# Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions in the space Cn of n-tuples of complex numbers.

${\displaystyle f(z_{1},z_{2},\ldots ,z_{n})}$

As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi. Equivalently, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, the any domain was the domain of holomorphy, but for several complex variables, the any domain is not the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and projective complex varieties and has a different flavour to complex analytic geometry in ${\displaystyle \mathbb {C} ^{n}}$ or on Stein manifolds.

## Historical perspective

Many examples of such functions were familiar in nineteenth-century mathematics: abelian functions, theta functions, and some hypergeometric series. Naturally also any 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 area in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory.

With work of Friedrich Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for any analytic function

${\displaystyle f:\mathbb {C} ^{n}\to \mathbb {C} }$

whenever n > 1. Naturally the analogues of contour integrals will be harder to handle: when n = 2 an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface. This means that the residue calculus will have to take a very different character.

After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory: while for any open connected set D in C we can find a function that will nowhere continue analytically over the boundary, that cannot be said for n > 1. In fact the D of that kind are rather special in nature (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. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied to analytic geometry, [note 1] automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper GAGA of Serre[ref 1] pinned down the crossover point from géometrie analytique to géometrie algébrique.

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

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

## The Cn space

${\displaystyle \mathbb {C} ^{n}}$ is defined as the Cartesian product of n copies of ${\displaystyle \mathbb {C} }$, and when ${\displaystyle \mathbb {C} ^{n}}$ is a domain of holomorphy, ${\displaystyle \mathbb {C} ^{n}}$ can be regarded as a Stein manifold. It can be considered as an n-dimensional vector space over complex numbers, which gives its dimension 2n over R.[note 2] Hence, as a set, and as topological space, Cn is identical to R2n and its topological dimension is 2n.

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

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

${\displaystyle {\begin{pmatrix}u&-v\\v&u\end{pmatrix}},}$

with determinant

${\displaystyle u^{2}+v^{2}=|w|^{2}.}$

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 Cn to Cn.

### Connected space

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

### Compact

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

## Holomorphic functions

A function ${\displaystyle f(z)}$ defined on a domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ is called holomorphic if ${\displaystyle f(z)}$ satisfies the following two conditions.[note 3][ref 2]

1. ${\displaystyle f(z)}$ is continuous[note 4] on D[note 5]
2. For each variable ${\displaystyle z_{\lambda }}$, ${\displaystyle f(z)}$ is holomorphic, namely,
${\displaystyle {\frac {\partial f}{\partial {\overline {z}}_{\lambda }}}=0}$

(1)

which is a generalization of the Cauchy–Riemann equations (using a partial Wirtinger derivative), and has the origin of Riemann's differential equation methods.

### Cauchy–Riemann equations

For each index λ let

${\displaystyle z_{\lambda }=x_{\lambda }+iy_{\lambda },\quad f(z_{1},\dots ,z_{n})=u(x_{1},\dots ,x_{n},y_{1},\dots ,y_{n})+iv(x_{1},\dots ,x_{n},y_{1},\dots y_{n}),}$

and generalize the usual Cauchy–Riemann equation for one variable for each index λ, then we obtain

${\displaystyle {\frac {\partial u}{\partial x_{\lambda }}}={\frac {\partial v}{\partial y_{\lambda }}},\ \ \ \ {\frac {\partial u}{\partial y_{\lambda }}}=-{\frac {\partial v}{\partial x_{\lambda }}}.}$

(2)

Let

{\displaystyle {\begin{aligned}dz_{\lambda }&=dx_{\lambda }+i\,dy_{\lambda },&d{\bar {z}}_{\lambda }&=dx_{\lambda }-i\,dy_{\lambda }\\{\frac {\partial }{\partial z_{\lambda }}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{\lambda }}}-i{\frac {\partial }{\partial y_{\lambda }}}\right),&{\frac {\partial }{\partial {\bar {z}}_{\lambda }}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x_{\lambda }}}+i{\frac {\partial }{\partial y_{\lambda }}}\right)\end{aligned}}}

through

${\displaystyle \operatorname {Re} \left({\frac {\partial f}{\partial {\bar {z}}_{\lambda }}}\right)={\frac {\partial u}{\partial x_{\lambda }}}-{\frac {\partial v}{\partial y_{\lambda }}}=0,\quad \operatorname {Im} \left({\frac {\partial f}{\partial {\bar {z}}_{\lambda }}}\right)={\frac {\partial u}{\partial y_{\lambda }}}+{\frac {\partial v}{\partial x_{\lambda }}}=0}$

the above equations (1) and (2) turn to be equivalent.

