Function of several complex variables#Top

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem.{{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 }} Naturally also same function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged field in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory.

With work of Friedrich Hartogs, {{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, Wilhelm Wirtinger and Francesco Severi. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function

$$f\; :\; \backslash mathbb\; C^n\; \backslash to\; \backslash Complex$$

whenever {{math|1=n > 1}}. Naturally the analogues of 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 $\backslash Complex$ 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 $\backslash mathbb\; C^n$ and Stein manifolds, satisfying a condition called pseudoconvexity). The natural domains of definition of functions, continued to the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish, 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 analytic geometry, {{refn|group=note|A name adopted, confusingly, for the geometry of zeroes of analytic functions; this is not the analytic geometry learned at school. (In other words, in the sense of GAGA on Serre.){{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}}}} automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper GAGA of Serre{{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 forms. The classical candidates are the Hilbert modular forms and Siegel modular forms. These days these are associated to algebraic groups (respectively the Weil restriction from a totally real number field of {{math|GL(2)}}, and the symplectic group), for which it happens that automorphic representations can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

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

The complex coordinate space $\backslash mathbb\; C^n$ is the Cartesian product of {{mvar|n}} copies of $\backslash mathbb\; C$, and when $\backslash mathbb\; C^n$ is a domain of holomorphy, $\backslash mathbb\; C^n$ can be regarded as a Stein manifold, and more generalized Stein space. $\backslash mathbb\; C^n$ is also considered to be a complex projective variety, a Kähler manifold,{{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}} etc. It is also an dimension (vector space) over the complex numbers, which gives its dimension {{math|2n}} over $\backslash mathbb\; R$.The field of complex numbers is a 2-dimensional vector space over real numbers. Hence, as a set and as a topological space, $\backslash mathbb\; C^n$ may be identified to the real coordinate space $\backslash mathbb\; R^\{2n\}$ and its topological dimension is thus {{math|2n}}.

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

Any such space, as a real space, is oriented. On the complex plane thought of as a Cartesian plane, multiplication by a complex number {{math|1=w = u + iv}} may be represented by the real matrix

:$\backslash begin\{pmatrix\}$

u & -v \\

v & u

\end{pmatrix},

with determinant

:$u^2\; +\; v^2\; =\; |w|^2.$

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be composed from 2 × 2 blocks of the aforementioned form), then its determinant equals to the square of absolute value of the corresponding complex determinant. It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator. The same applies to Jacobians of holomorphic functions from $\backslash mathbb\; C^n$ to $\backslash mathbb\; C^n$.

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

f(z+h) = f(z) + L(h) + o(\lVert h\rVert)

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 :

$z\; \backslash mapsto\; f(z\_1,\backslash dots,z\_\{i-1\},z,z\_\{i+1\},\backslash dots,z\_n)$

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 Hartog's theorem, or as Osgood's lemma under the additional hypothesis that f is continuous.

In one complex variable, a function $f:\backslash mathbb\{C\}\backslash to\; \backslash mathbb\{C\}$ defined on the plane is holomorphic at a point $p\backslash in\; \backslash mathbb\{C\}$ if and only if its real part $u$ and its imaginary part $v$ satisfy the so-called Cauchy-Riemann equations at $p$ :

$$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}(p)\; =\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; y\}(p)\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; (p)=-\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}(p)$$

In several variables, a function $f:\backslash mathbb\{C\}^n\backslash to\; \backslash mathbb\{C\}$ is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part $u$ and the imaginary part $v$ of $f$ satisfiy the Cauchy Riemann equations :

$$\backslash forall\; i\backslash in\; \backslash \{1,\backslash dots,n\backslash \},\backslash quad\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\_i\}\; =\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; y\_i\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\_i\}\; =\; -\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\_i\}$$

Using the formalism of Wirtinger derivatives, this can be reformulated as :

$$\backslash forall\; i\backslash in\; \backslash \{1,\backslash dots,n\backslash \},\backslash quad\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash overline\{z\_i\}\}\; =\; 0,$$

or even more compactly using the formalism of complex differential forms, as :

$$\backslash bar\backslash partial\; f=0.$$

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 $\backslash gamma$, $\backslash gamma\_\backslash nu$ is piecewise smoothness, class $\backslash mathcal\{C\}^1$ Jordan closed curve. ($\backslash nu=1,2,\backslash ldots,n$) Let $D\_\backslash nu$ be the domain surrounded by each $\backslash gamma\_\backslash nu$. Cartesian product closure $\backslash overline\{D\_1\backslash times\; D\_2\backslash times\backslash cdots\backslash times\; D\_n\}$ is $\backslash overline\{D\_1\}\backslash times\; \backslash overline\{D\_2\}\backslash times\backslash cdots\backslash times\; \backslash overline\{D\_n\}\; \backslash in\; D$. Also, take the closed polydisc $\backslash overline\{\backslash Delta\}$ so that it becomes $\backslash overline\{\backslash Delta\}\backslash subset\{D\_1\; \backslash times\; D\_2\; \backslash times\; \backslash cdots\; \backslash times\; D\_n\}$. $\backslash overline\{\backslash Delta\}(z,r)\; =\; \backslash left\backslash \{\backslash zeta=(\backslash zeta\_1,\; \backslash zeta\_2,\; \backslash dots,\; \backslash zeta\_n)\backslash in\; \backslash Complex^n\; ;\; \backslash left|\backslash zeta\_\backslash nu\; -\; z\_\backslash nu\backslash right|\; \backslash leq\; r\_\backslash nu\; \backslash text\{\; for\; all\; \}\; \backslash nu\; =\; 1,\backslash dots,n\backslash right\backslash \}$ and let $\backslash \{z\_\backslash nu\; \backslash \}^n\_\{\backslash nu=1\}$ be the center of each disk.) Using the Cauchy's integral formula of one variable repeatedly, Note that this formula only holds for polydisc. See §Bochner–Martinelli formula for the Cauchy's integral formula on the more general domain.

