

From Hartogs's extension theorem the domain of convergence extends from <math>H_\varepsilon</math> to <math>\Delta^2</math>. Looking at this from the perspective of the Reinhardt domain, <math>H_\varepsilon</math> is the Reinhardt domain containing the center z = 0, and the domain of convergence of <math>H_\varepsilon</math> has been extended to the smallest complete Reinhardt domain <math>\Delta^2</math> containing <math>H_\varepsilon</math>.<ref>{{Cite journal|first=Henri|last=Cartan|title= Les fonctions de deux variables complexes et le problème de la représentation analytique|journal=[[Journal de Mathématiques Pures et Appliquées]]|volume=10|year=1931|pages=1–116|zbl =0001.28501}}</ref>

==== Thullen's classic results ====

[[Peter Thullen|Thullen]]'s<ref>{{cite journal |doi=10.1007/bf01457933|doi-access=free|title=Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veränderlichen die Invarianz des Mittelpunktes von Kreiskörpern|year=1931|last1=Thullen|first1=Peter|journal=Mathematische Annalen|volume=104|pages=244–259|s2cid=121072397}}</ref> classical result says that a 2-dimensional bounded Reinhard domain containing the origin is [[biholomorphic]] to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

# <math>\{(z,w)\in \Complex^2;~|z| < 1,~|w| < 1\}</math> (polydisc);
# <math>\{(z,w)\in \Complex^2;~|z|^2 + |w|^2 < 1\}</math> (unit ball);
# <math>\{(z,w)\in \Complex^2;~|z|^2 + |w|^{\frac{2}{p}} < 1\}\, (p > 0,\neq 1)</math> (Thullen domain).

==== Sunada's results ====
[[Toshikazu Sunada]] (1978)<ref>{{cite journal |doi=10.1007/BF01405009|doi-access=free|title=Holomorphic equivalence problem for bounded Reinhardt domains|year=1978|last1=Sunada|first1=Toshikazu|journal=Mathematische Annalen|volume=235|issue=2|pages=111–128|s2cid=124324696}}</ref> established a generalization of Thullen's result:
:Two ''n''-dimensional bounded Reinhardt domains <math>G_1</math> and <math>G_2</math> are mutually biholomorphic if and only if there exists a transformation <math>\varphi:\Complex^n\to \Complex^n</math> given by <math>z_i\mapsto r_iz_{\sigma(i)} (r_i>0)</math>, <math>\sigma</math> being a permutation of the indices), such that <math>\varphi(G_1)=G_2</math>.

== Natural domain of the holomorphic function (domain of holomorphy) ==
When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of the domain are natural boundaries (In the complex coordinate space <math>\Complex^n</math> call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of ''H''. Cartan and Thullen.<ref>{{cite journal | last1=Cartan | first1=Henri | last2=Thullen | first2=Peter | date = 1932 | title =Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen Regularitäts-und Konvergenzbereiche| journal = Mathematische Annalen | volume = 106 | pages = 617–647 | doi =10.1007/BF01455905 | doi-access=free}}</ref> Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for <math>\Complex^2</math>,<ref name="Oka'sVI">{{Citation | last1=Oka | first1=Kiyoshi | title=Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes | year=1943 | journal=[[Tohoku Mathematical Journal]] | series=First Series | issn=0040-8735 | volume=49 | pages=15–52 |zbl = 0060.24006 | url=https://www.jstage.jst.go.jp/article/tmj1911/49/0/49_0_15/_article/-char/en}}</ref> later extended to <math>\Complex^n</math>.<ref name="Oka'sIX">{{Citation | last1=Oka | first1=Kiyoshi | title=Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur | year=1953 | journal= Japanese Journal of Mathematics: Transactions and Abstracts| issn=0075-3432 | volume=23 | pages=97–155|doi=10.4099/jjm1924.23.0_97| doi-access=free }}</ref><ref>{{Citation | author = Hans J. Bremermann | year = 1954 | title =Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum vonn komplexen Veränderlichen| journal = Mathematische Annalen | volume = 106 | pages = 63–91 | doi =10.1007/BF01360125 | doi-access = free | s2cid = 119837287 }}</ref>)<ref name="Huckleberry2013" >{{cite journal |last1=Huckleberry |first1=Alan |title=Hans Grauert (1930–2011) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |year=2013 |volume=115 |pages=21–45 |doi=10.1365/s13291-013-0061-7|arxiv=1303.6933|s2cid=119685542 }}</ref> [[Kiyoshi Oka]]'s{{refn|name=Oka'sVII|1={{cite journal |doi=10.24033/bsmf.1408|title=Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques|year=1950|last1=Oka|first1=Kiyoshi|journal=Bulletin de la Société Mathématique de France |volume=2|pages=1–27|doi-access=free}}, {{cite journal |title=Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques|year=1961|last1=Oka|first1=Kiyoshi|journal=Iwanami Shoten, Tokyo (Oka's Original Version)|url=http://www.nara-wu.ac.jp/aic/gdb/nwugdb/oka/ko_ron/pdf/ko-f77.pdf}}{{refn|group=note|1=Oka says that<ref>{{cite web |last1=Oka |first1=Kiyoshi |title=Sur les formes objectives et les contenus subjectifs dans les sciences math'ematiques; Propos post'erieur |url=https://www.ms.u-tokyo.ac.jp/~noguchi/oka/oka-propos-posterieurs-v6.pdf|editor-last1=Merker|editor-first1=j.|editor-last2=Noguchi|editor-first2=j.|year=1953}}</ref> the contents of these two papers are different.<ref>{{cite web |last1=Noguchi |first1=J.|url=https://www.ms.u-tokyo.ac.jp/~noguchi/oka/|title=Related to Works of Dr. Kiyoshi OKA}}</ref>}}}}<ref name="Oka'sVIII">{{Citation|last1=Oka | first1=Kiyoshi |title = Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental|journal=Journal of the Mathematical Society of Japan |volume=3|issue=1| year=1951|pages=204–214|doi = 10.2969/jmsj/00310204|doi-access=free}}, {{Citation| last1=Oka | first1=Kiyoshi |title = Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental (Suite)|journal=Journal of the Mathematical Society of Japan|issue=2|year=1951| volume=3 |pages=259–278|doi = 10.2969/jmsj/00320259|doi-access=free}}</ref> notion of [[#Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))|''idéal de domaines indéterminés'']] is interpreted theory of [[sheaf cohomology]] by 
''H''. Cartan and more development Serre.<ref group="note">The idea of the [[Sheaf (mathematics)|sheaf]] itself is by [[Jean Leray]].</ref><ref name=Cartan1950>{{cite journal |doi=10.24033/bsmf.1409|title=Idéaux et modules de fonctions analytiques de variables complexes|year=1950|last1=Cartan|first1=Henri|journal=Bulletin de la Société Mathématique de France|volume=2|pages=29–64|doi-access=free}}</ref><ref name = Cartan1953>{{cite journal |last1=Cartan |first1=Henri |title=Variétés analytiques complexes et cohomologie |journal=Colloque sur les fonctions de plusieurs variables, Bruxelles<!-- --> |year=1953 |pages=41–55|zbl=0053.05301|mr=64154}}</ref><ref name="CT3">{{cite journal|website=numdam.org |last1=Cartan |first1=H. |last2=Eilenberg |first2=Samuel |last3=Serre |first3=J-P. |title=Séminaire Henri Cartan, Tome 3 (1950-1951) |url=http://www.numdam.org/volume/SHC_1950-1951__3/}}</ref><ref name=Chorlay2010>{{cite journal |last1=Chorlay |first1=Renaud |title=From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept |journal=Archive for History of Exact Sciences |date=January 2010 |volume=64 |issue=1 |pages=1–73 |doi=10.1007/s00407-009-0052-3|jstor=41342411|s2cid=73633995 }}</ref><ref>{{cite book |doi=10.1007/978-3-662-02661-8|title=Sheaves on Manifolds|series=Grundlehren der mathematischen Wissenschaften|year=1990|volume=136|isbn=978-3-642-08082-1|url={{Google books|EyzqCAAAQBAJ|Sheaves on Manifolds|page=12|plainurl=yes|chapterA Short History: Les débuts de la théorie des faisceaux|chapter-url=https://doi.org/10.1007/978-3-662-02661-8_2}}}}</ref><ref>{{cite journal |last1=Serre |first1=Jean-Pierre |title=Quelques problèmes globaux rélatifs aux variétés de Stein |journal=Centre Belge Rech. Math., Colloque Fonctions Plusieurs Variables, Bruxelles du 11 Au 14 Mars |date=1953|pages=67–58|zbl=0053.05302|url={{Google books|eaUoAKOAbUsC|Oeuvres - Collected Papers I: 1949 - 1959|page=259|plainurl=yes}}}}</ref>{{R|IWS}} In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.<ref name="CT4">{{cite web |last1=Cartan |first1=H. |last2=Bruhat |first2=F. |last3=Cerf |first3=Jean. |last4=Dolbeault |first4=P. |last5=Frenkel |first5=Jean. |last6=Hervé |first6=Michel |last7=Malatian. |last8=Serre |first8=J-P. |title=Séminaire Henri Cartan, Tome 4 (1951-1952) |url=http://www.numdam.org/volume/SHC_1951-1952__4/ |archive-url=https://web.archive.org/web/20201020181016/http://www.numdam.org/volume/SHC_1951-1952__4/ |url-status=dead |archive-date=October 20, 2020 }}</ref> The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.{{R|Siu1978}}

=== Domain of holomorphy ===
{{Main|Domain of holomorphy}}
[[Image:Domain of holomorphy illustration.svg|thumb|right|The sets in the definition. Note: On this section, replace <math>\Omega</math> in the figure with ''D'']]
When a function ''f'' is holomorpic on the domain <math>D\subset \Complex^n</math> and cannot directly connect to the domain outside ''D'', including the point of the domain boundary <math>\partial D</math>, the domain ''D'' is called the domain of holomorphy of ''f'' and the boundary is called the natural boundary of ''f''. In other words, the domain of holomorphy ''D'' is the supremum of the domain where the holomorphic function ''f'' is holomorphic, and the domain ''D'', which is holomorphic, cannot be extended any more. For several complex variables, i.e. domain <math>D\subset \Complex^n\ (n\geq 2)</math>, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.<ref name="Forstnerič2011§SteinManifolds">{{cite book |doi=10.1007/978-3-642-22250-4|title=Stein Manifolds and Holomorphic Mappings |year=2011 |last1=Forstnerič |first1=Franc |isbn=978-3-642-22249-8 |chapter=Stein Manifolds|series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |volume=56 |chapter-url={{Google books|5ibv2wzZ2IQC|An Introduction to Complex Analysis in Several Variables|page=43|plainurl=yes}}}}</ref>

Formally, a domain ''D'' in the ''n''-dimensional complex coordinate space <math>\Complex^n</math> is called a ''domain of holomorphy'' if there do not exist non-empty domain <math>U \subset D</math> and <math>V \subset \Complex^n</math>, <math>V \not\subset D</math> and <math>U \subset D \cap V</math> such that for every holomorphic function ''f'' on ''D'' there exists a holomorphic function ''g'' on ''V'' with <math>f = g</math> on ''U''.