### Cauchy's integral formula

f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve ${\displaystyle \gamma }$, ${\displaystyle \gamma _{\nu }}$ is piecewise smoothness, class ${\displaystyle C^{1}}$ Jordan closed curve. (${\displaystyle \nu =1,2,\ldots ,n}$) Let ${\displaystyle D_{\nu }}$ be the domain surrounded by each ${\displaystyle \gamma _{\nu }}$. Cartesian product closure ${\displaystyle {\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}}$ is ${\displaystyle {\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}\in D}$. Also, take the polydisc ${\displaystyle {\overline {\Delta }}}$ so that it becomes ${\displaystyle {\overline {\Delta }}\subset {D_{1}\times D_{2}\times \cdots \times D_{n}}}$. (${\displaystyle {\overline {\Delta }}(z,r)=\left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in C^{n}:\left|\zeta _{\nu }-z_{\nu }\right|\leq r_{\nu }{\text{ for all }}\nu =1,\dots ,n\right\}}$ and let ${\displaystyle \{z\}_{\nu =1}^{n}}$ be the center of each disk.) Using Cauchy's integral formula of one variable repeatedly,

{\displaystyle {\begin{aligned}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{aligned}}}

Because ${\displaystyle \partial D}$ is a rectifiable Jordanian closed curve[note 6] and f is continuous, so the order of products and sums can be exchanged so the iterated integral can be calculated as a multiple integral. Therefore,

${\displaystyle f(z_{1},\dots ,z_{n})={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-z_{1})\cdots (\zeta _{n}-z_{n})}}\,d\zeta _{1}\cdots d\zeta _{n}}$

(3)

While in the one-variable case Cauchy's integral formula is an integral over the circumference of a disc with some radius r, in several variables case over the surface of a polydisc with radii ${\displaystyle r_{\nu }}$'s as in (3).

#### Cauchy's evaluation formula

Because the order of products and sums is interchangeable, from (3) we get

${\displaystyle {\frac {\partial ^{k_{1}+\cdots +k_{n}}f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}={\frac {k_{1}\cdots k_{n}!}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-z_{1})^{k_{1}+1}\cdots (\zeta _{n}-z_{n})^{k_{n}+1}}}\,d\zeta _{1}\cdots d\zeta _{n}.}$

(4)

f is differentiable any number of times and the derivative is continuous.

From (4), if ${\displaystyle f(z_{1},\ldots ,z_{n})}$ is holomorphic, on polydisc ${\displaystyle \left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in \mathbb {C} ^{n}\mid |\zeta _{\nu }-z_{\nu }|\leq r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}}$ and ${\displaystyle |f|\leq {M}}$, the following evaluation equation is obtained.

${\displaystyle \left|{\frac {\partial ^{k_{1}+\cdots +k_{n}}f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{{\partial z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}\right|\leq {\frac {Mk_{1}\cdots k_{n}!}{{r_{1}}^{k_{1}}\cdots {r_{n}}^{k_{n}}}}}$

Therefore, Liouville's theorem hold.

#### Power series expansion of holomorphic functions

If ${\displaystyle f(z_{1},\ldots ,z_{n})}$ is holomorphic, on polydisc ${\displaystyle \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|, from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

{\displaystyle {\begin{aligned}&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{aligned}}}

(5)

In addition, ${\displaystyle f(z)}$ that satisfies the following conditions is called an analytic function.

For each point ${\displaystyle a=(a_{1},\dots ,a_{n})\in D\subset \mathbb {C} ^{n}}$, ${\displaystyle f(z)}$ is expressed as a power series expansion that is convergent on D :

${\displaystyle f(z)=\sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ ,}$

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

If a sequence of functions ${\displaystyle f_{1},\ldots ,f_{n}}$ which converges uniformly on compacta inside a domain D, the limit function f of ${\displaystyle f_{v}}$ also uniformly on compacta inside a domain D. Also, respective partial derivative of ${\displaystyle f_{v}}$ also compactly converges on domain D to the corresponding derivative of f.
${\displaystyle {\frac {\partial ^{k_{1}+\cdots +k_{n}}f}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}=\sum _{v=1}^{\infty }{\frac {\partial ^{k_{1}+\cdots +k_{n}}f_{v}}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}}$[ref 3]
##### Radius of convergence of power series

It is possible to define a combination of positive real numbers ${\displaystyle \{r_{\nu }\ (\nu =1,\dots ,n)\}}$ such that the power series ${\textstyle \sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ }$ converges uniformly at ${\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n}\mid |z_{\nu }-a_{\nu }| and does not converge uniformly at ${\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n}\mid |z_{\nu }-a_{\nu }|>r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}}$.