: $$

\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}

Because $\backslash partial\; D$ is a rectifiable Jordanian closed curveAccording to the Jordan curve theorem, domain D is bounded closed set, that is, each domain $D\_\backslash nu$ is compact. 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|::|$f(z\_1,\backslash dots,z\_n)=\backslash frac\{1\}\{(2\backslash pi\; i)^n\}\backslash int\_\{\backslash partial\; D\_1\}\backslash cdots\backslash int\_\{\backslash partial\; D\_n\}\backslash frac\{f(\backslash zeta\_1,\backslash dots,\backslash zeta\_n)\}\{(\backslash zeta\_1\; -\; z\_1)\; \backslash cdots\; (\backslash zeta\_n\; -\; z\_n)\}\; \backslash ,\; d\backslash zeta\_1\backslash cdots\; d\backslash zeta\_n$|{{EquationRef|1}}}}

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

{{NumBlk|::|$\backslash frac\{\backslash partial^\{k\_1\; +\; \backslash cdots\; +\; k\_n\}f(z\_1,z\_2,\backslash ldots,z\_n)\}\{\backslash partial\{z\_1\}^\{k\_1\}\; \backslash cdots\; \backslash partial\{z\_n\}^\{k\_n\}\; \}\; =\; \backslash frac\{k\_1!\; \backslash cdots\; k\_n!\}\{(2\backslash pi\; i)^n\}\; \backslash int\_\{\backslash partial\; D\_1\}\; \backslash cdots\; \backslash int\_\{\backslash partial\; D\_n\}\; \backslash frac\{f(\backslash zeta\_1,\backslash dots,\backslash zeta\_n)\}\{(\backslash zeta\_1\; -\; z\_1)^\{k\_1+1\}\; \backslash cdots\; (\backslash zeta\_n\; -\; z\_n)^\{k\_n\; +\; 1\}\; \}\; \backslash ,\; d\backslash zeta\_1\backslash cdots\; d\backslash zeta\_n.$|{{EquationRef|2}}}}

f is class $\backslash mathcal\{C\}^\{\backslash infty\}$-function.

From (2), if f is holomorphic, on polydisc $\backslash left\backslash \{\; \backslash zeta=(\backslash zeta\_1,\; \backslash zeta\_2,\; \backslash dots,\; \backslash zeta\_n)\; \backslash in\; \backslash Complex^n\; ;\; |\; \backslash zeta\_\backslash nu\; -\; z\_\backslash nu\; |\; \backslash leq\; r\_\backslash nu,\; \backslash text\{\; for\; all\; \}\; \backslash nu\; =\; 1,\backslash dots,n\; \backslash right\backslash \}$ and $|f|\; \backslash leq\; \{M\}$, the following evaluation equation is obtained.

: $\backslash left|\backslash frac\{\backslash partial^\{k\_1\; +\; \backslash cdots\; +\; k\_n\}\; f(\backslash zeta\_1,\backslash zeta\_2,\backslash ldots,\backslash zeta\_n)\}\{\{\backslash partial\; z\_1\}^\{k\_1\}\; \backslash cdots\; \backslash partial\; \{z\_n\}^\{k\_n\}\}\; \backslash right|\; \backslash leq\; \backslash frac\{Mk\_1!\; \backslash cdots\; k\_n!\}\{\{r\_1\}^\{k\_1\}\; \backslash cdots\; \{r\_n\}^\{k\_n\}\}$

Therefore, Liouville's theorem hold.

If function f is holomorphic, on polydisc , from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

: $\backslash begin\{align\}$

& 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}

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

For each point $a=(a\_1,\backslash dots,a\_n)\backslash in\; D\; \backslash subset\; \backslash mathbb\; C^n$, $f(z)$ is expressed as a power series expansion that is convergent on D :

: $f(z)=\backslash sum\_\{k\_1,\backslash dots,k\_n=0\}^\backslash infty\; c\_\{k\_1,\backslash dots,k\_n\}(z\_1\; -\; a\_1)^\{k\_1\}\backslash cdots(z\_n\; -\; a\_n)^\{k\_n\}\backslash \; ,$

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 $f\_1,\backslash ldots,f\_n$ which converges uniformly on compacta inside a domain D, the limit function f of $f\_v$ also uniformly on compacta inside a domain D. Also, respective partial derivative of $f\_v$ also compactly converges on domain D to the corresponding derivative of f.

:$\backslash frac\{\backslash partial^\{k\_1\; +\; \backslash cdots\; +\; k\_n\}f\}\{\backslash partial\{z\_1\}^\{k\_1\}\; \backslash cdots\; \backslash partial\{z\_n\}^\{k\_n\}\}\; =\; \backslash sum\_\{v=1\}^\backslash infty\; \backslash frac\{\backslash partial^\{k\_1\; +\; \backslash cdots\; +\; k\_n\}\; f\_v\}\{\backslash partial\{z\_1\}^\{k\_1\}\; \backslash cdots\; \backslash partial\{z\_n\}^\{k\_n\}\}${{Eom| title = Weierstrass theorem | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 49192}}

It is possible to define a combination of positive real numbers $\backslash \{r\_\backslash nu\; \backslash \; (\backslash nu\; =\; 1,\backslash dots,n)\; \backslash \}$ such that the power series $\backslash sum\_\{k\_1,\backslash dots,k\_n=0\}^\backslash infty\; c\_\{k\_1,\backslash dots,k\_n\}(z\_1-a\_1)^\{k\_1\}\backslash cdots(z\_n-a\_n)^\{k\_n\}\backslash$ converges uniformly at and does not converge uniformly at $\backslash left\backslash \{\; z=(z\_1,\; z\_2,\; \backslash dots,\; z\_n)\; \backslash in\; \backslash Complex^n\; ;\; |\; z\_\backslash nu\; -\; a\_\backslash nu\; |\; >\; r\_\backslash nu,\; \backslash text\{\; for\; all\; \}\; \backslash nu\; =\; 1,\backslash dots,n\; \backslash right\backslash \}$.