For the <math>n=1</math> case, the every domain (<math>D\subset\mathbb{C}</math>) was the domain of holomorphy; we can define a holomorphic function with zeros [[accumulation point|accumulating]] everywhere on the [[boundary (topology)|boundary]] of the domain, which must then be a [[analytic continuation#Natural boundary|natural boundary]] for a domain of definition of its reciprocal.

==== Properties of the domain of holomorphy ====
* If <math>D_1, \dots, D_n</math> are domains of holomorphy, then their intersection <math display="inline">D = \bigcap_{\nu=1}^n D_\nu</math> is also a domain of holomorphy.
* If <math>D_1 \subseteq D_2 \subseteq \cdots</math> is an increasing sequence of domains of holomorphy, then their union <math display="inline">D = \bigcup_{n=1}^\infty D_n</math> is also a domain of holomorphy (see [[Behnke–Stein theorem]]).<ref>{{cite journal |last1=Behnke |first1=H. |last2=Stein |first2=K. |title=Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität |journal=Mathematische Annalen |year=1939 |volume=116 |pages=204–216 |doi=10.1007/BF01597355|s2cid=123982856 }}</ref>
* If <math>D_1</math> and <math>D_2</math> are domains of holomorphy, then <math> D_1 \times D_2</math> is a domain of holomorphy.
* The first [[Cousin problems|Cousin problem]] is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for <math>n\geq 3</math>.<ref>{{cite journal |last1=Kajiwara |first1=Joji |title=Relations between domains of holomorphy and multiple Cousin's problems |journal=Kodai Mathematical Journal |date=1 January 1965 |volume=17 |issue=4 |doi=10.2996/kmj/1138845123|doi-access=free }}</ref> this is also true, with additional topological assumptions, for the second Cousin problem.

=== Holomorphically convex hull ===
Let <math>G \subset \Complex^n</math> be a domain, or alternatively for a more general definition, let <math>G</math> be an <math>n</math> dimensional [[complex analytic manifold]]. Further let <math>{\mathcal{O}}(G)</math> stand for the set of holomorphic functions on ''G''. For a compact set <math>K \subset G</math>, the '''holomorphically convex hull''' of ''K'' is

:<math> \hat{K}_G := \left \{ z \in G ; |f(z)| \leq \sup_{w \in K} |f(w)| \text{ for all } f \in \mathcal{O}(G) . \right \} .</math>

One obtains a narrower concept of '''polynomially convex hull''' by taking <math>\mathcal O(G)</math> instead to be the set of complex-valued polynomial functions on ''G''. The polynomially convex hull contains the holomorphically convex hull.

The domain <math>G</math> is called '''holomorphically convex''' if for every compact subset <math>K, \hat{K}_G</math> is also compact in ''G''. Sometimes this is just abbreviated as ''holomorph-convex''.

When <math>n=1</math>, every domain <math>G</math> is holomorphically convex since then <math>\hat{K}_G</math> is the union of ''K'' with the relatively compact components of <math>G \setminus K \subset G</math>.

When <math>n\geq 1</math>, if ''f'' satisfies the above holomorphic convexity on ''D'' it has the following properties. <math>\text{dist} (K, D^c) = \text{dist} (\hat{K}_D, D^c
)</math> for every compact subset ''K'' in ''D'', where
<math>\text{dist} (K, D^c)</math> denotes the distance between K and <math>D^c = \mathbb{C}^n \setminus D</math>. Also, at this time, D is a domain of holomorphy. Therefore, every convex domain <math>(D\subset\Complex^n)</math> is domain of holomorphy.{{R|Chen2000}}

=== Pseudoconvexity ===
Hartogs showed that

{{blockquote|Hartogs (1906):{{R|Hartogs1906}} Let ''D'' be a Hartogs's domain on <math>\mathbb{C}</math> and ''R'' be a positive function on ''D'' such that the set <math>\Omega</math> in <math>\mathbb{C}^2</math> defined by <math>z_1 \in D</math> and <math>|z_2| < R (z_1)</math> is a domain of holomorphy. Then <math>-\log {R} (z_1)</math> is a subharmonic function on ''D''.{{R|Siu1978}}}}

If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.{{refn|group=note|1=In fact, this was proved by Kiyoshi Oka{{R|Oka'sVI|}} with respect to <math>\Complex^n</math> domain.See [[Oka's lemma]]. }} The [[subharmonic function]] looks like a kind of [[convex function]], so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.<ref>{{cite journal |doi=10.1090/noti798|title=WHAT IS...a Pseudoconvex Domain?|year=2012|last1=Range|first1=R. Michael|journal=Notices of the American Mathematical Society|volume=59|issue=2|page=1|doi-access=free}}</ref>

==== Definition of plurisubharmonic function ====
:A function
:<math>f \colon D \to {\mathbb{R}}\cup\{-\infty\},</math>
:with ''domain'' <math>D \subset {\mathbb{C}}^n</math>
is called '''plurisubharmonic''' if it is [[semi-continuous function|upper semi-continuous]], and for every complex line

:<math>\{ a + b z ; z \in \mathbb{C} \}\subset \mathbb{C}^n</math> with <math>a, b \in \mathbb{C}^n</math>

:the function <math>z \mapsto f(a + bz)</math> is a subharmonic function on the set

:<math>\{ z \in \mathbb{C} ; a + b z \in D \}.</math>

:In ''full generality'', the notion can be defined on an arbitrary complex manifold or even a Complex analytic space <math>X</math> as follows. An [[semi-continuity|upper semi-continuous function]]
:<math>f \colon X \to \mathbb{R} \cup \{ - \infty \}</math>
:is said to be plurisubharmonic if and only if for any [[holomorphic map]]
<math>\varphi\colon\Delta \to X</math> the function
:<math>f\circ\varphi \colon \Delta \to \mathbb{R} \cup \{ -\infty \}</math>
is subharmonic, where <math>\Delta \subset \mathbb{C}</math> denotes the unit disk.

In one-complex variable, necessary and sufficient condition that the real-valued function <math>u=u(z)</math>, that can be second-order differentiable with respect to ''z'' of one-variable complex function is subharmonic is <math>\Delta=4\left(\frac{\partial^2 u}{\partial z\,\partial\overline{z}}\right)\geq0</math>. Therefore, if <math>u</math> is of class <math>\mathcal{C}^2</math>, then <math>u</math> is plurisubharmonic if and only if the [[hermitian matrix]] <math>H_u=(\lambda_{ij}),\lambda_{ij}=\frac{\partial^2u}{\partial z_i\,\partial\bar z_j}</math> is positive semidefinite.

Equivalently, a <math>\mathcal{C}^2</math>-function ''u'' is plurisubharmonic if and only if <math>\sqrt{-1}\partial\bar\partial f</math> is a [[positive form|positive (1,1)-form]].<ref name="CAaDG" >[https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf Complex Analytic and Differential Geometry]</ref>{{rp|pages=39–40}}

===== Strictly plurisubharmonic function =====
When the hermitian matrix of ''u'' is positive-definite and class <math>\mathcal{C}^2</math>, we call ''u'' a strict plurisubharmonic function.

==== (Weakly) pseudoconvex (p-pseudoconvex) ====
Weak pseudoconvex is defined as : Let <math>X\subset {\mathbb{C}}^n</math> be a domain. One says that ''X'' is ''pseudoconvex'' if there exists a [[continuous function|continuous]] [[plurisuperharmonic function|plurisubharmonic function]] <math>\varphi</math> on ''X'' such that the set <math>\{ z \in X ; \varphi(z) \leq \sup x \}</math> is a [[relatively compact]] subset of ''X'' for all real numbers ''x''. <ref group="note" >This is a hullomorphically convex hull condition expressed by a plurisubharmonic function. For this reason, it is also called p-pseudoconvex or simply p-convex.</ref> i.e. there exists a smooth plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>. Often, the definition of pseudoconvex is used here and is written as; Let ''X'' be a complex ''n''-dimensional manifold. Then is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>.{{R|CAaDG|}}{{rp|page=49}}

==== Strongly (Strictly) pseudoconvex ====
Let ''X'' be a complex ''n''-dimensional manifold. '''Strongly (or Strictly) pseudoconvex''' if there exists a smooth '''strictly''' plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>, i.e., <math>H\psi</math> is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.{{R|CAaDG|}}{{rp|page=49}} Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete<ref>{{cite book |url={{Google books|I3rSBwAAQBAJ|pg=267|plainurl=yes}} | title=From Holomorphic Functions to Complex Manifolds | isbn=9781468492736 | last1=Fritzsche | first1=Klaus | last2=Grauert | first2=Hans | date=6 December 2012 | publisher=Springer }}</ref>) are often used interchangeably,<ref>{{cite book |url={{Google books|bL8jDwAAQBAJ|pg=136|plainurl=yes}}| title=Function Theory of Several Complex Variables | isbn=9780821827246 | last1=Krantz | first1=Steven George | year=2001 | publisher=American Mathematical Soc. }}</ref> see  Lempert<ref>{{cite journal |url=https://eudml.org/doc/87405 |title=La métrique de Kobayashi et la représentation des domaines sur la boule |journal=Bulletin de la Société Mathématique de France |year=1981 |volume=109 |pages=427–474 |last1=Lempert |first1=Laszlo |doi=10.24033/bsmf.1948 |doi-access=free }}</ref> for the technical difference.

==== Levi form ====

===== (Weakly) Levi(–Krzoska) pseudoconvexity =====
If <math>\mathcal{C}^2</math> boundary , it can be shown that ''D'' has a defining function; i.e., that there exists <math>\rho: \mathbb{C}^n \to \mathbb{R} </math> which is <math>\mathcal{C}^2</math> so that <math>D = \{\rho < 0 \}</math>, and <math>\partial D = \{\rho =0\}</math>. Now, ''D'' is pseudoconvex iff for every <math>p \in \partial D</math> and <math>w</math> in the complex tangent space at p, that is,

:<math> \nabla \rho(p) w = \sum_{i=1}^n \frac{\partial \rho (p)}{ \partial z_j }w_j =0 </math>, we have
:<math>H(\rho) = \sum_{i,j=1}^n \frac{\partial^2 \rho(p)}{\partial z_i \, \partial \bar{z_j} } w_i \bar{w_j} \geq 0.</math>{{R|Chen2000}}<ref>{{cite journal |doi=10.2206/kyushumfs.41.45 |title=Stein Neighborhood Bases for Product Sets of Polydiscs and Open Intervals |year=1987 |last1=Shon |first1=Kwang Ho |journal=Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics |volume=41 |pages=45–80 |doi-access=free }}</ref>

If ''D'' does not have a <math>\mathcal{C}^2</math> boundary, the following approximation result can be useful.