In this way it is possible to have a similar, combination of radius of convergence[note 7] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

##### Identity theorem

When the function f,g is holomorphic in the concatenated domain D,[note 8] even for several complex variables, the identity theorem[note 9] holds on the domain D, because it has a power series expansion the neighbourhood of holomorphic point. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold.

##### Analytic continuation

Let U, V be open subsets in ${\displaystyle \mathbb {C} ^{n}}$, ${\displaystyle f\in {\mathcal {O}}(U)}$ and ${\displaystyle g\in {\mathcal {O}}(V)}$. Assume that ${\displaystyle U\cap V\neq \varnothing }$ and ${\displaystyle W}$ is a connected component of ${\displaystyle U\cap V}$. If ${\displaystyle f|_{W}=g|_{W}}$ 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. Whether or not the definition of this analytic continuation is well-defined should be considered whether the domains U,V and W can be defined well. Several complex variables have restrictions on this domain, and depending on the shape of the domain , all holomorphic functions f belonging to U are connected to V, and there may be not exist function f with ${\displaystyle \partial U}$ as the natural boundary. In other words, U cannot be defined. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of Several complex variables. Also, in the general dimension, there may be multiple intersections between U and V. That is, f is not connected as a monovalent holomorphic function, but as an multivalued holomorphic function. This means that W is not unique and has different properties in the neighborhood of the branch point than in the case of one variable.

## Reinhardt domain

Power series expansion of several complex variables it is possible to define the combination of radius of convergence similar to that of one complex variable, but each variable cannot independently define a unique radius of convergence. The Reinhardt domain is considered in order to investigate the characteristics of the convergence domain of the power series, but when considering the Reinhardt domain, it can be seen that the convergence domain of the power series satisfies the convexity called Logarithmically-convex. There are various convexity for the convergence domain of Several complex variables.

A domain D in the complex space ${\displaystyle \mathbb {C} ^{n}}$, ${\displaystyle n\geq 1}$, with centre at a point ${\displaystyle a=(a_{1},\dots ,a_{n})\in \mathbb {C} ^{n}}$, with the following property: Together with any point ${\displaystyle z^{0}=(z_{1}^{0},\dots ,z_{n}^{0})\in D}$, the domain also contains the set

${\displaystyle \left\{z=(z_{1}\dots z_{n}):\left|z_{\nu }-a_{\nu }\right|=\left|z_{\nu }^{0}-a_{\nu }\right|,\ \nu =1\dots n\right\}.}$

A Reinhardt domain D with ${\displaystyle a=0}$ is invariant under the transformations ${\displaystyle \left\{z^{0}\right\}\to \left\{z_{\nu }^{0}e^{i\theta _{\nu }}\right\}}$, ${\displaystyle 0\leq \theta _{\nu }<2\pi }$, ${\displaystyle \nu =1,\dots ,n}$. The Reinhardt domains constitute a subclass of the Hartogs domains [ref 4] and a subclass of the circular domains, which are defined by the following condition: Together with any ${\displaystyle z^{0}\in D}$, the domain contains the set

${\displaystyle \left\{z=(z_{1}\cdots z_{n}):z=a+\left(z^{0}-a\right)e^{i\theta },\ 0\leq \theta <2\pi \right\},}$

i.e. all points of the circle with center ${\displaystyle a}$ and radius ${\textstyle \left|z^{0}-a\right|=\left(\sum _{\nu =1}^{n}\left|z_{\nu }^{0}-a_{\nu }\right|^{2}\right)^{1/2}}$ that lie on the complex line through ${\displaystyle a}$ and ${\displaystyle z^{0}}$.

A Reinhardt domain D is called a complete Reinhardt domain if together with any point ${\displaystyle z^{0}\in D}$ it also contains the polydisc

${\displaystyle \left\{z=(z_{1}\dots z_{n}):\left|z_{\nu }-a_{\nu }\right|\leq \left|z_{\nu }^{0}-a_{\nu }\right|,\ \nu =1\dots n\right\}.}$

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

### Logarithmically-convex

A Reinhardt domain D is called logarithmically convex if the image ${\displaystyle \lambda (D^{*})}$ of the set