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

Let $\backslash omega(z)$ be holomorphic in the annulus

: $$

\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}

The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus , where $r\text{'}\_\backslash nu\; >\; r\_\backslash nu$ and , and so it is possible to integrate term.{{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}}

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 $\backslash mathbb\; C^n$ with piecewise smooth boundary $\backslash partial\; D$, and let the symbol $\backslash land$ 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 $\backslash zeta$, z in $\backslash mathbb\; C^n$ the Bochner–Martinelli kernel $\backslash omega(\backslash zeta,z)$ is a differential form in $\backslash zeta$ of bidegree $(n,n-1)$, defined by

:$\backslash omega(\backslash zeta,z)\; =\; \backslash frac\{(n-1)!\}\{(2\backslash pi\; i)^n\}\backslash frac\{1\}\{|z-\backslash zeta|^\{2n\}\}$

\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

:$\backslash displaystyle\; f(z)\; =\; \backslash int\_\{\backslash partial\; D\}f(\backslash zeta)\backslash omega(\backslash zeta,\; z)\; -\; \backslash int\_D\backslash overline\backslash partial\; f(\backslash zeta)\backslash land\backslash omega(\backslash zeta,z).$

In particular if f is holomorphic the second term vanishes, so

:$\backslash displaystyle\; f(z)\; =\; \backslash int\_\{\backslash partial\; D\}f(\backslash zeta)\backslash omega(\backslash zeta,\; z).$

Holomorphic functions of several complex variables satisfy an identity theorem, as in one variable : two holomorphic functions defined on the same connected open set $D\backslash subset\; \backslash mathbb\{C\}^n$ 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 $f(z\_1,z\_2)=0$ and $g(z\_1,z\_2)=z\_1$coincide on the whole complex line of $\backslash mathbb\{C\}^2$ defined by the equation $z\_1=0$.

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.

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 $\backslash Complex^n$, the bijective holomorphic function $\backslash phi:U\backslash to\; V$ and the inverse mapping $\backslash phi^\{-1\}:V\backslash to\; U$ is also holomorphic. At this time, $\backslash phi$ is called a U, V biholomorphism also, we say that U and V are biholomorphically equivalent or that they are biholomorphic.

When $n\; >\; 1$, open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two.{{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}} This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.{{R|Chen2000}}{{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 }} 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.{{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}}

Let U, V be domain on $\backslash mathbb\{C\}^n$, such that $f\; \backslash in\; \backslash mathcal\{O\}(U)$ and $g\; \backslash in\; \backslash mathcal\{O\}(V)$, ($\backslash mathcal\{O\}(U)$ is the set/ring of holomorphic functions on U.) assume that $U,\backslash \; V,\backslash \; U\; \backslash cap\; V\; \backslash ne\; \backslash varnothing$ and $W$ is a connected component of $U\; \backslash cap\; V$. If $f|\_W\; =g|\_W$ then f is said to be connected to V, and g is said to be analytic continuation of f. 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 $\backslash partial\; U$: there exists domain U, V, such that all holomorphic functions $f$ over the domain U, have an analytic continuation $g\; \backslash in\; \backslash mathcal\{O\}(V)$. In other words, there may not exist a function $f\; \backslash in\; \backslash mathcal\{O\}(U)$ such that $\backslash partial\; U$ as the natural boundary. This is called the Hartogs's phenomenon. Therefore, investigating when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. In addition, if $n\; \backslash geq\; 2$, 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.

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 $D\; \backslash subset\; \backslash Complex^n$ ($n\; \backslash geq\; 1$) to be a domain, with centre at a point $a\; =\; (a\_1,\backslash dots,a\_n)\; \backslash in\; \backslash Complex^n$, such that, together with each point $z^0\; =\; (z\_1^0,\backslash dots,z\_n^0)\backslash in\; D$, the domain also contains the set

: $\backslash left\backslash \{\; z\; =\; (z\_1,\; \backslash dots,\; z\_n)\; ;\; \backslash left|z\_\backslash nu\; -\; a\_\backslash nu\; \backslash right|\; =\; \backslash left|z\_\backslash nu^0\; -\; a\_\backslash nu\backslash right|,\backslash \; \backslash nu\; =\; 1,\; \backslash dots,\; n\; \backslash right\backslash \}\; .$

A domain D is called a Reinhardt domain if it satisfies the following conditions:{{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}}}}{{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}}

Let $\backslash theta\_\backslash nu\; \backslash ;(\backslash nu\; =\; 1,\backslash dots,n)$ is a arbitrary real numbers, a domain D is invariant under the rotation: $\backslash left\backslash \{z^0\; -\; a\_\backslash nu\; \backslash right\backslash \}\; \backslash to\; \backslash left\backslash \{e^\{i\backslash theta\_\backslash nu\}\; (z\_\backslash nu^0\; -\; a\_\backslash nu)\; \backslash right\backslash \}$.

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

:

A Reinhardt domain D is called a complete Reinhardt domain with centre at a point a if together with all point $z^0\backslash in\; D$ it also contains the polydisc

: $$

\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\}.

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

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 $\backslash lambda(D^\{*\})$ of the set

: $$

D ^{*} = \{ z = (z_1, \dots, z_n) \in D ; z_1, \dots, z_n \neq 0 \}

under the mapping

: $$

\lambda ; z \rightarrow \lambda(z) = (\ln|z_1|, \dots, \ln |z_n|)

is a convex set in the real coordinate space $\backslash R^n$.

Every such domain in $\backslash Complex^n$ is the interior of the set of points of absolute convergence of some power series in $\backslash sum\_\{k\_1,\backslash dots,k\_n=0\}^\backslash infty\; c\_\{k\_1,\backslash dots,k\_n\}(z\_1\; -\; a\_1)^\{k\_1\}\backslash cdots(z\_n\; -\; a\_n)^\{k\_n\}\backslash$, and conversely; The domain of convergence of every power series in $z\_1,\backslash dots,z\_n$ is a logarithmically-convex Reinhardt domain with centre $a\; =\; 0$.

When described using the domain of holomorphy, which is a generalization of the convergence domain, a Reinhardt domain is a domain of holomorphy if and only if logarithmically convex. But, there is an example of a complete Reinhardt domain D which is not logarithmically convex.{{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}}}}

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 $\backslash mathbb\{C\}^n$ were all connected to larger domain.{{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 }}

:On the polydisk consisting of two disks when .

