Several complex variables

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

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

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

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

through

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

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

f is meets condition 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 ${\displaystyle {\overline {\Delta }}\subset {D_{1}\times D_{2}\times \cdots \times D_{n}}}$. (${\displaystyle {\overline {\Delta }}(z,r)=\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in C^{n}:\left|\zeta _{\nu }-z_{\nu }\right|\leq r_{\nu }{\text{ for all }}\nu =1,\dots ,n\}}$ 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).

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 ${\displaystyle \{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in {\mathbb {C} }^{n}\mid |\zeta _{\nu }-z_{\nu }|\leq r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}}$ 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.

If ${\displaystyle f(z_{1},\ldots ,z_{n})}$ is holomorphic, on polydisc ${\displaystyle \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|<r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}}$, 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]

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

In this way it is possible to have a similar radius of convergence[note 7] for a single complex variable, but there is a point where it converges outside the circle of convergence.[note 8]

When the function f,g is holomorphic in the concatenated domain D,[note 9] even for several complex variables, the identity theorem[note 10] 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.

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.

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 ${\displaystyle \mathbf {R} ^{n}}$. 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 ${\displaystyle \sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ }$, 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 11]

Thullen's[ref 4] 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).

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 5] any holomorphic functions f on ${\displaystyle H_{\varepsilon }}$ are analytically continued to ${\displaystyle \Delta ^{2}}$ . Namely, there is a holomorphic function F on ${\displaystyle \Delta ^{2}}$ such that ${\displaystyle F=f}$ on ${\displaystyle H_{\varepsilon }}$ .

The convergence domain extends from ${\displaystyle H_{\varepsilon }}$ to ${\displaystyle \Delta ^{2}}$. i.e. The convergent domain of ${\displaystyle H_{\varepsilon }}$ is extended to the smallest complete Reinhardt domain ${\displaystyle \Delta ^{2}}$ that can cover ${\displaystyle H_{\varepsilon }}$.

Toshikazu Sunada (1978)[ref 6] 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 ${\displaystyle \varphi$ :\mathbf {C} ^{n}\longrightarrow \mathbf {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 ${\displaystyle \varphi (G_{1})=G_{2}}$.

Function f is holomorpic on the domain ${\displaystyle D\subset \mathbb {C} ^{n}}$, when f 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.

The first result on the properties of the domain of holomorphy is the regular convexity of Henri Cartan and Peter Thullen (1932).[ref 13]

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

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

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 ${\displaystyle \Delta }$.

For every sequence ${\displaystyle S_{n}\subseteq D}$ of analytic compact surfaces such that ${\displaystyle S_{n}\rightarrow S,\partial S_{n}\rightarrow \Gamma }$ for some set ${\displaystyle \Gamma }$ we have ${\displaystyle S\subseteq D}$. i.e. Approximate from the inside by analytic compact surfaces (${\displaystyle \partial D}$ cannot be "touched from inside" by a sequence of analytic surfaces).

Pseudoconvex Hartogs showed that ${\displaystyle -\log R}$ is subharmonic for the radius of convergence ${\displaystyle R(z)}$ 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.

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.

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 ${\displaystyle \mathbf {C} ^{2}}$-class, we call u a strict plural subharmonic function.

Weak pseudoconvex[ref 14] 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 12] i.e there exists a smooth plurisubharmonic exhaustion function ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$.

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 14]

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.

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 15]

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 ${\displaystyle Q(t)\subset D\ (0<t)}$ and ${\displaystyle B(0)\subset D}$, Q(t) is called Family of Oka's disk.[ref 15]

When ${\displaystyle Q(0)\subset D}$ holds on any Family of Oka's disk, D is called Oka pseudoconvex.[ref 15]

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 15]

For a domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ the following conditions are equivalent.[note 13]:

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

Implications ${\displaystyle 1\Leftrightarrow 2,}$[note 14] ${\displaystyle 3\Leftrightarrow 4,1\Rightarrow 4,3\Rightarrow 5}$ are standard results (for ${\displaystyle 1\Rightarrow 3}$, see Oka's lemma). 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, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\displaystyle {\bar {\partial }}}$-problem).

• If ${\displaystyle D_{1},\dots ,D_{n}}$ are domains of holomorphy, then their intersection ${\displaystyle D=\bigcap _{\nu =1}^{n}D_{\nu }}$ 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 ${\displaystyle D=\bigcup _{n=1}^{\infty }D_{n}}$ 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.

The definition of the coherent sheaf is as follows.[ref 21]

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 21] 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 ${\displaystyle J}$.

Kiyoshi Oka (1950)[ref 10] proved the following

Sheaf of holomorphic function germ ${\displaystyle {\mathcal {O}}}$ on the analytic variety is the coherent sheaf. Therefore, ${\displaystyle {\mathcal {O}}^{p}}$ is also a coherent sheaf. This theorem is also used to prove Cartan's theorems A and B.

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 analytic manifold. 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 22] 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.

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 ${\displaystyle {\mathcal {O}}(X)}$.

Let X be a connected, non-compact Riemann surface. A deep theorem of Heinrich Behnke and Stein (1948)[ref 23] 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.

• The standard[note 15] 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 24] 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 25]
• 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 X_c 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 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.

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".