${\displaystyle D^{*}=\{z=(z_{1}\dots z_{n})\in D:z_{1}\dots z_{n}\neq 0\}}$

under the mapping

${\displaystyle \lambda :z\rightarrow \lambda (z)=(\ln |z_{1}|\cdots \ln |z_{n}|)}$

is a convex set in the real space ${\displaystyle \mathbb {R} ^{n}}$. An important property of logarithmically-convex Reinhardt domains is the following: Every such domain in ${\displaystyle \mathbb {C} ^{n}}$ is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in ${\textstyle \sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ }$, and conversely: The domain of convergence of any power series in ${\displaystyle z_{1},\dots ,z_{n}}$ is a logarithmically-convex Reinhardt domain with centre ${\displaystyle a=0}$. [note 10]

### Some results

#### Thullen's classic results

Thullen's[ref 5] 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. ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|<1,~|w|<1\}}$ (polydisc);
2. ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|^{2}+|w|^{2}<1\}}$ (unit ball);
3. ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|^{2}+|w|^{2/p}<1\}(p>0,\neq 1)}$ (Thullen domain).

#### Hartogs's phenomenon

Let's look at the example on the Hartogs's extension theorem page in terms of the Reinhardt domain.

On the polydisk consisting of two disks ${\displaystyle \Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}}$ when ${\displaystyle 0<\varepsilon <1}$.

Internal domain of

${\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2}:|z_{1}|<\varepsilon \ \cup \ 1-\varepsilon <|z_{2}|\}\ (0<\varepsilon <1)}$

Theorem Hartogs (1906)[ref 6] Let f be a holomorphic function on a set G \ K, where G is an open subset of Cn (n ≥ 2) and K is a compact subset of G. If the complement G \ K is connected, then f can be extended to a unique holomorphic function on G.

From Hartogs's extension theorem the convergence domain extends from ${\displaystyle H_{\varepsilon }}$ to ${\displaystyle \Delta ^{2}}$. Looking at this from the perspective of the Reinhardt domain, ${\displaystyle H_{\varepsilon }}$ is the Reinhardt domain containing the center z = 0, and the convergence domain of ${\displaystyle H_{\varepsilon }}$ has been extended to the smallest complete Reinhardt domain ${\displaystyle \Delta ^{2}}$ containing ${\displaystyle H_{\varepsilon }}$.[ref 7]

Toshikazu Sunada (1978)[ref 8] established a generalization of Thullen's result:

Two n-dimensional bounded Reinhardt domains ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ are mutually biholomorphic if and only if there exists a transformation ${\displaystyle \varphi :\mathbb {C} ^{n}\to \mathbb {C} ^{n}}$ given by ${\displaystyle z_{i}\mapsto r_{i}z_{\sigma (i)}(r_{i}>0)}$, ${\displaystyle \sigma }$ being a permutation of the indices), such that ${\displaystyle \varphi (G_{1})=G_{2}}$.

## Domain of holomorphy

The sets in the definition. Note: On this page, replace ${\displaystyle \Omega }$ in the figure with D

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 (i.e. domain of holomorphy), the first result in the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen. Levi's problem shows that the pseudoconvex domain was a domain of holomorphy.[ref 9][ref 10][ref 11][ref 12] Also Kiyoshi Oka's idéal de domaines indéterminés[ref 13] is interpreted by Cartan.[note 11] In sheaf[ref 14] theory, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[ref 15]

### Definition

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

Formally, an open set D in the n-dimensional complex space ${\displaystyle \mathbb {C} ^{n}}$ is called a domain of holomorphy if there do not exist non-empty open sets ${\displaystyle U\subset D}$ and ${\displaystyle V\subset \mathbb {C} ^{n}}$ where V is connected, ${\displaystyle V\not \subset D}$ and ${\displaystyle U\subset D\cap V}$ such that for every holomorphic function f on D there exists a holomorphic function g on V with ${\displaystyle f=g}$ on U.

In the ${\displaystyle n=1}$ case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal.

### Holomorphically convex hull

The first result on the properties of the domain of holomorphy is the holomorphic convexity of Henri Cartan and Peter Thullen (1932).[ref 16]

The holomorphically convex hull of a given compact set in the n-dimensional complex space ${\displaystyle \mathbb {C} ^{n}}$ is defined as follows.

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

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

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

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

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

If ${\displaystyle f}$ satisfies the above holomorphically convexity it has the following properties. The radius ${\displaystyle \rho }$ of the polydisc ${\displaystyle \Delta =\{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|<\rho _{\nu },{\text{ for all }}\nu =1,\dots ,n\}}$ satisfies condition ${\displaystyle \rho _{\nu }=|K,\partial D|=\inf\{|z_{\nu }-z_{\nu }^{'}|\mid z_{\nu }\in K,z_{\nu }^{'}\in \partial D\}}$ also the compact set satisfies ${\displaystyle K\subset D}$ and ${\displaystyle D\subset \mathbb {C} ^{n}}$ is the domain. In the time that, any holomorphic function on the domain ${\displaystyle D}$ can be direct analytic continuated up to ${\displaystyle \Delta }$.