:Internal domain of

::Hartogs's extension theorem (1906);{{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

}} Let f be a holomorphic function on a set {{math|G \ 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.{{cite arXiv|eprint=1608.00950|last1=Simonič|first1=Aleksander|title=Elementary approach to the Hartogs extension theorem|year=2016|class=math.CV }}}} on $\backslash Complex^n$ ({{math|n ≥ 2}}) and K is a compact subset of G. If the complement {{math|G \ K}} is connected, then every holomorphic function f regardless of how it is chosen can be each extended to a unique holomorphic function on G.{{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}}{{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.{{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}}{{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}}

From Hartogs's extension theorem the domain of convergence extends from $H\_\backslash varepsilon$ to $\backslash Delta^2$. Looking at this from the perspective of the Reinhardt domain, $H\_\backslash varepsilon$ is the Reinhardt domain containing the center z = 0, and the domain of convergence of $H\_\backslash varepsilon$ has been extended to the smallest complete Reinhardt domain $\backslash Delta^2$ containing $H\_\backslash varepsilon$.{{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}}

Thullen's{{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}} classical result says that a 2-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

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

Toshikazu Sunada (1978){{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}} established a generalization of Thullen's result:

:Two n-dimensional bounded Reinhardt domains $G\_1$ and $G\_2$ are mutually biholomorphic if and only if there exists a transformation $\backslash varphi:\backslash Complex^n\backslash to\; \backslash Complex^n$ given by $z\_i\backslash mapsto\; r\_iz\_\{\backslash sigma(i)\}\; (r\_i>0)$, $\backslash sigma$ being a permutation of the indices), such that $\backslash varphi(G\_1)=G\_2$.

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 $\backslash Complex^n$ call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen.{{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}} Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for $\backslash Complex^2$,{{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}} later extended to $\backslash Complex^n$.{{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 }}{{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 }}){{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 }} 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{{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}} the contents of these two papers are different.{{cite web |last1=Noguchi |first1=J.|url=https://www.ms.u-tokyo.ac.jp/~noguchi/oka/|title=Related to Works of Dr. Kiyoshi OKA}}}}}}{{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}} notion of idéal de domaines indéterminés is interpreted theory of sheaf cohomology by

H. Cartan and more development Serre.The idea of the sheaf itself is by Jean Leray.{{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}}{{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}}{{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/}}{{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 }}{{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}}}}{{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}}}}{{R|IWS}} In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.{{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 }} 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}}

{{Main|Domain of holomorphy}}

Image:Domain of holomorphy illustration.svg

When a function f is holomorpic on the domain $D\backslash subset\; \backslash Complex^n$ and cannot directly connect to the domain outside D, including the point of the domain boundary $\backslash partial\; D$, 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 $D\backslash subset\; \backslash Complex^n\backslash \; (n\backslash geq\; 2)$, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.{{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}}}}

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

For the $n=1$ case, every domain ($D\backslash subset\backslash mathbb\{C\}$) is a domain of holomorphy; we can find a holomorphic function that is not identically 0, but whose zeros accumulate everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal.

• If $D\_1,\; \backslash dots,\; D\_n$ are domains of holomorphy, then their intersection $D\; =\; \backslash bigcap\_\{\backslash nu=1\}^n\; D\_\backslash nu$ is also a domain of holomorphy.
• If $D\_1\; \backslash subseteq\; D\_2\; \backslash subseteq\; \backslash cdots$ is an increasing sequence of domains of holomorphy, then their union $D\; =\; \backslash bigcup\_\{n=1\}^\backslash infty\; D\_n$ is also a domain of holomorphy (see Behnke–Stein theorem).{{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 }}
• If $D\_1$ and $D\_2$ are domains of holomorphy, then $D\_1\; \backslash times\; D\_2$ is a domain of holomorphy.
• The first Cousin problem is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for $n\backslash geq\; 3$.{{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 }} this is also true, with additional topological assumptions, for the second Cousin problem.

Let $G\; \backslash subset\; \backslash Complex^n$ be a domain, or alternatively for a more general definition, let $G$ be an $n$ dimensional complex analytic manifold. Further let $\{\backslash mathcal\{O\}\}(G)$ stand for the set of holomorphic functions on G. For a compact set $K\; \backslash subset\; G$, the holomorphically convex hull of K is

:$\backslash hat\{K\}\_G\; :=\; \backslash left\; \backslash \{\; z\; \backslash in\; G\; ;\; |f(z)|\; \backslash leq\; \backslash sup\_\{w\; \backslash in\; K\}\; |f(w)|\; \backslash text\{\; for\; all\; \}\; f\; \backslash in\; \backslash mathcal\{O\}(G)\; .\; \backslash right\; \backslash \}\; .$

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

The domain $G$ is called holomorphically convex if for every compact subset $K,\; \backslash hat\{K\}\_G$ is also compact in G. Sometimes this is just abbreviated as holomorph-convex.

When $n=1$, every domain $G$ is holomorphically convex since then $\backslash hat\{K\}\_G$ is the union of K with the relatively compact components of $G\; \backslash setminus\; K\; \backslash subset\; G$.

When $n\backslash geq\; 1$, if f satisfies the above holomorphic convexity on D it has the following properties. $\backslash text\{dist\}\; (K,\; D^c)\; =\; \backslash text\{dist\}\; (\backslash hat\{K\}\_D,\; D^c$

) for every compact subset K in D, where

$\backslash text\{dist\}\; (K,\; D^c)$ denotes the distance between K and $D^c\; =\; \backslash mathbb\{C\}^n\; \backslash setminus\; D$. Also, at this time, D is a domain of holomorphy. Therefore, every convex domain $(D\backslash subset\backslash Complex^n)$ is domain of holomorphy.{{R|Chen2000}}

Hartogs showed that

{{blockquote|Hartogs (1906):{{R|Hartogs1906}} Let D be a Hartogs's domain on $\backslash mathbb\{C\}$ and R be a positive function on D such that the set $\backslash Omega$ in $\backslash mathbb\{C\}^2$ defined by $z\_1\; \backslash in\; D$ and is a domain of holomorphy. Then $-\backslash log\; \{R\}\; (z\_1)$ 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 $\backslash Complex^n$ 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.{{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}}

:A function

:$f\; \backslash colon\; D\; \backslash to\; \{\backslash mathbb\{R\}\}\backslash cup\backslash \{-\backslash infty\backslash \},$