'''Proposition 1''' ''If ''D'' is pseudoconvex, then there exist [[bounded set|bounded]], strongly Levi pseudoconvex domains <math>D_k \subset D</math> with class <math>\mathcal{C}^\infty</math>-boundary which are relatively compact in ''D'', such that''

:<math>D = \bigcup_{k=1}^\infty D_k.</math>

This is because once we have a <math>\varphi</math> as in the definition we can actually find a <math>\mathcal{C}^\infty</math> exhaustion function.

===== Strongly (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex) =====
When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.{{R|Chen2000}}

==== Levi total pseudoconvex ====
If for every boundary point <math>\rho</math> of ''D'', there exists an [[analytic variety]] <math>\mathcal{B}</math> passing <math>\rho</math> which lies entirely outside ''D'' in some neighborhood around <math>\rho</math>, except the point <math>\rho</math> itself. Domain ''D'' that satisfies these conditions is called Levi total pseudoconvex.<ref name="Hitomatsu1958" >{{Citation | author = Sin Hitomatsu | title=On some conjectures concerning pseudo-convex domains | year=1958 | journal=Journal of the Mathematical Society of Japan| volume=6 | issue=2 | pages=177–195| zbl=0057.31503|doi=10.2969/jmsj/00620177 | doi-access=free}}</ref>

==== Oka pseudoconvex ====
===== Family of Oka's disk =====
Let ''n''-functions <math>\varphi:z_j = \varphi_j(u, t)</math> be continuous on <math>\Delta:|U|\leq1, 0\leq t\leq1</math>, holomorphic in <math>|u|< 1</math> when the parameter ''t'' is fixed in [0, 1], and assume that <math>\frac{\partial\varphi_j}{\partial u}</math> are not all zero at any point on <math>\Delta</math>. Then the set <math>Q(t):= \{Z_j= \varphi_j(u, t);|u|\leq 1\}</math> is called an analytic disc de-pending on a parameter ''t'', and <math>B(t):= \{Z_j= \varphi_j(u, t);|u|= 1\}</math> is called its shell. If <math>Q(t)\subset D \ (0 <t)</math> and <math>B(0)\subset D</math>, ''Q(t)'' is called Family of Oka's disk.{{R|Hitomatsu1958|}}<ref name="Kajiwara1959">{{cite journal |doi=10.2206/kyushumfs.13.37|title=Some Results on the Equivalence of Complex-Analytic Fibre Bundles|year=1959|last1=Kajiwara|first1=Joji|journal=Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics|volume=13|pages=37–48|doi-access=free}}</ref>

===== Definition =====
When <math>Q(0)\subset D</math> holds on any family of Oka's disk, ''D'' is called Oka pseudoconvex.{{R|Hitomatsu1958|}} Oka's proof of Levi's problem was that when the [[unramified]] Riemann domain over <math>\mathbb{C}^n</math><ref>{{Eom| title = Riemannian domain | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 44356}}</ref> 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.{{R|Oka'sIX|}}{{R|Kajiwara1959|}}

==== Locally pseudoconvex (a.k.a. locally Stein, Cartan pseudoconvex, local Levi property) ====
For every point <math>x \in \partial D</math> there exist a neighbourhood ''U'' of ''x'' and ''f'' holomorphic. ( i.e. <math>U \cap D</math> be holomorphically convex.) such that ''f'' cannot be extended to any neighbourhood of ''x''. i.e., let <math>\psi : X \to Y</math> be a holomorphic map, if every point <math>y\in Y</math> has a neighborhood U such that <math>\psi^{-1}(U)</math> admits a <math> \mathcal{C}^{\infty}</math>-plurisubharmonic exhaustion function (weakly 1-complete<ref>{{cite journal |doi=10.5802/aif.3226|title=On the local pseudoconvexity of certain analytic families of <math>\mathbb{C}</math>|year=2018|last1=Ohsawa|first1=Takeo|journal=Annales de l'Institut Fourier|volume=68|issue=7|pages=2811–2818|doi-access=free}}</ref>), in this situation, we call that ''X'' is locally pseudoconvex (or locally Stein) over ''Y''. As an old name, it is also called Cartan pseudoconvex. In <math>\Complex^n</math> the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.<ref name=Ohsawa2021>{{cite journal |hdl=2433/263965|title=NISHIno's Rigidity, Locally pseudoconvex maps, and holomorphic motions (Topology of pseudoconvex domains and analysis of reproducing kernels)|journal=RIMS Kôkyûroku|date=February 2021|volume=2175|pages=27–46|last1=Ohsawa|first1=Takeo}}</ref>{{R|Hitomatsu1958|}} For example, Diederich–Fornæss<ref>{{cite journal |doi=10.1007/BF01312449|title=A smooth pseudoconvex domain without pseudoconvex exhaustion|year=1982|last1=Diederich|first1=Klas|last2=Fornæss|first2=John Erik|journal=Manuscripta Mathematica|volume=39|pages=119–123|s2cid=121224216 |url=http://eudml.org/doc/154876}}</ref> found local pseudoconvex bounded domains <math>\Omega</math> with smooth boundary on non-Kähler manifolds such that <math>\Omega</math> is not weakly 1-complete.<ref>{{cite journal |doi=10.4064/ap106-0-19| url=http://eudml.org/doc/281083 | title=Hartogs type extension theorems on some domains in Kähler manifolds | year=2012 | last1=Ohsawa | first1=Takeo | journal=[[Annales Polonici Mathematici]] | volume=106 | pages=243–254 | s2cid=123827662 | doi-access=free }}</ref>{{refn|group=note|1=Definition of weakly 1-complete.<ref>{{cite journal |doi=10.2977/prims/1195186709 |title=Weakly 1-Complete Manifold and Levi Problem|year=1981 |last1=Ohsawa |first1=Takeo |journal=Publications of the Research Institute for Mathematical Sciences |volume=17 |pages=153–164 |doi-access=free }}</ref>}}

=== Conditions equivalent to domain of holomorphy ===
For a domain <math>D \subset \mathbb C^n</math> the following conditions are equivalent:{{refn|group=note|1=In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification<ref>{{Citation | author = Heinrich Behnke & Karl Stein | date = 1951 | title =Modifikationen komplexer Mannigfaltigkeiten und Riernannscher Gebiete| journal = Mathematische Annalen | volume = 124 | pages = 1–16 | doi =10.1007/BF01343548|zbl=0043.30301| s2cid = 120455177|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00228264X }}</ref><ref>{{Eom| title = Modification | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47868}}</ref> on the complex analytic space (when n = 2, the result by Hirzebruch,<ref>{{Citation | author = Friedrich Hirzebruch | date = 1953 | title =Über vierdimensionaleRIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen| journal = Mathematische Annalen | volume = 126 | pages = 1–22 | doi =10.1007/BF01343146| hdl = 21.11116/0000-0004-3A47-C | s2cid = 122862268 | hdl-access = free }}</ref> when n = 3 the result by Zariski<ref>{{Citation | author = Oscar Zariski | date = 1944 | title =Reduction of the Singularities of Algebraic Three Dimensional Varieties| journal = Annals of Mathematics |series=Second Series| volume = 45 | issue = 3 | pages = 472–542 | doi =10.2307/1969189| jstor = 1969189 }}</ref> for algebraic varietie.), but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy:
<ref>{{Citation | author = Hans Grauert & Reinhold Remmert | date = 1956 | title = Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie | journal = [[Commentarii Mathematici Helvetici]]| volume = 31 | pages = 152–183 | doi =10.1007/BF02564357|zbl =0073.30301| s2cid = 117913713 }}</ref>}}
<ol type="1">
<li> ''D'' is a domain of holomorphy.</li>
<li> ''D'' is holomorphically convex.</li>
<li> ''D'' is the union of an increasing sequence of [[analytic polyhedron]]s in ''D''.</li>
<li> ''D'' is pseudoconvex.</li>
<li> ''D'' is Locally pseudoconvex.</li></ol>

The implications <math>1 \Leftrightarrow 2 \Leftrightarrow 3 </math>,{{refn|group=note|1=This relation is called the Cartan–Thullen theorem.<ref>{{cite journal |jstor=43698735|title=Some properties of holomorphic convexity in general function algebras|last1=Tsurumi|first1=Kazuyuki|last2=Jimbo|first2=Toshiya|journal=Science Reports of the Tokyo Kyoiku Daigaku, Section A|year=1969|volume=10|issue=249/262|pages=178–183}}</ref>}} <math>1 \Rightarrow 4</math>,<ref group=note>See [[Oka's lemma]]</ref> and <math>4\Rightarrow 5</math> are standard results. Proving <math>5 \Rightarrow 1</math>, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the [[Levi problem]] (after [[Eugenio Elia Levi|E. E. Levi]]) and was solved for unramified Riemann domains over <math>\mathbb{C}^n</math> by Kiyoshi Oka,<ref group=note>Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.</ref> but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,<ref>{{cite journal |doi=10.1007/BF01420649|title=A counterexample for the Levi problem for branched Riemann domains over <math>\mathbb{C}^n</math> |year=1978 |last1=Fornæss |first1=John Erik |journal=Mathematische Annalen |volume=234 |issue=3 |pages=275–277 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002315858}}</ref> and then by [[Lars Hörmander]] using methods from functional analysis and partial differential equations (a consequence of <math>\bar{\partial}</math>-problem(equation) with a [[Ohsawa–Takegoshi L2 extension theorem|L<sup>2</sup> methods]]).{{R|Hörmander1965|}}{{R|Forstnerič2011§SteinManifolds|}}{{R|Błocki2014|}}{{R|Noguchi2019}}

== Sheaves ==
The introduction of [[Sheaf (mathematics)|sheaves]] into several complex variables allowed the reformulation of and solution to several important problems in the field.

=== Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf)) ===
Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains".{{R|Oka'sVII}}{{R|Oka'sVIII}} Specifically, it is a set <math>(I)</math> of pairs <math>(f, \delta)</math>, <math>f</math> holomorphic on a non-empty open set <math>\delta</math>, such that
<ol type="1">
<li> If <math>(f, \delta) \in (I)</math> and <math>(a, \delta')</math> is arbitrary, then <math>(af, \delta \cap \delta') \in (I)</math>.</li>
<li> For each <math>(f, \delta), (f', \delta') \in (I)</math>, then <math>(f + f', \delta \cap \delta') \in (I).</math></li>
</ol>

The origin of indeterminate domains comes from the fact that domains change depending on the pair <math>(f, \delta)</math>. Cartan{{R|Cartan1950}}{{R|Cartan1953}} translated this notion into the notion of the '''[[coherent sheaf|coherent]]''' ('''[[Sheaf (mathematics)|sheaf]]''') (Especially, coherent analytic sheaf) in sheaf cohomology.<ref name=Noguchi2019>{{cite journal |doi=10.4310/ICCM.2019.V7.N2.A2|title=A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem |year=2019 |last1=Noguchi |first1=Junjiro |journal=Notices of the International Congress of Chinese Mathematicians |volume=7 |issue=2 |pages=19–24 |arxiv=1807.08246 |s2cid=119619733 }}</ref><ref>{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.x) |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 |url={{Google books|4gHdDAAAQBAJ|Analytic Function Theory of Several Variables: Elements of Oka's Coherence|page=PR10|plainurl=yes}}}}</ref> This name comes from 
H. Cartan.<ref>{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.33) |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 |url={{Google books|4gHdDAAAQBAJ|Analytic Function Theory of Several Variables: Elements of Oka's Coherence|page=33|plainurl=yes}}}}</ref> Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.<ref name="Serre1955">{{Citation|author1-first=Jean-Pierre|author1-last=Serre|author1-link=Jean-Pierre Serre|title=Faisceaux algébriques cohérents|journal=Annals of Mathematics|volume=61|pages=197–278|year=1955|issue=2|doi=10.2307/1969915|jstor=1969915|mr=0068874|url=https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf}}</ref> The notion of coherent ([[coherent sheaf cohomology]]) helped solve the problems in several complex variables.{{R|Chorlay2010|}}

=== Coherent sheaf ===

==== Definition ====
The definition of the coherent sheaf is as follows.{{R|Serre1955|}}<ref>{{cite journal| last1 = Grothendiec | first1 = Alexander | last2 =Dieudonn | first2 = Jean  | year = 1960| title = Éléments de géométrie algébrique: I. Le langage des schémas (ch.0 § 5. FAISCEAUX QUASI-COHÉRENTS ET FAISCEAUX COHÉRENTS (0.5.1–0.5.3))| journal = Publications Mathématiques de l'IHÉS | volume = 4   | mr = 0217083 | url = http://www.numdam.org/item/PMIHES_1960__4__5_0| doi = 10.1007/bf02684778 | s2cid = 121855488 }}</ref><ref>{{cite book |doi=10.1007/978-3-662-09873-8_2|chapter=Local Theory of Complex Spaces |title=Several Complex Variables VII §6. Calculs of Coherent sheaves |series=Encyclopaedia of Mathematical Sciences |year=1994 |last1=Remmert |first1=R. |volume=74 |pages=7–96 |isbn=978-3-642-08150-7 | url={{Google books|WqnvCAAAQBAJ|page=40|plainurl=yes}}}}</ref><ref>{{cite book |isbn=9784431568513|doi=10.1007/978-4-431-55747-0|title=L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds|last1=Ohsawa|first1=Takeo|series=Springer Monographs in Mathematics|date=10 December 2018|url={{Google books|DIJ8DwAAQBAJ|page=25|plainurl=yes}}}}</ref>
{{R|CAaDG}}{{rp|pages=83–89}}<!--that is: R|CaDG|pages=83-89
 -->
A '''quasi-coherent sheaf''' on a [[ringed space]] <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> of <math>\mathcal O_X</math>-[[sheaf of modules|modules]] which has a local presentation, that is, every point in <math>X</math> has an open neighborhood <math>U</math> in which there is an [[exact sequence]]
:<math>\mathcal{O}_X^{\oplus I}|_{U} \to \mathcal{O}_X^{\oplus J}|_{U} \to \mathcal{F}|_{U} \to 0</math>
for some (possibly infinite) sets <math>I</math> and <math>J</math>.

A '''coherent sheaf''' on a ringed space <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> satisfying the following two properties:
<ol type="1">
<li> <math>\mathcal F</math> is of ''finite type'' over <math>\mathcal O_X</math>, that is, every point in <math>X</math> has an [[open neighborhood]] <math>U</math> in <math>X</math> such that there is a surjective morphism <math>\mathcal{O}_X^{\oplus n}|_{U} \to \mathcal{F}|_{U} </math> for some natural number <math>n</math>;</li>
<li> for each open set <math>U\subseteq X</math>, integer <math>n > 0</math>, and arbitrary morphism <math>\varphi: \mathcal{O}_X^{\oplus n}|_{U} \to \mathcal{F}|_{U} </math> of <math>\mathcal O_X</math>-modules, the kernel of <math>\varphi</math> is of finite type.</li>
</ol>

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of <math>\mathcal O_X</math>-modules.

Also, [[Jean-Pierre Serre]] (1955){{R|Serre1955|}} proves that

:If in an exact sequence <math>0\to \mathcal{F}_1|_U\to\mathcal{F}_2|_U\to\mathcal{F}_3|_U\to 0</math> of sheaves of <math>\mathcal{O}</math>-modules two of the three sheaves <math>\mathcal{F}_{j}</math> are coherent, then the third is coherent as well.

==== (Oka–Cartan) coherent theorem ====
(Oka–Cartan) coherent theorem{{R|Oka'sVII|}} says that each sheaf that meets the following conditions is a coherent.<ref name="Noguchi">{{Citation | last1=Noguchi | first1=Junjiro | title=A Weak Coherence Theorem and Remarks to the Oka Theory |journal=Kodai Math. J.|volume=42|url=https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf | year=2019 | issue=3 | arxiv = 1704.07726|doi =10.2996/kmj/1572487232|pages=566–586| s2cid=119697608 }}</ref>

<ol type="i">
<li> the sheaf <math>\mathcal{O} := \mathcal{O}_{\mathbb{C}_n}</math> of [[germ (mathematics)|germs]] of holomorphic functions on <math>\mathbb{C}_n</math>, or the structure sheaf <math>\mathcal{O}_X</math> of complex submanifold or every complex analytic space <math>(X, \mathcal{O}_X)</math><ref>{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|page=60|publisher=Springer |url={{Google books|title=Coherent Analytic Sheaves|blPxCAAAQBAJ|page=60|plainurl=yes}}}}</ref></li>
<li> the ideal sheaf <math>\mathcal{I} \langle A \rangle</math> of an analytic subset A of an open subset of <math>\mathbb{C}_n</math>. (Cartan 1950{{R|Cartan1950}})<ref>{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|page=84|publisher=Springer |url={{Google books|title=Coherent Analytic Sheaves|blPxCAAAQBAJ|page=84|plainurl=yes}}}}</ref><ref>{{cite web |last1=Demailly |first1=Jean-Pierre |title=Basic results on Sheaves and Analytic Sets |url=https://www-fourier.ujf-grenoble.fr/~demailly/analytic_geometry_2019/sheaves_and_analytic_sets.pdf |publisher=Institut Fourier}}</ref></li>
<li> the normalization of the structure sheaf of a complex analytic space<ref>{{cite book |doi=10.1007/978-3-642-69582-7_8|chapter=Normalization of Complex Spaces |title=Coherent Analytic Sheaves |series=Grundlehren der mathematischen Wissenschaften |year=1984 |last1=Grauert |first1=Hans |last2=Remmert |first2=Reinhold |volume=265 |pages=152–166 |isbn=978-3-642-69584-1 }}</ref></li>
</ol>
From the above Serre(1955) theorem, <math>\mathcal{O}^p</math> is a coherent sheaf, also, (i) is used to prove [[Cartan's theorems A and B]].

=== Cousin problem ===
In the case of one variable complex functions, [[Mittag-Leffler's theorem]] was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and [[Weierstrass factorization theorem]] was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). However, these theorems do not hold in several complex variables because the singularities of [[Analytic function#Analytic functions of several variables|analytic function in several complex variables]] are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology. They were first introduced in special cases by Pierre Cousin in 1895.<ref>{{cite journal |last1=Cousin |first1=Pierre |title=Sur les fonctions de ''n'' variables complexes |journal=Acta Mathematica |volume=19 |year=1895|pages=1–61 |doi=10.1007/BF02402869|doi-access=free }}</ref> It was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy{{refn|group=note|There are some counterexamples in the domain of holomorphicity regarding second Cousin problem.{{R|OkaIII|}}<ref>{{cite book |last1=Serre |first1=Jean-Pierre |title=Oeuvres - Collected Papers I |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-39815-5|page=XXIII, 598|chapter=Quelques problèmes globaux rélatifs aux variétés de Stein|chapter-url={{Google books|eaUoAKOAbUsC|Quelques problèmes globaux rélatifs aux variétés de Stein|page=265|plainurl=yes}}|language=fr}}</ref>
 }} on the complex coordinate space,<ref>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles|journal=Journal of Science of the Hiroshima University|volume=6|year=1936|pages=245–255|doi=10.32917/hmj/1558749869|doi-access=free}}</ref><ref>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie|journal=Journal of Science of the Hiroshima University|volume=7|year=1937|pages=115–130|doi=10.32917/hmj/1558576819|doi-access=free}}</ref><ref name=OkaIII>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin|journal=Journal of Science of the Hiroshima University|volume=9|year=1939|pages=7–19|doi=10.32917/hmj/1558490525|doi-access=free}}</ref>{{refn|group=note|This is called the classic Cousin problem.{{R|Chorlay2010|}}}} also solving the second Cousin problem with additional topological assumptions. The Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological;{{R|OkaIII|}}{{R|Chorlay2010|}}{{R|Huckleberry2013|}} Serre called this the [[#Definition using sheaf cohomology words|Oka principle]].<ref>{{cite journal |url=http://www.numdam.org/item/SHC_1951-1952__4__A20_0/|title=Applications de la théorie générale à divers problèmes globaux|journal=Séminaire Henri Cartan|volume=4|pages=1–26|last1=Serre|first1=J. -P}}</ref> They are now posed, and solved, for arbitrary complex manifold ''M'', in terms of conditions on ''M''. ''M'', which satisfies these conditions, is one way to define a Stein manifold. The study of the cousin's problem made us realize that in the study of several complex variables, it is possible to study of global properties from the patching of local data,{{R|Cartan1950|}} that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.{{R|CT4|}}){{R|Chorlay2010|}}

==== First Cousin problem ====
Without the language of sheaves, the problem can be formulated as follows. On a complex manifold ''M'', one is given several meromorphic functions <math>f_i</math> along with domains <math>U_i</math> where they are defined, and where each difference <math>f_i-f_j</math> is holomorphic (wherever the difference is defined). The first Cousin problem then asks for a meromorphic function <math>f</math> on ''M'' such that <math>f-f_i</math> is ''holomorphic'' on <math>U_i</math>; in other words, that <math>f</math> shares the [[mathematical singularity|singular]] behaviour of the given local function.