### Levi convex (Approximate from the inside on the analytic polyhedron domain)

${\textstyle \bigcup _{n=1}^{\infty }S_{n}\subseteq D}$ is union of ascending sequence of analytic compact surfaces with paracompact and Holomorphically convex properties such that ${\textstyle \bigcup _{n=1}^{\infty }S_{n}\rightarrow D,S_{n}\subset S_{n+1}}$. i.e. Approximate from the inside by analytic polyhedron. [note 12]

### Pseudoconvex

Pseudoconvex Hartogs showed that ${\displaystyle -\log R}$ is subharmonic for the radius of convergence in the Hartogs series ${\displaystyle R(z)}$ when the Hartogs series is a one-variable ${\displaystyle f(z,\omega )\ \omega =0}$. If such a relationship holds in the domain of holomorphy of Several complex variables, it looks like a more manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain. Pseudoconvex domain are important, as they allow for classification of domains of holomorphy.

#### Definition of plurisubharmonic function

A function
${\displaystyle f\colon D\to {\mathbb {R} }\cup \{-\infty \},}$
with domain ${\displaystyle D\subset {\mathbb {C} }^{n}}$

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

${\displaystyle \{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n}}$ with ${\displaystyle a,b\in {\mathbb {C} }^{n}}$
the function ${\displaystyle z\mapsto f(a+bz)}$ is a subharmonic function on the set
${\displaystyle \{z\in {\mathbb {C} }\mid a+bz\in D\}.}$
In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space ${\displaystyle X}$ as follows. An upper semi-continuous function
${\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}}$
is said to be plurisubharmonic if and only if for any holomorphic map

${\displaystyle \varphi \colon \Delta \to X}$ the function

${\displaystyle f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}}$

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

##### Strictly plurisubharmonic function

Necessary and sufficient condition that the real-valued function u(z), that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is ${\displaystyle \Delta =4\left({\frac {\partial ^{2}u}{\partial z\partial {\overline {z}}}}\right)\geq 0}$. When the Hermitian matrix of u is positive-definite and class ${\displaystyle \mathbf {C} ^{2}}$, we call u a strict plural subharmonic function.

#### (Weakly) pseudoconvex (p-pseudoconvex)

Weak pseudoconvex[ref 17] is defined as : Let ${\displaystyle X\subset {\mathbb {C} }^{n}}$ be a domain, that is, an open connected subset. One says that X is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function ${\displaystyle \varphi }$ on X such that the set ${\displaystyle \{z\in X\mid \varphi (z)\leq \sup x\}}$ is a relatively compact subset of X for all real numbers x. [note 13] i.e there exists a smooth plurisubharmonic exhaustion function ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$.

#### Strongly pseudoconvex

Strongly pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$,i.e. ${\displaystyle H\psi }$ is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.[ref 17]

#### Levi–Krzoska pseudoconvexity

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

${\displaystyle \nabla \rho (p)w=\sum _{i=1}^{n}{\frac {\partial \rho (p)}{\partial z_{j}}}w_{j}=0}$, we have
${\displaystyle \sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}$

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

Proposition 1 If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains ${\displaystyle D_{k}\subset D}$ with ${\displaystyle C^{\infty }}$ (smooth) boundary which are relatively compact in D, such that

${\displaystyle D=\bigcup _{k=1}^{\infty }D_{k}.}$

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

#### Levi Strongly Pseudoconvex (Levi total Pseudoconvex)

If for every boundary point ${\displaystyle \rho }$ of D, there exists an analytic variety ${\displaystyle {\mathcal {B}}}$ passing ${\displaystyle \rho }$ which lies entirely outside D in some neighborhood around ${\displaystyle \rho }$, except the point ${\displaystyle \rho }$ itself. Domain D that satisfies these conditions is called Levi Strongly Pseudoconvex or Levi total Pseudoconvex.[ref 18]

#### Oka pseudoconvex

##### Family of Oka's disk

Let n-functions ${\displaystyle \varphi _{j}(u,t)}$ be continuous on ${\displaystyle \Delta :|U|\leq 1,0\leq t\leq 1}$, holomorphic in ${\displaystyle |u|<1}$ when the parameter t is fixed in [0, 1], and assume that ${\displaystyle {\frac {\partial \varphi _{j}}{\partial u}}}$ are not all zero at any point on ${\displaystyle \Delta }$. Then the set ${\displaystyle Q(t):=\{Z_{j}=\varphi _{j}(u,t)\mid ||u|\leq 1\}}$ is called an analytic disc de-pending on a parameter t, and ${\displaystyle B(t):=\{Z_{j}=\varphi _{j}(u,t)\mid ||u|=1\}}$ is called its shell. If ${\displaystyle Q(t)\subset D\ (0 and ${\displaystyle B(0)\subset D}$, Q(t) is called Family of Oka's disk.[ref 18]