:with domain $D\; \backslash subset\; \{\backslash mathbb\{C\}\}^n$

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

:$\backslash \{\; a\; +\; b\; z\; ;\; z\; \backslash in\; \backslash mathbb\{C\}\; \backslash \}\backslash subset\; \backslash mathbb\{C\}^n$ with $a,\; b\; \backslash in\; \backslash mathbb\{C\}^n$

:the function $z\; \backslash mapsto\; f(a\; +\; bz)$ is a subharmonic function on the set

:$\backslash \{\; z\; \backslash in\; \backslash mathbb\{C\}\; ;\; a\; +\; b\; z\; \backslash in\; D\; \backslash \}.$

:In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space $X$ as follows. An upper semi-continuous function

:$f\; \backslash colon\; X\; \backslash to\; \backslash mathbb\{R\}\; \backslash cup\; \backslash \{\; -\; \backslash infty\; \backslash \}$

:is said to be plurisubharmonic if and only if for any holomorphic map

$\backslash varphi\backslash colon\backslash Delta\; \backslash to\; X$ the function

:$f\backslash circ\backslash varphi\; \backslash colon\; \backslash Delta\; \backslash to\; \backslash mathbb\{R\}\; \backslash cup\; \backslash \{\; -\backslash infty\; \backslash \}$

is subharmonic, where $\backslash Delta\; \backslash subset\; \backslash mathbb\{C\}$ denotes the unit disk.

In one-complex variable, necessary and sufficient condition that the real-valued function $u=u(z)$, that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is $\backslash Delta=4\backslash left(\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; z\backslash ,\backslash partial\backslash overline\{z\}\}\backslash right)\backslash geq0$. Therefore, if $u$ is of class $\backslash mathcal\{C\}^2$, then $u$ is plurisubharmonic if and only if the hermitian matrix $H\_u=(\backslash lambda\_\{ij\}),\backslash lambda\_\{ij\}=\backslash frac\{\backslash partial^2u\}\{\backslash partial\; z\_i\backslash ,\backslash partial\backslash bar\; z\_j\}$ is positive semidefinite.

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

When the hermitian matrix of u is positive-definite and class $\backslash mathcal\{C\}^2$, we call u a strict plurisubharmonic function.

Weak pseudoconvex is defined as : Let $X\backslash subset\; \{\backslash mathbb\{C\}\}^n$ be a domain. One says that X is pseudoconvex if there exists a continuous plurisubharmonic function $\backslash varphi$ on X such that the set $\backslash \{\; z\; \backslash in\; X\; ;\; \backslash varphi(z)\; \backslash leq\; \backslash sup\; x\; \backslash \}$ is a relatively compact subset of X for all real numbers x. 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. i.e. there exists a smooth plurisubharmonic exhaustion function $\backslash psi\; \backslash in\; \backslash text\{Psh\}(X)\backslash cap\backslash mathcal\{C\}^\{\backslash infty\}(X)$. 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 $\backslash psi\; \backslash in\; \backslash text\{Psh\}(X)\backslash cap\backslash mathcal\{C\}^\{\backslash infty\}(X)$.{{R|CAaDG|}}{{rp|page=49}}

Let X be a complex n-dimensional manifold. Strongly (or Strictly) pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function $\backslash psi\; \backslash in\; \backslash text\{Psh\}(X)\backslash cap\backslash mathcal\{C\}^\{\backslash infty\}(X)$, i.e., $H\backslash psi$ 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{{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 }}) are often used interchangeably,{{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. }} see Lempert{{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 }} for the technical difference.

If $\backslash mathcal\{C\}^2$ boundary , it can be shown that D has a defining function; i.e., that there exists $\backslash rho:\; \backslash mathbb\{C\}^n\; \backslash to\; \backslash mathbb\{R\}$ which is $\backslash mathcal\{C\}^2$ so that , and $\backslash partial\; D\; =\; \backslash \{\backslash rho\; =0\backslash \}$. Now, D is pseudoconvex iff for every $p\; \backslash in\; \backslash partial\; D$ and $w$ in the complex tangent space at p, that is,

:$\backslash nabla\; \backslash rho(p)\; w\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; \backslash rho\; (p)\}\{\; \backslash partial\; z\_j\; \}w\_j\; =0$, we have

:$H(\backslash rho)\; =\; \backslash sum\_\{i,j=1\}^n\; \backslash frac\{\backslash partial^2\; \backslash rho(p)\}\{\backslash partial\; z\_i\; \backslash ,\; \backslash partial\; \backslash bar\{z\_j\}\; \}\; w\_i\; \backslash bar\{w\_j\}\; \backslash geq\; 0.${{R|Chen2000}}{{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 }}

If D does not have a $\backslash mathcal\{C\}^2$ boundary, the following approximation result can be useful.

Proposition 1 If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains $D\_k\; \backslash subset\; D$ with class $\backslash mathcal\{C\}^\backslash infty$-boundary which are relatively compact in D, such that

:$D\; =\; \backslash bigcup\_\{k=1\}^\backslash infty\; D\_k.$

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

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}}

If for every boundary point $\backslash rho$ of D, there exists an analytic variety $\backslash mathcal\{B\}$ passing $\backslash rho$ which lies entirely outside D in some neighborhood around $\backslash rho$, except the point $\backslash rho$ itself. Domain D that satisfies these conditions is called Levi total pseudoconvex.{{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}}

Let n-functions $\backslash varphi:z\_j\; =\; \backslash varphi\_j(u,\; t)$ be continuous on $\backslash Delta:|U|\backslash leq1,\; 0\backslash leq\; t\backslash leq1$, holomorphic in when the parameter t is fixed in [0, 1], and assume that $\backslash frac\{\backslash partial\backslash varphi\_j\}\{\backslash partial\; u\}$ are not all zero at any point on $\backslash Delta$. Then the set $Q(t):=\; \backslash \{Z\_j=\; \backslash varphi\_j(u,\; t);|u|\backslash leq\; 1\backslash \}$ is called an analytic disc de-pending on a parameter t, and $B(t):=\; \backslash \{Z\_j=\; \backslash varphi\_j(u,\; t);|u|=\; 1\backslash \}$ is called its shell. If Q(t) is called Family of Oka's disk.{{R|Hitomatsu1958|}}{{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}}