Now, let '''K''' be the sheaf of meromorphic functions and '''O''' the sheaf of holomorphic functions on ''M''. The first Cousin problem can always be solved if the following map is surjective:

:<math>H^0(M,\mathbf{K}) \xrightarrow{\phi} H^0(M,\mathbf{K}/\mathbf{O}).</math>

By the [[long exact sequence in homology|long exact cohomology sequence]],

:<math>H^0(M,\mathbf{K}) \xrightarrow{\phi} H^0(M,\mathbf{K}/\mathbf{O})\to H^1(M,\mathbf{O})</math>

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group ''H''<sup>1</sup>(''M'','''O''') vanishes. In particular, by [[Cartan's theorems A and B|Cartan's theorem B]], the Cousin problem is always solvable if ''M'' is a Stein manifold.

==== Second Cousin problem ====
The second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio <math>f_i/f_j</math> is a non-vanishing holomorphic function (where said difference is defined). It asks for a meromorphic function <math>f</math> on ''M'' such that <math>f/f_i</math> is holomorphic and non-vanishing.

Let <math>\mathbf{O}^*</math> be the sheaf of holomorphic functions that vanish nowhere, and <math>\mathbf{K}^*</math> the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of [[abelian group]]s, and the quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is well-defined. If the following map <math>\phi</math> is surjective, then Second Cousin problem can be solved:

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*).</math>

The long exact sheaf cohomology sequence associated to the quotient is

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*)\to H^1(M,\mathbf{O}^*)</math>

so the second Cousin problem is solvable in all cases provided that <math>H^1(M,\mathbf{O}^*)=0.</math>

The cohomology group <math>H^1(M,\mathbf{O}^*)</math> for the multiplicative structure on <math>\mathbf{O}^*</math> can be compared with the cohomology group <math>H^1(M,\mathbf{O})</math> with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

:<math>0\to 2\pi i\Z\to \mathbf{O} \xrightarrow{\exp} \mathbf{O}^* \to 0</math>

where the leftmost sheaf is the locally constant sheaf with fiber <math>2\pi i\Z</math>. The obstruction to defining a logarithm at the level of ''H''<sup>1</sup> is in <math>H^2(M,\Z)</math>, from the long exact cohomology sequence

:<math>H^1(M,\mathbf{O})\to H^1(M,\mathbf{O}^*)\to 2\pi i H^2(M,\Z) \to H^2(M, \mathbf{O}).</math>

When ''M'' is a Stein manifold, the middle arrow is an isomorphism because <math>H^q(M,\mathbf{O}) = 0</math> for <math>q > 0</math> so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that <math>H^2(M,\Z)=0.</math> (This condition called Oka principle.)

== Manifolds and analytic varieties with several complex variables ==

=== Stein manifold (non-compact Kähler manifold) ===
Since a non-compact (open) Riemann surface<ref name =Weyl1913>{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The concept of a Riemann surface | orig-year=1913 | url=https://archive.org/details/dieideederrieman00weyluoft | publisher=[[Dover Publications]] | location=New York | edition=3rd | isbn=978-0-486-47004-7 | year=2009 | mr=0069903}}</ref> always has a non-constant single-valued holomorphic function,<ref name ="Behnke–Stein1948">{{Citation | author = Heinrich Behnke & Karl Stein | title=Entwicklung analytischer Funktionen auf Riemannschen Flächen | year=1948 | journal=Mathematische Annalen| volume=120 | pages=430–461|doi=10.1007/BF01447838|zbl =0038.23502 | s2cid=122535410 }}</ref> and satisfies the [[second axiom of countability]], the open Riemann surface is in fact a ''1''-dimensional complex manifold possessing a holomorphic mapping into the complex plane <math>\mathbb C</math>. (In fact, Gunning and Narasimhan have shown (1967)<ref>{{cite journal |doi=10.1007/BF01360812|title=Immersion of open Riemann surfaces |year=1967 |last1=Gunning |first1=R. C. |last2=Narasimhan |first2=Raghavan |journal=Mathematische Annalen |volume=174 |issue=2 |pages=103–108 |s2cid=122162708 }}</ref> 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.)<ref>{{cite book |first1=J.E.|last1 =Fornaess |last2=Forstneric |first2=F |last3=Wold |first3=E.F |editor1-first=Daniel |editor1-last=Breaz |editor2-first=Michael Th. |editor2-last=Rassias |title=Advancements in Complex Analysis – Holomorphic Approximation |chapter=The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan |date=2020 |publisher=[[Springer Nature]] |pages=133–192|doi=10.1007/978-3-030-40120-7|arxiv=1802.03924 |isbn =978-3-030-40119-1 |s2cid =220266044 }}</ref> The [[Whitney embedding theorem]] tells us that every smooth ''n''-dimensional manifold can be [[Embedding|embedded]] as a smooth submanifold of <math>\mathbb{R}^{2n}</math>, whereas it is "rare" for a complex manifold to have a holomorphic embedding into <math>\mathbb C^n</math>. For example, for an arbitrary compact connected complex manifold ''X'', every holomorphic function on it is constant by Liouville's theorem, and so it cannot have any embedding into complex n-space. That is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants. So, consider the conditions under which a complex manifold has a holomorphic function that is not a constant. Now if we had a holomorphic embedding of ''X'' into <math>\mathbb C^n</math>, then the coordinate functions of <math>\mathbb C^n</math> would restrict to nonconstant holomorphic functions on ''X'', contradicting compactness, except in the case that ''X'' is just a point. Complex manifolds that can be holomorphic embedded into <math>\mathbb C^n</math> are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.<ref>{{cite journal |doi=10.1016/j.crma.2010.11.020|title=On complex Banach manifolds similar to Stein manifolds|year=2011|last1=Patyi|first1=Imre|journal=Comptes Rendus Mathematique|volume=349|issue=1–2|pages=43–45|arxiv=1010.3738|s2cid=119631664}}</ref>

A '''Stein manifold''' is a complex [[submanifold]] of the [[vector space]] of ''n'' complex dimensions. They were introduced by and named after Karl Stein (1951).<ref>{{citation|mr=0043219|last=Stein|first= Karl|title=Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem|language=German|journal=Math. Ann. |volume=123|year=1951|pages=201–222|doi=10.1007/bf02054949|s2cid=122647212}}</ref> A '''Stein space''' is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of [[affine variety|affine varieties]] or [[affine scheme]]s in algebraic geometry. If the univalent domain on <math>\mathbb C^n</math> is connection to a manifold, can be regarded as a [[complex manifold]] and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) [[analytic continuation]] of an analytic function.

==== Definition ====
Suppose ''X'' is a [[paracompact]] [[complex manifold]]s of complex dimension <math>n</math> and let <math>\mathcal O(X)</math> denote the ring of holomorphic functions on ''X''. We call ''X'' a '''Stein manifold''' if the following conditions hold:<ref>{{cite journal |arxiv=1108.2078|last1=Noguchi|first1=Junjiro|title=Another Direct Proof of Oka's Theorem (Oka IX)|year=2011|mr=3086750|journal=J. Math. Sci. Univ. Tokyo|url=https://www.ms.u-tokyo.ac.jp/journal/pdf/jms190407.pdf|volume=19|issue=4}}</ref>
<ol type="1">
<li>''X'' is holomorphically convex, i.e. for every compact subset <math>K \subset X</math>, the so-called ''holomorphically convex hull'',
:<math>\bar K = \left \{z \in X ; |f(z)| \leq \sup_{w \in K} |f(w)|, \ \forall f \in \mathcal O(X) \right \},</math>
is also a ''compact'' subset of ''X''.</li>
<li>''X'' is [[holomorphically separable]],<ref group = note>From this condition, we can see that the Stein manifold is not compact.</ref> i.e. if <math>x \neq y</math> are two points in ''X'', then there exists <math>f \in \mathcal O(X)</math> such that <math>f(x) \neq f(y).</math></li>
<li> The open neighborhood of every point on the manifold has a holomorphic [[Atlas (topology)#Charts|chart]] to the <math>\mathcal O(X)</math>.</li></ol>

Note that condition (3) can be derived from conditions (1) and (2).<ref>{{cite journal |doi=10.1007/BF01362369|title=Charakterisierung der holomorph vollständigen komplexen Räume |year=1955 |last1=Grauert |first1=Hans |journal=Mathematische Annalen |volume=129 |pages=233–259 |s2cid=122840967|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002284383}}</ref>

==== Every non-compact (open) Riemann surface is a Stein manifold ====

Let ''X'' be a connected, non-compact (open) [[Riemann surface]]. A [[Behnke–Stein theorem on Stein manifolds#Method of proof|deep theorem]] of Behnke and Stein (1948){{R|Behnke–Stein1948|}} asserts that ''X'' is a Stein manifold.

Another result, attributed to [[Hans Grauert]] and [[Helmut Röhrl]] (1956), states moreover that every [[holomorphic vector bundle]] on ''X'' is trivial. In particular, every line bundle is trivial, so <math>H^1(X, \mathcal O_X^*) =0 </math>. The [[exponential sheaf sequence]] leads to the following exact sequence:

:<math>H^1(X, \mathcal O_X) \longrightarrow H^1(X, \mathcal O_X^*) \longrightarrow H^2(X, \Z) \longrightarrow H^2(X, \mathcal O_X) </math>

Now [[Cartan's theorems A and B|Cartan's theorem B]] shows that <math>H^1(X,\mathcal{O}_X) = H^2(X,\mathcal{O}_X)=0 </math>, therefore <math>H^2(X,\Z) = 0</math>.

This is related to the solution of the [[Cousin problems|second (multiplicative) Cousin problem]].