##### Definition

When ${\displaystyle Q(0)\subset D}$ holds on any Family of Oka's disk, D is called Oka pseudoconvex.[ref 18] Oka's proof of Levi's problem was proved by the fact that each boundary point of the domain of holomorphy is an Oka pseudoconvex.[ref 10]

#### Cartan pseudoconvex (Local Levi property)

For every point ${\displaystyle x\in \partial D}$ there exist a neighbourhood U of x and f holomorphic on ${\displaystyle U\cap D}$ such that f cannot be extended to any neighbourhood of x. Such a property is called local Levi property, and the domain that satisfies this property is called the Cartan pseudoconvex domain. The Cartan pseudoconvex domain is itself a pseudoconvex domain and is a domain of holomorphy.[ref 18]

### Equivalent conditions (In connection with Levi problem)

For a domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ the following conditions are equivalent.[note 14]:

1. D is a domain of holomorphy.
2. D is holomorphically convex.
3. D is Levi convex.
4. D is pseudoconvex.
5. D is Cartan pseudoconvex.

The implications ${\displaystyle 1\Leftrightarrow 2\Leftrightarrow 3}$,[note 15] ${\displaystyle 1\Rightarrow 4}$,[note 16] and ${\displaystyle 4\Rightarrow 5}$ are standard results. Proving ${\displaystyle 5\Rightarrow 1}$, 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 first solved by Kiyoshi Oka,[note 17] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\displaystyle {\bar {\partial }}}$-problem).

### Properties of the domain of holomorphy

• If ${\displaystyle D_{1},\dots ,D_{n}}$ are domains of holomorphy, then their intersection ${\textstyle D=\bigcap _{\nu =1}^{n}D_{\nu }}$ is also a domain of holomorphy.
• If ${\displaystyle D_{1}\subseteq D_{2}\subseteq \cdots }$ is an ascending sequence of domains of holomorphy, then their union ${\textstyle D=\bigcup _{n=1}^{\infty }D_{n}}$ is also a domain of holomorphy (see Behnke–Stein theorem).
• If ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ are domains of holomorphy, then ${\displaystyle D_{1}\times D_{2}}$ is a domain of holomorphy.
• The first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem.

## Sheaf

### Coherent sheaf

#### Definition

The definition of the coherent sheaf is as follows.[ref 24]

A coherent sheaf on a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is a sheaf ${\displaystyle {\mathcal {F}}}$ satisfying the following two properties:

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

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of ${\displaystyle {\mathcal {O}}_{X}}$-modules.

Also, Jean-Pierre Serre (1955)[ref 24] proves that

If in an exact sequence ${\displaystyle 0\to {\mathcal {F}}_{1}|_{U}\to {\mathcal {F}}_{2}|_{U}\to {\mathcal {F}}_{3}|_{U}\to 0}$ of sheaves of ${\displaystyle {\mathcal {O}}}$-modules two of the three sheaves ${\displaystyle {\mathcal {F}}_{j}}$ are coherent, then the third is coherent as well.

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

${\displaystyle {\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0}$

for some (possibly infinite) sets ${\displaystyle I}$ and ${\displaystyle J}$.

#### Oka's coherent theorem for sheaf of holomorphic function germ

Kiyoshi Oka (1950)[ref 13][ref 25] proved the following

Sheaf of holomorphic function germ ${\displaystyle {\mathcal {O}}:={\mathcal {O}}_{\mathbf {C^{n}} }}$ on the complex manifold is the coherent sheaf. Therefore, from the above Serre(1955) theorem, ${\displaystyle {\mathcal {O}}^{p}}$ is also a coherent sheaf. This theorem is also used to prove Cartan's theorems A and B.

#### Ideal sheaf

If ${\displaystyle Z}$ is a closed subscheme of a locally Noetherian scheme ${\displaystyle X}$, the sheaf ${\displaystyle {\mathcal {I}}_{Z/X}}$ of all regular functions vanishing on ${\displaystyle Z}$ is coherent. Likewise, if ${\displaystyle Z}$ is a closed analytic subspace of a complex analytic space ${\displaystyle X}$, the ideal sheaf ${\displaystyle {\mathcal {I}}_{Z/X}}$ is coherent.