When $Q(0)\backslash subset\; D$ 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 $\backslash mathbb\{C\}^n${{Eom| title = Riemannian domain | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 44356}} 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|}}

For every point $x\; \backslash in\; \backslash partial\; D$ there exist a neighbourhood U of x and f holomorphic. ( i.e. $U\; \backslash cap\; D$ be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. i.e., let $\backslash psi\; :\; X\; \backslash to\; Y$ be a holomorphic map, if every point $y\backslash in\; Y$ has a neighborhood U such that $\backslash psi^\{-1\}(U)$ admits a $\backslash mathcal\{C\}^\{\backslash infty\}$-plurisubharmonic exhaustion function (weakly 1-complete{{cite journal |doi=10.5802/aif.3226|title=On the local pseudoconvexity of certain analytic families of $\backslash mathbb\{C\}$|year=2018|last1=Ohsawa|first1=Takeo|journal=Annales de l'Institut Fourier|volume=68|issue=7|pages=2811–2818|doi-access=free}}), 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 $\backslash Complex^n$ the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.{{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}}{{R|Hitomatsu1958|}} For example, Diederich–Fornæss{{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}} found local pseudoconvex bounded domains $\backslash Omega$ with smooth boundary on non-Kähler manifolds such that $\backslash Omega$ is not weakly 1-complete.{{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 }}{{refn|group=note|1=Definition of weakly 1-complete.{{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 }}}}

For a domain $D\; \backslash subset\; \backslash mathbb\; C^n$ 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{{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 }}{{Eom| title = Modification | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47868}} on the complex analytic space (when n = 2, the result by Hirzebruch,{{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 }} when n = 3 the result by Zariski{{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 }} 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:

{{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 }}}}

1. D is a domain of holomorphy.
2. D is holomorphically convex.
3. D is the union of an increasing sequence of analytic polyhedrons in D.
4. D is pseudoconvex.
5. D is Locally pseudoconvex.

The implications $1\; \backslash Leftrightarrow\; 2\; \backslash Leftrightarrow\; 3$,{{refn|group=note|1=This relation is called the Cartan–Thullen theorem.{{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}}}} $1\; \backslash Rightarrow\; 4$,See Oka's lemma and $4\backslash Rightarrow\; 5$ are standard results. Proving $5\; \backslash Rightarrow\; 1$, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was solved for unramified Riemann domains over $\backslash mathbb\{C\}^n$ by Kiyoshi Oka,Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex. but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,{{cite journal |doi=10.1007/BF01420649|title=A counterexample for the Levi problem for branched Riemann domains over $\backslash mathbb\{C\}^n$ |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}} and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of $\backslash bar\{\backslash partial\}$-problem(equation) with a L² methods).{{R|Hörmander1965|}}{{R|Forstnerič2011§SteinManifolds|}}{{R|Błocki2014|}}{{R|Noguchi2019}}

The introduction of sheaves into several complex variables allowed the reformulation of and solution to several important problems in the field.

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 $(I)$ of pairs $(f,\; \backslash delta)$, $f$ holomorphic on a non-empty open set $\backslash delta$, such that

1. If $(f,\; \backslash delta)\; \backslash in\; (I)$ and $(a,\; \backslash delta\text{'})$ is arbitrary, then $(af,\; \backslash delta\; \backslash cap\; \backslash delta\text{'})\; \backslash in\; (I)$.
2. For each $(f,\; \backslash delta),\; (f\text{'},\; \backslash delta\text{'})\; \backslash in\; (I)$, then $(f\; +\; f\text{'},\; \backslash delta\; \backslash cap\; \backslash delta\text{'})\; \backslash in\; (I).$

The origin of indeterminate domains comes from the fact that domains change depending on the pair $(f,\; \backslash delta)$. Cartan{{R|Cartan1950}}{{R|Cartan1953}} translated this notion into the notion of the coherent (sheaf) (Especially, coherent analytic sheaf) in sheaf cohomology.{{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 }}{{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}}}} This name comes from

H. Cartan.{{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}}}} Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.{{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}} The notion of coherent (coherent sheaf cohomology) helped solve the problems in several complex variables.{{R|Chorlay2010|}}

The definition of the coherent sheaf is as follows.{{R|Serre1955|}}{{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 }}{{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}}}}{{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}}}}

{{R|CAaDG}}{{rp|pages=83–89}}

A quasi-coherent sheaf on a ringed space $(X,\; \backslash mathcal\; O\_X)$ is a sheaf $\backslash mathcal\; F$ of $\backslash mathcal\; O\_X$-modules which has a local presentation, that is, every point in $X$ has an open neighborhood $U$ in which there is an exact sequence

:$\backslash mathcal\{O\}\_X^\{\backslash oplus\; I\}|\_\{U\}\; \backslash to\; \backslash mathcal\{O\}\_X^\{\backslash oplus\; J\}|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}\; \backslash to\; 0$

for some (possibly infinite) sets $I$ and $J$.

A coherent sheaf on a ringed space $(X,\; \backslash mathcal\; O\_X)$ is a sheaf $\backslash mathcal\; F$ satisfying the following two properties:

1. $\backslash mathcal\; F$ is of finite type over $\backslash mathcal\; O\_X$, that is, every point in $X$ has an open neighborhood $U$ in $X$ such that there is a surjective morphism $\backslash mathcal\{O\}\_X^\{\backslash oplus\; n\}|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}$ for some natural number $n$;
2. for each open set $U\backslash subseteq\; X$, integer $n\; >\; 0$, and arbitrary morphism $\backslash varphi:\; \backslash mathcal\{O\}\_X^\{\backslash oplus\; n\}|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}$ of $\backslash mathcal\; O\_X$-modules, the kernel of $\backslash varphi$ is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of $\backslash mathcal\; O\_X$-modules.