==== Levi problems ====
Cartan extended Levi's problem to Stein manifolds.<ref>{{cite journal |last1=Cartan |first1=Henri |title=Variétés analytiques réelles et variétés analytiques complexes |journal=Bulletin de la Société Mathématique de France |year=1957 |volume=85 |pages=77–99 |doi=10.24033/bsmf.1481|doi-access=free }}</ref>
:If the [[Relatively compact subspace|relative compact open subset]] <math>D\subset X</math> of the Stein manifold X is a Locally pseudoconvex, then ''D'' is a Stein manifold, and conversely, if ''D'' is a Locally pseudoconvex, then ''X'' is a Stein manifold. i.e. Then ''X'' is a Stein manifold if and only if ''D'' is locally the Stein manifold.<ref>{{cite journal |last1=Barth |first1=Theodore J. |title=Families of nonnegative divisors |journal=Trans. Amer. Math. Soc. |year=1968 |volume=131 |pages=223–245 |doi=10.1090/S0002-9947-1968-0219751-3|doi-access=free }}</ref>

This was proved by Bremermann<ref>{{cite journal |last1=Bremermann |first1=Hans J. |title=On Oka's theorem for Stein manifolds.|journal=Seminars on Analytic Functions. Institute for Advanced Study (Princeton, N.J.) |year=1957 |volume=1 |pages=29–35|zbl=0192.18304}}</ref> by embedding it in a sufficiently high dimensional <math>\mathbb{C}^n</math>, and reducing it to the result of Oka.{{R|Oka'sIX|}}

Also, Grauert proved for arbitrary '''complex''' manifolds ''M''.{{refn|group=note|1=Levi problem is not true for domains in arbitrary manifolds.{{R|Huckleberry2013|}}<ref name="Sibony2018">{{cite journal |doi=10.1007/s00208-017-1539-x|title=Levi problem in complex manifolds|year=2018|last1=Sibony|first1=Nessim|journal=Mathematische Annalen|volume=371|issue=3–4|pages=1047–1067|arxiv=1610.07768|s2cid=119670805}}</ref><ref name="Grauert1963">{{cite journal |last1=Grauert |first1=Hans |title=Bemerkenswerte pseudokonvexe Mannigfaltigkeiten |journal=Mathematische Zeitschrift |year=1963 |volume=81 |issue=5 |pages=377–391 |doi=10.1007/BF01111528|s2cid=122214512 }}</ref>}}<ref name="Grauert1958" >{{Citation | author = Hans Grauert | title=On Levi's Problem and the Imbedding of Real-Analytic Manifolds | year=1958 | journal=Annals of Mathematics |series=Second Series| volume=68 | issue=2 | pages=460–472| zbl=0108.07804|doi=10.2307/1970257 | jstor=1970257 }}</ref>{{R|Huckleberry2013|}}{{R|Sibony2018|}}
:If the relative compact subset <math>D\subset M</math> of a arbitrary complex manifold ''M'' is a '''strongly pseudoconvex''' on ''M'', then ''M'' is a holomorphically convex (i.e. Stein manifold). Also, ''D'' is itself a Stein manifold.

And Narasimhan<ref name=NarasimhanI>{{cite journal |last1=Narasimhan |first1=Raghavan |title=The Levi problem for complex spaces |journal=Mathematische Annalen |year=1961 |volume=142 |issue=4 |pages=355–365 |doi=10.1007/BF01451029|s2cid=120565581 }}</ref><ref name=NarasimhanII>{{cite journal |last1=Narasimhan |first1=Raghavan |title=The Levi problem for complex spaces II|journal=Mathematische Annalen |year=1962 |volume=146 |issue=3 |pages=195–216 |doi=10.1007/BF01470950|s2cid=179177434 }}</ref> extended Levi's problem to [[complex analytic space]], a generalized in the singular case of complex manifolds.
:A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space.{{R|Siu1978}}

Levi's problem remains unresolved in the following cases;	

:Suppose that ''X'' is a singular Stein space,{{refn|group=note|1=In the case of Stein space with isolated singularities, it has already been positively solved by Narasimhan.{{R|Siu1978}}{{R|Coltoiu2009}}}} <math>D \subset\subset X</math> . Suppose that for all <math>p\in \partial D</math> there is an open neighborhood <math>U (p)</math> so that <math>U\cap D</math> is Stein space. Is ''D'' itself Stein?{{R|Siu1978}}<ref>{{cite journal |last1=Fornæss |first1=John Erik |last2=Sibony |first2=Nessim |title=Some open problems in higher dimensional complex analysis and complex dynamics |year=2001 |journal = Publicacions Matemàtiques |volume=45 |issue=2 |pages=529–547 |doi=10.5565/PUBLMAT_45201_11|jstor =43736735 |url=http://ddd.uab.cat/record/1973 }}</ref><ref name = Coltoiu2009>{{cite arXiv |eprint=0905.2343|last1=Coltoiu|first1=Mihnea|title=The Levi problem on Stein spaces with singularities. A survey|year=2009|class=math.CV}}</ref>

more generalized

:Suppose that ''N'' be a Stein space and ''f'' an injective, and also <math>f :M \to N</math> a Riemann unbranched domain, such that map ''f'' is a locally pseudoconvex map (i.e. Stein morphism). Then ''M'' is itself Stein ?{{R|Coltoiu2009}}<ref>{{cite book |isbn=9784431568513|doi=10.1007/978-4-431-55747-0|url=https://books.google.com/books?id=DIJ8DwAAQBAJ&pg=PA109|title=L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds|last1=Ohsawa|first1=Takeo|series=Springer Monographs in Mathematics|date=10 December 2018}}</ref>{{rp|page=109}}

and also,

:Suppose that ''X'' be a Stein space and <math>D = \bigcup_{n\in\mathbb{N}} D_n</math> an increasing union of Stein open sets. Then ''D'' is itself Stein ?

This means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space. {{R|Coltoiu2009}}

===== K-complete =====
Grauert introduced the concept of K-complete in the proof of Levi's problem.

Let ''X'' is complex manifold, ''X'' is K-complete if, to each point <math>x_0\in X</math>, there exist finitely many holomorphic map <math>f_1,\dots,f_k</math> of ''X'' into <math>\Complex^p</math>, <math>p = p(x_0)</math>, such that <math>x_0</math> is an isolated point of the set <math>A = \{x\in X;f^{-1}f(x_0)\ (v=1,\dots,k)\}</math>.{{R|Grauert1958|}} This concept also applies to complex analytic space.<ref name=Andreotti&Narasimhan1964>{{cite journal |doi=10.1090/S0002-9947-1964-0159961-3|jstor=1994247|title=Oka's Heftungslemma and the Levi Problem for Complex Spaces|last1=Andreotti|first1=Aldo|last2=Narasimhan|first2=Raghavan|journal=Transactions of the American Mathematical Society|year=1964|volume=111|issue=2|pages=345–366|doi-access=free}}</ref>

==== Properties and examples of Stein manifolds ====
* The standard<ref group="note"><math>\Complex^n \times \mathbb{P}_m</math> (<math>\mathbb{P}_m</math> is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.</ref> complex space <math>\Complex^n</math> is a Stein manifold.
* Every domain of holomorphy in <math>\Complex^n</math> is a Stein manifold.{{R|FieldCM1982}}
* It can be shown quite easily that every closed complex submanifold '''of a Stein manifold''' is a Stein manifold, too.
* The embedding theorem for Stein manifolds states the following: Every Stein manifold ''X'' of complex dimension ''n'' can be embedded into <math>\Complex^{2 n+1}</math> by a [[biholomorphic]] [[proper map]].<ref>{{cite journal |last1=Raghavan |first1=Narasimhan |title=Imbedding of Holomorphically Complete Complex Spaces |journal=American Journal of Mathematics |year=1960 |volume=82 |issue=4 |pages=917–934 |doi=10.2307/2372949|jstor=2372949 }}</ref><ref>{{cite journal |last1=Eliashberg |first1=Yakov |last2=Gromov |first2=Mikhael |title=Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 +1 |journal=Annals of Mathematics |series=Second Series |year=1992 |volume=136 |issue=1 |pages=123–135|doi =10.2307/2946547 |jstor=2946547 }}</ref><ref>{{cite journal |title =Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes |journal= Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris| pages=118–121| last1=Remmert|first1=Reinhold|year=1956|volume=243|url=https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.item|language=fr|zbl=0070.30401}}</ref>

These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the [[ambient space]] (because the embedding is biholomorphic).
* Every Stein manifold of (complex) dimension ''n'' has the homotopy type of an ''n''-dimensional CW-Complex.<ref>{{cite journal |doi=10.1090/S0002-9904-1967-11839-1|title=Some remarks on parallelizable Stein manifolds |year=1967 |last1=Forster |first1=Otto |journal=Bulletin of the American Mathematical Society |volume=73 |issue=5 |pages=712–716 |doi-access=free }}</ref>
* In one complex dimension the Stein condition can be simplified: a connected [[Riemann surface]] is a Stein manifold [[if and only if]] it is not compact. This can be proved using a version of the [[Runge theorem]]<ref>{{cite journal |first=R. R. |last=Simha |title= The Behnke-Stein Theorem for Open Riemann Surfaces |journal=[[Proceedings of the American Mathematical Society]] |volume=105 |issue=4 |year=1989 |pages=876–880 |doi=10.1090/S0002-9939-1989-0953748-X |jstor=2047046|doi-access=free }}</ref> for Riemann surfaces,<ref group = note>The proof method uses an approximation by the [[Analytic polyhedron|polyhedral domain]], as in [[Oka-Weil theorem]].</ref> due to Behnke and Stein.{{R|Behnke–Stein1948|}}
* Every Stein manifold ''X'' is holomorphically spreadable, i.e. for every point <math>x \in X</math>, there are ''n'' holomorphic functions defined on all of ''X'' which form a local coordinate system when restricted to some open neighborhood of ''x''.
* The first Cousin problem can always be solved on a Stein manifold.
* Being a Stein manifold is equivalent to being a (complex) ''strongly pseudoconvex manifold''. The latter means that it has a strongly pseudoconvex (or [[plurisubharmonic function|plurisubharmonic]]) exhaustive function,{{R|Grauert1958|}} i.e. a smooth real function <math>\psi</math> on ''X'' (which can be assumed to be a [[Morse theory|Morse function]]) with <math>i \partial \bar \partial \psi >0</math>,{{R|Grauert1958|}} such that the subsets <math>\{z \in X \mid \psi (z)\leq c \}</math> are compact in ''X'' for every real number ''c''. This is a solution to the so-called '''Levi problem''',<ref>{{Eom| title = Levi problem | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47620}}</ref> named after [[Eugenio Elia Levi|E. E. Levi]] (1911). The function <math>\psi</math> invites a generalization of ''Stein manifold'' to the idea of a corresponding class of compact complex manifolds with boundary called '''Stein domain'''.<ref>{{cite journal |doi=10.2977/prims/1195183303|title=A Stein domain with smooth boundary which has a product structure|year=1982|last1=Ohsawa|first1=Takeo|journal=Publications of the Research Institute for Mathematical Sciences|volume=18|issue=3|pages=1185–1186|doi-access=free}}</ref> A Stein domain is the preimage <math>\{z \mid -\infty\leq\psi(z)\leq c\}</math>. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
*Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface ''X'' with a real-valued Morse function ''f'' on ''X'' such that, away from the critical points of ''f'', the field of complex tangencies to the preimage <math>X_c=f^{-1}(c)</math> is a [[Contact geometry|contact structure]] that induces an orientation on ''X<sub>c</sub>'' agreeing with the usual orientation as the boundary of <math>f^{-1}(-\infty, c).</math> That is, <math>f^{-1}(-\infty, c)</math> is a Stein [[Symplectic filling|filling]] of ''X<sub>c</sub>''.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" [[holomorphic function]]s taking values in the complex numbers. See for example [[Cartan's theorems A and B]], relating to [[sheaf cohomology]].

In the [[GAGA]] set of analogies, Stein manifolds correspond to [[affine variety|affine varieties]].<ref>{{cite journal |doi=10.2307/2007052|jstor=2007052 |last1=Neeman |first1=Amnon |title=Steins, Affines and Hilbert's Fourteenth Problem |journal=Annals of Mathematics |year=1988 |volume=127 |issue=2 |pages=229–244 }}</ref>

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is [[fibrant object|fibrant]] in the sense of so-called "holomorphic homotopy theory".