### Cousin problem

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given pole, and Weierstrass factorization theorem was able to create a global meromorphic function from a given zero. The theory of Riemann's surface suggests that in multivariate complex functions, the similar theorem that holds for one-variable complex functions does not hold unless Several restrictions are added to the open Complex manifold. This problem is called the Cousin problem and is formulated in Sheaf cohomology terms. They were introduced in special cases by Pierre Cousin in 1895. It was Kiyoshi Oka who gave the complete answer to this question.[ref 26][ref 27][ref 28]

#### First Cousin problem

##### Definition without Sheaf words

Each difference ${\displaystyle f_{i}-f_{j}}$ is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that ${\displaystyle f-f_{i}}$ is holomorphic on Ui; in other words, that f shares the singular behaviour of the given local function.

##### Definition using Sheaf words

Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. If the next map is surjective, Cousin first problem can be solved.

${\displaystyle H^{0}(M,\mathbf {K} ){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} /\mathbf {O} ).}$
${\displaystyle H^{0}(M,\mathbf {K} ){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} /\mathbf {O} )\to H^{1}(M,\mathbf {O} )}$

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.

#### Second Cousin problem

##### Definition without Sheaf words

Each ratio ${\displaystyle f_{i}/f_{j}}$ is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that ${\displaystyle f/f_{i}}$ is holomorphic and non-vanishing.

##### Definition using Sheaf words

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

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

The long exact sheaf cohomology sequence associated to the quotient is

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

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

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

${\displaystyle 0\to 2\pi i\mathbb {Z} \to \mathbf {O} {\xrightarrow {\exp }}\mathbf {O} ^{*}\to 0}$

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

${\displaystyle H^{1}(M,\mathbf {O} )\to H^{1}(M,\mathbf {O} ^{*})\to 2\pi iH^{2}(M,\mathbb {Z} )\to H^{2}(M,\mathbf {O} ).}$

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

## Manifolds considered with Several complex variables

### Stein manifold

Since the open Riemann surface always has a non-constant monovalent holomorphic function and satisfies the second axiom of countability, the Riemann surface was considered for embedding the one-dimensional complex plane into a complex manifolds. In fact, taking one point at infinity on the one-dimensional complex plane ${\displaystyle \mathbb {C} }$ extended it to the Riemann sphere. The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of ${\displaystyle \mathbb {R} ^{2n}}$, whereas it is "rare" for a complex manifold to have a holomorphic embedding into ${\displaystyle \mathbb {C} ^{n}}$. Consider for example any compact connected complex manifold X: any holomorphic function on it is constant by Liouville's theorem. Now that we know that for Several complex variables, complex manifolds do not always have holomorphic functions that are not constants, consider the conditions that have holomorphic functions. Now if we had a holomorphic embedding of X into ${\displaystyle \mathbb {C} ^{n}}$, then the coordinate functions of ${\displaystyle \mathbb {C} ^{n}}$ 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 embedded in Cn are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.

Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951).[ref 29] 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 ${\displaystyle \mathbb {C} ^{n}}$ 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.

#### Definition

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

• X is holomorphically convex, i.e. for every compact subset ${\displaystyle K\subset X}$, the so-called holomorphically convex hull,
${\displaystyle {\bar {K}}=\left\{z\in X\left||f(z)|\leq \sup _{w\in K}|f(w)|\ \forall f\in {\mathcal {O}}(X)\right.\right\},}$
is also a compact subset of X.
• X is holomorphically separable, i.e. if ${\displaystyle x\neq y}$ are two points in X, then there exists ${\displaystyle f\in {\mathcal {O}}(X)}$ such that ${\displaystyle f(x)\neq f(y).}$
• The open neighborhood of any point on the manifold has a holomorphic Chart to the ${\displaystyle {\mathcal {O}}(X)}$.

#### Non-compact Riemann surfaces are Stein

Let X be a connected, non-compact Riemann surface. A deep theorem of Heinrich Behnke and Stein (1948)[ref 30] 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 ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0}$. The exponential sheaf sequence leads to the following exact sequence:

${\displaystyle H^{1}(X,{\mathcal {O}}_{X})\longrightarrow H^{1}(X,{\mathcal {O}}_{X}^{*})\longrightarrow H^{2}(X,\mathbb {Z} )\longrightarrow H^{2}(X,{\mathcal {O}}_{X})}$

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

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

#### Levi problem

Cartan extended Levi's problem to Stein manifolds.[ref 31]