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

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

(Oka–Cartan) coherent theorem{{R|Oka'sVII|}} says that each sheaf that meets the following conditions is a coherent.{{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 }}

1. the sheaf $\backslash mathcal\{O\}\; :=\; \backslash mathcal\{O\}\_\{\backslash mathbb\{C\}\_n\}$ of germs of holomorphic functions on $\backslash mathbb\{C\}\_n$, or the structure sheaf $\backslash mathcal\{O\}\_X$ of complex submanifold or every complex analytic space $(X,\; \backslash mathcal\{O\}\_X)${{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}}}}
2. the ideal sheaf $\backslash mathcal\{I\}\; \backslash langle\; A\; \backslash rangle$ of an analytic subset A of an open subset of $\backslash mathbb\{C\}\_n$. (Cartan 1950{{R|Cartan1950}}){{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}}}}{{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}}
3. the normalization of the structure sheaf of a complex analytic space{{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 }}

From the above Serre(1955) theorem, $\backslash mathcal\{O\}^p$ is a coherent sheaf, also, (i) is used to prove Cartan's theorems A and B.

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 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.{{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 }} 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|}}{{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}}

}} on the complex coordinate space,{{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}}{{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}}{{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}}{{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 Oka principle.{{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}} 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|}}

Without the language of sheaves, the problem can be formulated as follows. On a complex manifold M, one is given several meromorphic functions $f\_i$ along with domains $U\_i$ where they are defined, and where each difference $f\_i-f\_j$ is holomorphic (wherever the difference is defined). The first Cousin problem then asks for a meromorphic function $f$ on M such that $f-f\_i$ is holomorphic on $U\_i$; in other words, that $f$ shares the singular 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:

:$H^0(M,\backslash mathbf\{K\})\; \backslash xrightarrow\{\backslash phi\}\; H^0(M,\backslash mathbf\{K\}/\backslash mathbf\{O\}).$

By the long exact cohomology sequence,

:$H^0(M,\backslash mathbf\{K\})\; \backslash xrightarrow\{\backslash phi\}\; H^0(M,\backslash mathbf\{K\}/\backslash mathbf\{O\})\backslash to\; H^1(M,\backslash mathbf\{O\})$

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group H¹(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold.

The second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio $f\_i/f\_j$ is a non-vanishing holomorphic function (where said difference is defined). It asks for a meromorphic function $f$ on M such that $f/f\_i$ is holomorphic and non-vanishing.

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

:$H^0(M,\backslash mathbf\{K\}^*)\backslash xrightarrow\{\backslash phi\}\; H^0(M,\backslash mathbf\{K\}^*/\backslash mathbf\{O\}^*).$

The long exact sheaf cohomology sequence associated to the quotient is

:$H^0(M,\backslash mathbf\{K\}^*)\backslash xrightarrow\{\backslash phi\}\; H^0(M,\backslash mathbf\{K\}^*/\backslash mathbf\{O\}^*)\backslash to\; H^1(M,\backslash mathbf\{O\}^*)$

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

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

:$0\backslash to\; 2\backslash pi\; i\backslash Z\backslash to\; \backslash mathbf\{O\}\; \backslash xrightarrow\{\backslash exp\}\; \backslash mathbf\{O\}^*\; \backslash to\; 0$

where the leftmost sheaf is the locally constant sheaf with fiber $2\backslash pi\; i\backslash Z$. The obstruction to defining a logarithm at the level of H¹ is in $H^2(M,\backslash Z)$, from the long exact cohomology sequence

:$H^1(M,\backslash mathbf\{O\})\backslash to\; H^1(M,\backslash mathbf\{O\}^*)\backslash to\; 2\backslash pi\; i\; H^2(M,\backslash Z)\; \backslash to\; H^2(M,\; \backslash mathbf\{O\}).$

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

Since a non-compact (open) Riemann surface{{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}} always has a non-constant single-valued holomorphic function,{{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 }} 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 $\backslash mathbb\; C$. (In fact, Gunning and Narasimhan have shown (1967){{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 }} 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.){{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 }} The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of $\backslash mathbb\{R\}^\{2n\}$, whereas it is "rare" for a complex manifold to have a holomorphic embedding into $\backslash mathbb\; C^n$. 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 $\backslash mathbb\; C^n$, then the coordinate functions of $\backslash mathbb\; C^n$ 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 $\backslash mathbb\; C^n$ are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.{{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}}

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).{{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}} A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. If the univalent domain on $\backslash mathbb\; C^n$ is connection to a manifold, can be regarded as a complex manifold and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.

Suppose X is a paracompact complex manifolds of complex dimension $n$ and let $\backslash mathcal\; O(X)$ denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold:{{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}}

1. X is holomorphically convex, i.e. for every compact subset $K\; \backslash subset\; X$, the so-called holomorphically convex hull,

   :$\backslash bar\; K\; =\; \backslash left\; \backslash \{z\; \backslash in\; X\; ;\; |f(z)|\; \backslash leq\; \backslash sup\_\{w\; \backslash in\; K\}\; |f(w)|,\; \backslash \; \backslash forall\; f\; \backslash in\; \backslash mathcal\; O(X)\; \backslash right\; \backslash \},$

   is also a compact subset of X.

2. X is holomorphically separable,From this condition, we can see that the Stein manifold is not compact. i.e. if $x\; \backslash neq\; y$ are two points in X, then there exists $f\; \backslash in\; \backslash mathcal\; O(X)$ such that $f(x)\; \backslash neq\; f(y).$
3. The open neighborhood of every point on the manifold has a holomorphic chart to the $\backslash mathcal\; O(X)$.

Note that condition (3) can be derived from conditions (1) and (2).{{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}}

Let X be a connected, non-compact (open) Riemann surface. A 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 $H^1(X,\; \backslash mathcal\; O\_X^*)\; =0$. The exponential sheaf sequence leads to the following exact sequence:

:$H^1(X,\; \backslash mathcal\; O\_X)\; \backslash longrightarrow\; H^1(X,\; \backslash mathcal\; O\_X^*)\; \backslash longrightarrow\; H^2(X,\; \backslash Z)\; \backslash longrightarrow\; H^2(X,\; \backslash mathcal\; O\_X)$

Now Cartan's theorem B shows that $H^1(X,\backslash mathcal\{O\}\_X)\; =\; H^2(X,\backslash mathcal\{O\}\_X)=0$, therefore $H^2(X,\backslash Z)\; =\; 0$.