=== Complex projective varieties (compact complex manifold) ===
Meromorphic function in one-variable complex function were studied in a 
compact (closed) Riemann surface, because since the [[Riemann-Roch theorem]] ([[Riemann's inequality]]) holds for compact Riemann surfaces (Therefore the theory of compact Riemann surface can be regarded as the theory of (smooth (non-singular) projective) [[algebraic curve]] over <math>\mathbb{C}</math><ref>{{cite book |doi=10.1090/gsm/005|title=Algebraic Curves and Riemann Surfaces |series=Graduate Studies in Mathematics |year=1995 |volume=5 |isbn=9780821802687|first=Rick|last= Miranda|url={{Google books|aN4bfzgHvvkC|page=195|plainurl=yes}}}}</ref><ref>{{cite book |url={{Google books|FQslb7pH8EgC|page=130|plainurl=yes}} | title=Algebraic Geometry over the Complex Numbers | isbn=9781461418092 | last1=Arapura | first1=Donu | date=15 February 2012 | publisher=Springer }}</ref>). In fact, compact Riemann surface had a non-constant single-valued meromorphic function{{R|Weyl1913|}}, and also a compact Riemann surface had enough meromorphic functions. A compact one-dimensional complex manifold was a Riemann sphere <math>\widehat\mathbb{C} \cong \mathbb{CP}^1</math>. However, the abstract notion of a compact Riemann surface is always algebraizable (The [[Riemann's existence theorem]], [[Kodaira embedding theorem]].),<ref group=note>Note that the Riemann extension theorem and its references explained in the linked article includes a generalized version of the Riemann extension theorem by Grothendieck that was proved using the GAGA principle, also every one-dimensional compact complex manifold is a Hodge manifold.</ref> but it is not easy to verify which compact complex analytic spaces are algebraizable.<ref>{{cite book |doi=10.1007/978-3-642-60925-1_1|chapter=Cohomology of Algebraic Varieties |title=Algebraic Geometry II |series=Encyclopaedia of Mathematical Sciences |year=1996 |last1=Danilov |first1=V. I. |volume=35 |pages=1–125 |isbn=978-3-642-64607-2| url={{Google books|nDAiCQAAQBAJ|page=70|plainurl=yes}}}}</ref> In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.{{R|Ohsawa2021|}} However, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic.<ref>{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52 | url={{Google books|7z4mBQAAQBAJ|Algebraic Geometry|page=442|plainurl=yes}}|doi=10.1007/978-1-4757-3849-0| s2cid=197660097 }}</ref> The generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira,<ref>{{cite journal |doi=10.2307/2372120|jstor=2372120 |title=The Theorem of Riemann-Roch on Compact Analytic Surfaces |last1=Kodaira |first1=Kunihiko |journal=American Journal of Mathematics |year=1951 |volume=73 |issue=4 |pages=813–875 }}</ref> Kodaira also extended the theorem to three-dimensional,<ref>{{cite journal |doi=10.2307/1969802|jstor=1969802 |last1=Kodaira |first1=Kunihiko |title=The Theorem of Riemann-Roch for Adjoint Systems on 3-Dimensional Algebraic Varieties |journal=Annals of Mathematics |year=1952 |volume=56 |issue=2 |pages=298–342 }}</ref> and n-dimensional Kähler varieties.<ref>{{cite journal |jstor=88542 |last1=Kodaira |first1=Kunihiko |title=On the Theorem of Riemann-Roch for Adjoint Systems on Kahlerian Varieties |journal=Proceedings of the National Academy of Sciences of the United States of America |year=1952 |volume=38 |issue=6 |pages=522–527 |doi=10.1073/pnas.38.6.522 |pmid=16589138 |pmc=1063603 |bibcode=1952PNAS...38..522K |doi-access=free }}</ref> Serre formulated the  Riemann–Roch theorem as a problem of dimension of [[coherent sheaf cohomology]],{{R|IWS}} and also Serre proved [[Serre duality]].<ref>{{Citation | author1-last=Serre | author1-first=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Un théorème de dualité | journal=[[Commentarii Mathematici Helvetici]] | volume=29 | year=1955 | pages=9–26 | mr=0067489 | doi=10.1007/BF02564268| s2cid=123643759 |url=https://doi.org/10.5169/seals-23275}}</ref> Cartan and Serre proved the following property:<ref>{{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k3189t/f128.item| zbl=0050.17701 | title=Un théorème de finitude concernant les variétés analytiques compactes | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris | year=1953 | volume=237 | pages=128–130 | last1=Cartan | first1=Henri | last2=Serre | first2=Jean-Pierre }}</ref> the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M.<ref>{{cite book |doi=10.1007/BFb0093697|chapter=Vector bundles over complex manifolds |title=Holomorphic Vector Bundles over Compact Complex Surfaces |series=Lecture Notes in Mathematics |year=1996 |last1=Brînzănescu |first1=Vasile |volume=1624 |pages=1–27 |isbn=978-3-540-61018-2 }}</ref> Riemann–Roch on a Riemann surface for a vector bundle was proved by [[Vector bundles on algebraic curves|Weil]] in 1938.<ref>{{cite journal |doi=10.1515/crll.1938.179.129|url=http://eudml.org/doc/150043 |title=Zur algebraischen Theorie der algebraischen Funktionen. (Aus einem Brief an H. Hasse.) |year=1938 |last1=Weil |first1=A. |journal=Journal für die reine und angewandte Mathematik |volume=179 |pages=129–133|s2cid=116472982 }}</ref> 
[[Hirzebruch–Riemann–Roch theorem|Hirzebruch]] generalized the theorem to compact complex manifolds in 1994<ref>{{cite book |doi=10.1007/978-3-642-62018-8|title=Topological Methods in Algebraic Geometry |year=1966 |last1=Hirzebruch |first1=Friedrich |isbn=978-3-540-58663-0 }}</ref> and [[Grothendieck–Hirzebruch–Riemann–Roch theorem|Grothendieck]] generalized it to a relative version (relative statements about [[morphism]]s.).<ref>{{cite book | last = Berthelot | first = Pierre |editor=Alexandre Grothendieck |editor2=Luc Illusie | title = Théorie des Intersections et Théorème de Riemann-Roch | series = Lecture Notes in Mathematics | year = 1971 | volume = 225 | publisher = Springer Science+Business Media| pages = xii+700 |doi=10.1007/BFb0066283 |isbn= 978-3-540-05647-8}}</ref><ref>
{{Cite journal | last1=Borel | first1=Armand | last2=Serre | first2=Jean-Pierre | title=Le théorème de Riemann–Roch | mr=0116022 | year=1958 | journal=Bulletin de la Société Mathématique de France | volume=86 | pages=97–136 | doi=10.24033/bsmf.1500 | doi-access=free }}</ref> Next, the generalization of the result that "the compact Riemann surfaces are projective" to the high-dimension. In particular, consider the conditions that when embedding of compact complex submanifold ''X'' into the complex projective space <math>\mathbb{CP}^n</math>. <ref group=note>This is the standard method for compactification of <math>\mathbb{C}^n</math>, but not the only method like the Riemann sphere that was compactification of <math>\mathbb{C}</math>.</ref> The [[vanishing theorem]] (was first introduced by [[Kodaira vanishing theorem|Kodaira]] in 1953) gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of [[Positive form|positivity]]. As an application of this theorem, the [[Kodaira embedding theorem]]<ref>{{cite journal |last1=Kodaira |first1=K. |title=On Kahler Varieties of Restricted Type (An Intrinsic Characterization of Algebraic Varieties)|journal=Annals of Mathematics |series=Second Series |year=1954 |volume=60 |issue=1 |pages=28–48 |doi=10.2307/1969701|jstor=1969701 }}</ref> says that a compact [[Kähler manifold]] ''M'', with a Hodge metric, there is a complex-analytic embedding of ''M'' into [[complex projective space]] of enough high-dimension ''N''. In addition the [[GAGA#Chow's theorem|Chow's theorem]]<ref>{{cite journal |last1=Chow |first1=Wei-Liang |title=On Compact Complex Analytic Varieties |journal=American Journal of Mathematics |year=1949 |volume=71 |issue=2 |pages=893–914 |doi=10.2307/2372375|jstor=2372375 }}</ref> 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 [[Algebraic geometry and analytic geometry|GAGA principle]].{{R|GAGA}} The complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties. Then combined with Kodaira's result, a compact Kähler manifold ''M'' embeds as an algebraic variety. This result gives an example of a complex manifold with enough meromorphic functions. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as [[Hodge theory]]. Also, the [[deformation theory]] of compact complex manifolds has developed as Kodaira–Spencer theory. However, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic.<ref>{{cite journal |doi=10.2307/1969750|jstor=1969750|last1=Calabi|first1=Eugenio|last2=Eckmann|first2=Beno|title=A Class of Compact, Complex Manifolds Which are not Algebraic|journal=Annals of Mathematics|year=1953|volume=58|issue=3|pages=494–500}}</ref> Analogy of the Levi problems on the complex projective space <math>\mathbb{CP}^n</math> by Takeuchi.{{R|Siu1978}}<ref>{{cite journal |doi=10.1215/21562261-1625181|title=On the complement of effective divisors with semipositive normal bundle|year=2012|last1=Ohsawa|first1=Takeo|journal=Kyoto Journal of Mathematics|volume=52|issue=3|s2cid=121799985 |doi-access=free}}</ref><ref>{{cite journal |last1=Matsumoto |first1=Kazuko |title=Takeuchi's equality for the levi form of the Fubini–Study distance to complex submanifolds in complex projective spaces |journal=Kyushu Journal of Mathematics |date=2018 |volume=72 |issue=1 |pages=107–121 |doi=10.2206/kyushujm.72.107|doi-access=free }}</ref><ref>{{cite journal |doi=10.2969/jmsj/01620159|title=Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projecti|year=1964|last1=Takeuchi|first1=Akira|journal=Journal of the Mathematical Society of Japan|volume=16|issue=2|s2cid=122894640 |doi-access=free}}</ref>

==See also==
*[[Bicomplex number]]
*[[Complex geometry]]
*[[CR manifold]]
*[[Dolbeault cohomology]]
*[[Harmonic map]]s
*[[Harmonic morphism]]s
*[[Infinite-dimensional holomorphy]]
*[[Oka–Weil theorem]]