If the relative compact open subset ${\displaystyle D\subset X}$ of the Stein manifold X is a Cartan pseudoconvex, then D is a Stein manifold, and conversely, if D is a Cartan pseudoconvex, then X is a Stein manifold. i.e. Then X is a Stein manifold if and only if D is locally the Stein manifold.[ref 32]

This was proved by Bremermann[ref 33] by embedding it in a sufficiently high dimensional ${\displaystyle \mathbb {C} ^{m}}$, and reducing it to the result of Oka.[ref 10]

Also, Grauert proved for arbitrary complex manifolds.[ref 34][ref 12]

If the relative compact subset ${\displaystyle D\subset X}$ of a arbitrary complex manifold X is a strongly pseudoconvex on X, then X is a holomorphically convex (i.e. Stein manifold). Also, D is itself a Stein manifold.

#### Properties and examples of Stein manifolds

• The standard[note 18] complex space ${\displaystyle \mathbb {C} ^{n}}$ is a Stein manifold.
• Every domain of holomorphy in ${\displaystyle \mathbb {C} ^{n}}$ is a Stein manifold.
• 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 ${\displaystyle n}$ can be embedded into ${\displaystyle \mathbb {C} ^{2n+1}}$ by a biholomorphic proper map.

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.
• 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 for Riemann surfaces, due to Behnke and Stein.
• Every Stein manifold ${\displaystyle X}$ is holomorphically spreadable, i.e. for every point ${\displaystyle x\in X}$, there are ${\displaystyle n}$ holomorphic functions defined on all of ${\displaystyle X}$ 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, i.e. a smooth real function ${\displaystyle \psi }$ on ${\displaystyle X}$ (which can be assumed to be a Morse function) with ${\displaystyle i\partial {\bar {\partial }}\psi >0}$, such that the subsets ${\displaystyle \{z\in X\mid \psi (z)\leq c\}}$ are compact in X for every real number ${\displaystyle c}$. This is a solution to the so-called Levi problem,[ref 35] named after E. E. Levi (1911). The function ${\displaystyle \psi }$ invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains. A Stein domain is the preimage ${\displaystyle \{z\mid -\infty \leq \psi (z)\leq c\}}$. Some authors call such manifolds therefore strictly pseudoconvex manifolds.[ref 34]
• 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 ${\displaystyle X_{c}=f^{-1}(c)}$ is a contact structure that induces an orientation on Xc agreeing with the usual orientation as the boundary of ${\displaystyle f^{-1}(-\infty ,c).}$ That is, ${\displaystyle f^{-1}(-\infty ,c)}$ is a Stein filling of Xc.

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

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

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".

## Annotation

1. ^ a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the analytic geometry learned at school
2. ^ The field of complex numbers is a 2-dimensional vector space over real numbers.
3. ^ This may seem nontrivial, but it's known as Osgood's lemma. Osgood's lemma can be proved from the establishment of Cauchy's integral formula, also Cauchy's integral formula can be proved by assuming separate holomorphicity and continuity, so it is appropriate to define it in this way.
4. ^ It is not separate continuous.
5. ^ Using Hartogs's theorem on separate holomorphicity, If condition (ii) is met, it will be derived to be continuous. But, there is no theorem similar to several real variables, and there is no theorem that indicates the continuity of the function, assuming differentiability.
6. ^ According to the Jordan curve theorem, domain D is bounded closed set.
7. ^ But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge.
8. ^ For several variables, the boundary of any domain is not always the natural boundary, so depending on how the domain is taken, there may not be a holomorphic function that makes that domain the natural boundary. See domain of holomorphy for an example of a condition where the boundary of a domain is a natural boundary.
9. ^ Note that from Hartogs' extension theorem, the zeros of holomorphic functions of several variables are not isolated points. Therefore, for several variables it is not enough that ${\displaystyle f=g}$ is satisfied at the accumulation point.
10. ^ The final paragraph reduces to: A Reinhardt domain is a domain of holomorphy if and only if it is logarithmically convex.
11. ^ The idea of the sheaf itself is by Jean Leray.
12. ^ ${\displaystyle \partial D}$ cannot be "touched from inside" by a sequence of analytic surfaces
13. ^ 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.
14. ^ In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification[ref 19][ref 20] on the complex analytic space (when n = 2, the result by Hirzebruch,[ref 21] when n = 3 the result by Zariski[ref 22] for algebraic varietie.), but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy. [ref 23]
15. ^ The Cartan–Thullen theorem
16. ^ See Oka's lemma
17. ^ Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.
18. ^ ${\displaystyle \mathbb {C} ^{n}\times P_{m}}$ (${\displaystyle P_{m}}$ is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.

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