This is related to the solution of the second (multiplicative) Cousin problem.

Cartan extended Levi's problem to Stein manifolds.{{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 }}

:If the relative compact open subset $D\backslash subset\; X$ 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.{{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 }}

This was proved by Bremermann{{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}} by embedding it in a sufficiently high dimensional $\backslash mathbb\{C\}^n$, 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|}}{{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}}{{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 }}}}{{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 }}{{R|Huckleberry2013|}}{{R|Sibony2018|}}

:If the relative compact subset $D\backslash subset\; M$ 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{{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 }}{{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 }} 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}}}} $D\; \backslash subset\backslash subset\; X$ . Suppose that for all $p\backslash in\; \backslash partial\; D$ there is an open neighborhood $U\; (p)$ so that $U\backslash cap\; D$ is Stein space. Is D itself Stein?{{R|Siu1978}}{{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 }}{{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}}

more generalized

:Suppose that N be a Stein space and f an injective, and also $f\; :M\; \backslash to\; N$ a Riemann unbranched domain, such that map f is a locally pseudoconvex map (i.e. Stein morphism). Then M is itself Stein ?{{R|Coltoiu2009}}{{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}}{{rp|page=109}}

and also,

:Suppose that X be a Stein space and $D\; =\; \backslash bigcup\_\{n\backslash in\backslash mathbb\{N\}\}\; D\_n$ 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}}

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 $x\_0\backslash in\; X$, there exist finitely many holomorphic map $f\_1,\backslash dots,f\_k$ of X into $\backslash Complex^p$, $p\; =\; p(x\_0)$, such that $x\_0$ is an isolated point of the set $A\; =\; \backslash \{x\backslash in\; X;f^\{-1\}f(x\_0)\backslash \; (v=1,\backslash dots,k)\backslash \}$.{{R|Grauert1958|}} This concept also applies to complex analytic space.{{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}}

• The standard$\backslash Complex^n\; \backslash times\; \backslash mathbb\{P\}\_m$ ($\backslash mathbb\{P\}\_m$ is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity. complex space $\backslash Complex^n$ is a Stein manifold.
• Every domain of holomorphy in $\backslash Complex^n$ 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 $\backslash Complex^\{2\; n+1\}$ by a biholomorphic proper map.{{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 }}{{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 }}{{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}}

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.{{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 }}
• 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{{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 }} for Riemann surfaces,The proof method uses an approximation by the polyhedral domain, as in Oka-Weil theorem. due to Behnke and Stein.{{R|Behnke–Stein1948|}}
• Every Stein manifold X is holomorphically spreadable, i.e. for every point $x\; \backslash in\; X$, there are n holomorphic functions defined on all of X which form a local coordinate system when restricted to some open neighborhood of x.
• The first Cousin problem can always be solved on a Stein manifold.
• Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function,{{R|Grauert1958|}} i.e. a smooth real function $\backslash psi$ on X (which can be assumed to be a Morse function) with $i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash psi\; >0$,{{R|Grauert1958|}} such that the subsets $\backslash \{z\; \backslash in\; X\; \backslash mid\; \backslash psi\; (z)\backslash leq\; c\; \backslash \}$ are compact in X for every real number c. This is a solution to the so-called Levi problem,{{Eom| title = Levi problem | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47620}} named after E. E. Levi (1911). The function $\backslash psi$ invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domain.{{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}} A Stein domain is the preimage $\backslash \{z\; \backslash mid\; -\backslash infty\backslash leq\backslash psi(z)\backslash leq\; c\backslash \}$. 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 $X\_c=f^\{-1\}(c)$ is a contact structure that induces an orientation on X_c agreeing with the usual orientation as the boundary of $f^\{-1\}(-\backslash infty,\; c).$ That is, $f^\{-1\}(-\backslash infty,\; c)$ is a Stein filling of X_c.

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

In the GAGA set of analogies, Stein manifolds correspond to affine varieties.{{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 }}

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

Meromorphic function in one-variable complex function were studied in 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 $\backslash mathbb\{C\}${{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}}}}{{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 }}). 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 $\backslash widehat\backslash mathbb\{C\}\; \backslash cong\; \backslash mathbb\{CP\}^1$. However, the abstract notion of a compact Riemann surface is always algebraizable (The Riemann's existence theorem, Kodaira embedding theorem.),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. but it is not easy to verify which compact complex analytic spaces are algebraizable.{{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}}}} 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.{{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 }} The generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira,{{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 }} Kodaira also extended the theorem to three-dimensional,{{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 }} and n-dimensional Kähler varieties.{{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 }} Serre formulated the Riemann–Roch theorem as a problem of dimension of coherent sheaf cohomology,{{R|IWS}} and also Serre proved Serre duality.{{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}} Cartan and Serre proved the following property:{{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 }} the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M.{{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 }} Riemann–Roch on a Riemann surface for a vector bundle was proved by Weil in 1938.{{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 }}

Hirzebruch generalized the theorem to compact complex manifolds in 1994{{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 }} and Grothendieck generalized it to a relative version (relative statements about morphisms.).{{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}}

{{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 }} 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 $\backslash mathbb\{CP\}^n$. This is the standard method for compactification of $\backslash mathbb\{C\}^n$, but not the only method like the Riemann sphere that was compactification of $\backslash mathbb\{C\}$. The vanishing theorem (was first introduced by Kodaira in 1953) gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of positivity. As an application of this theorem, the Kodaira embedding theorem{{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 }} 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 Chow's theorem{{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 }} shows that the complex analytic subspace (subvariety) of a closed complex projective space to be an algebraic that is, so it is the common zero of some homogeneous polynomials, such a relationship is one example of what is called Serre's GAGA principle.{{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.{{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}} Analogy of the Levi problems on the complex projective space $\backslash mathbb\{CP\}^n$ by Takeuchi.{{R|Siu1978}}{{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}}{{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 }}{{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}}

• Bicomplex number
• Complex geometry
• CR manifold
• Dolbeault cohomology
• Harmonic maps
• Harmonic morphisms
• Infinite-dimensional holomorphy
• Oka–Weil theorem

{{Reflist|group=note}}

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{{Refend}}

{{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}}

{{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}}

• [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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