== Annotation ==
{{Reflist|group=note}}

==References==

=== Inline citations ===
{{Reflist}}

=== Textbooks ===
{{Refbegin|2}}
*{{cite book |doi=10.1007/978-3-642-99659-7|title=Theorie der Funktionen mehrerer komplexer Veränderlichen|year=1934|last1=Behnke|first1=H.|last2=Thullen|first2=P.|isbn=978-3-642-98844-8}}
*{{cite book | last1=Bochner | first1=S. | last2=Martin | first2=W.T.| title=Several Complex Variables | publisher=Princeton University Press | series=Princeton mathematical series | year=1948 | isbn=978-0-598-34865-4 }}
*{{cite book|last1=Forster | first1=Otto | author1-link= Otto Forster | title=Lectures on Riemann surfaces | publisher=Springer Verlag | location=New-York | series=Graduate Text in Mathematics | isbn=0-387-90617-7 | year=1981 | volume=81}}
*{{cite book|title=Введение в теорию аналитических функции многих комплексных переменных|last1=Фукс|first1=Б.А.|oclc=896179082|year=1962|language=ru}}
**{{cite book |isbn=978-1-4704-4428-0|title=Theory of Analytic Functions of Several Complex Variables|last1=Fuks|first1=Boris Abramovich|year=1963|publisher=American Mathematical Society }}
*{{cite book |last1=Cartan |first1=Henri |last2=Takahashi |first2=Reiji |title=Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes |date=1992 |publisher=Paris : Hermann |isbn=9782705652159 |page=231 |edition=6é. ed., nouv. tir |language=French}}
** {{cite book |last1=Cartan |first1=Henri |title=Elementary Theory of Analytic Functions of One or Several Complex Variables |date=1992 |publisher=Courier Corporation |isbn=9780486318677 |page=228 |url={{Google books|xUHDAgAAQBAJ|Elementary Theory of Analytic Functions of One or Several Complex Variables|plainurl=yes}}}}
*{{cite book |last1=Freitag |first1=Eberhard |author1-link= Eberhard Freitag |title=Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions |series=Universitext |date=2011 |publisher=Springer |doi=10.1007/978-3-642-20554-5 |isbn=978-3-642-20554-5 |edition=2 |url=https://link.springer.com/book/10.1007/978-3-642-20554-5}}
*{{cite book |doi=10.1007/978-3-642-22250-4|title=Stein Manifolds and Holomorphic Mappings |year=2011 |last1=Forstnerič |first1=Franc |series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |volume=56 |isbn=978-3-642-22249-8 }}
*{{citation|mr=0580152|last1=Grauert|first1= Hans|last2= Remmert|first2= Reinhold|title=Theory of Stein spaces|series=Grundlehren der Mathematischen Wissenschaften |volume=236|publisher= Springer-Verlag|place= Berlin-New York|year= 1979|isbn= 3-540-90388-7 }}
*{{cite book |last1=Шабат |first1=Б.В. |title=Введение в комплексный анализ / Vvedenie v kompleksnyĭ analiz |publisher=Nauka, Glav. red. fiziko-matematicheskoĭ lit-ry, Moskva|oclc=14003250|year=1985|language=ru}}
**{{cite book|doi=10.1090/mmono/110|title=Introduction to Complex Analysis Part II. Functions of Several Variables|series=Translations of Mathematical Monographs|year=1992|volume=110|isbn=9780821819753}}
*{{Citation |author =[[Lars Hörmander]] |date=1990 |orig-year=1966 |title=An Introduction to Complex Analysis in Several Variables |edition=3rd |publisher=North Holland |isbn=978-1-493-30273-4 |url={{Google books|MaM7AAAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
*{{cite book |isbn=978-0-8218-2165-7|title=Analytic Functions of Several Complex Variables|last1=Gunning|first1=Robert Clifford|last2=Rossi|first2=Hugo|year=2009|publisher=American Mathematical Soc. |url={{Google books|title=Analytic Functions of Several Complex Variables|wsqFAwAAQBAJ|plainurl=yes}}}}
*{{cite book |isbn=9783110838350|title=Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|last1=Kaup|first1=Ludger|last2=Kaup|first2=Burchard|date=9 May 2011|publisher=Walter de Gruyter |url={{Google books|4YVXCgewhTIC|Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|plainurl=yes}}}}
*{{Cite book|last=Kodaira |first=Kunihiko |title=Complex Manifolds and Deformation of Complex Structures|doi=10.1007/b138372 |series=Classics in Mathematics|date=17 November 2004 |publisher=Springer |isbn=3-540-22614-1|url={{Google books|AmvdBwAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
*{{cite book|last1=Krantz|first1=Steven G.|title=Function Theory of Several Complex Variables |date=1992 |publisher=AMS Chelsea Publishing |isbn=978-0-8218-2724-6 |page=340 |edition=Second|doi =10.1090/chel/340}}
*{{cite book |doi=10.1007/978-1-4757-1918-5|title=Holomorphic Functions and Integral Representations in Several Complex Variables|series=Graduate Texts in Mathematics|year=1986|volume=108|isbn=978-1-4419-3078-1}}
*{{cite book |doi=10.1007/3-7643-7491-8|title=Introduction to Complex Analysis in Several Variables|year=2005|isbn=3-7643-7490-X}}
*{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 }}
*{{cite book |isbn=9780486458120|title=Methods of the Theory of Functions of Many Complex Variables|last1=Vladimirov|first1=Vasiliy Sergeyevich|last2=Technica|first2=Scripta|date=January 2007|publisher=Courier Corporation |url={{Google books|JO4dlEr79GYC|Methods of the Theory of Functions of Many Complex Variables|plainurl=yes}}}}
{{Refend}}

=== Encyclopedia of Mathematics ===
{{Refbegin|2}}
*{{Eom| title = Analytic function | author-last1 = Gonchar| author-first1 = A.A.| author-last2 = Shabat| author-first2 =B.V.| oldid = 51340}}
*{{Eom| title = Power series | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 44404}}
*{{Eom| title = Biholomorphic mapping | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 33098}}
*{{Eom| title = Reinhardt domain | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 48495}}
*{{Eom| title = Hartogs theorem | author-last1 = Chirka| author-first1 = E.M.| oldid = 40754}}
*{{Eom| title =Domain of holomorphy| author-last1 = Gonchar| author-first1 = A.A.| author-last2 = Vladimirov| author-first2 =V.S.| oldid = 46763}}
*{{Eom| title = Pseudo-convex and pseudo-concave | author-last1 = Onishchik| author-first1 = A.L.| oldid = 48344}}
*{{Eom| title = Plurisubharmonic function | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 48192}}
*{{Eom| title = Quasi-coherent sheaf | author-last1 = Danilov| author-first1 = V.I.| oldid = 48377}}
*{{Eom| title = Coherent sheaf | author-last1 = Onishchik| author-first1 = A.L.| oldid = 30768}}
*{{Eom| title = Coherent analytic sheaf | author-last1 = Onishchik| author-first1 = A.L.| oldid = 33071}}
*{{Eom| title = Coherent algebraic sheaf | author-last1 = Danilov| author-first1 = V.I.| oldid = 41096}}
*{{Eom| title = Oka theorems | author-last1 = Chirka| author-first1 = E.M.| oldid = 44640}}
*{{Eom| title = Cousin problems | author-last1 = Chirka| author-first1 = E.M.| oldid = 46538}}
*{{Eom| title = Stein manifold | author-last1 = Onishchik| author-first1 = A.L.| oldid = 48831}}
*{{Eom| title = Finiteness theorems | author-last1 = Parshin| author-first1 = A.N.| oldid = 44303}}
{{Refend}}

== Further reading ==
{{Refbegin|2}}
*{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52|doi=10.1007/978-1-4757-3849-0| s2cid=197660097 }}
*{{Citation|last1=Krantz |first1=Steven G. |title=What is Several Complex Variables? |journal=The American Mathematical Monthly |year=1987 |volume=94 |issue=3 |pages=236–256 |doi=10.2307/2323391|jstor=2323391 }}
*{{cite journal |doi=10.2307/2316199|jstor=2316199 |last1=Seebach |first1=J. Arthur |last2=Seebach |first2=Linda A. |last3=Steen |first3=Lynn A. |title=What is a Sheaf? |journal=The American Mathematical Monthly |year=1970 |volume=77 |issue=7 |pages=681–703 }}
*{{Citation|first1=Kiyoshi|last1=Oka | editor1-last=Remmert|editor1-first=R.|title=Collected Papers |date=1984 |publisher=Springer-Verlag Berlin Heidelberg |isbn=978-3-662-43412-3 |page=XIV, 226}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10013-X|title=Scientific report on the second summer institute, several complex variables. Part I. Report on the analysis seminar|year=1956|last1=Martin|first1=W. T.|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=79–102|doi-access=free}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10015-3|title=Scientific report on the second summer institute, several complex variables. Part II. Complex manifolds|year=1956|last1=Chern|first1=Shiing-Shen|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=101–118|doi-access=free}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10018-9|title=Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory|year=1956|last1=Zariski|first1=Oscar|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=117–142|doi-access=free}}
*{{cite journal |last1=Remmert |first1=Reinhold |title=From Riemann Surfaces to Complex Spaces |journal=Séminaires et Congrès |date=1998|zbl=1044.01520|url=https://www.emis.de/journals/SC/1998/3/pdf/smf_sem-cong_3_203-241.pdf}}
{{Refend}}

==External links==
* [https://www.jirka.org/scv/ Tasty Bits of Several Complex Variables] open source book by Jiří Lebl
* [https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf Complex Analytic and Differential Geometry] ([https://www-fourier.ujf-grenoble.fr/~demailly/documents.html OpenContent book See B2])
*{{cite book
 | last = Demailly | first = Jean-Pierre
 | editor1-last = Harinck | editor1-first = Pascale
 | editor2-last = Plagne | editor2-first = Alain
 | editor3-last = Sabbah | editor3-first = Claude
 | contribution = Henri Cartan et les fonctions holomorphes de plusieurs variables
 | contribution-url = https://www.math.polytechnique.fr/xups/xups12-03.pdf
 | isbn = 978-2-7302-1610-4
 | location = Palaiseau
 | pages = 99–168
 | publisher = Les Éditions de l'École Polytechnique
 | title = Henri Cartan et André Weil. Mathématiciens du XXesiècle. Journées mathématiques X-UPS, Palaiseau, France, May 3–4, 2012
 | year = 2012}}
* Victor Guillemin. 18.117 [https://ocw.mit.edu/courses/mathematics/18-117-topics-in-several-complex-variables-spring-2005/lecture-notes Topics in Several Complex Variables]. Spring 2005. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons [[BY-NC-SA]].
*{{PlanetMath attribution
 |urlname=ReinhardtDomain |title=Reinhardt domain
 |urlname2=Holomorphicallyconvex |title2=Holomorphically convex
 |urlname3=Domainofholomorphy |title3=Domain of holomorphy
 |urlname4=polydisc |title4=polydisc
 |urlname5=biholomorphicallyequivalent |title5=biholomorphically equivalent
 |urlname6=Levipseudoconvex |title6=Levi pseudoconvex
 |urlname7=pseudoconvex |title7=Pseudoconvex
 |urlname8=ExhaustionFunction |title8=exhaustion function
}}
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