# Function of several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space $\mathbb {C} ^{n}$ , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.

As in complex analysis 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 locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, every domain[note 1]($D\subset \mathbb {C}$ ), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains ($D\subset \mathbb {C} ^{n},\ n\geq 2$ ) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. 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 complex projective varieties ($\mathbb {CP} ^{n}$ ) and has a different flavour to complex analytic geometry in $\mathbb {C} ^{n}$ or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.

## Historical perspective

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. 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, Pierre Cousin [fr], 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

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

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 every open connected set D in $\mathbb {C}$  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 (especially in complex coordinate spaces $\mathbb {C} ^{n}$  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, also, the property that the sheaf cohomology group disappears is also found in other high-dimensional complex manifolds, indicating that the Hodge manifold is projective. 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 2] 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 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 complex coordinate space

The complex coordinate space $\mathbb {C} ^{n}$  is the Cartesian product of n copies of $\mathbb {C}$ , and when $\mathbb {C} ^{n}$  is a domain of holomorphy, $\mathbb {C} ^{n}$  can be regarded as a Stein manifold, and more generalized Stein space. $\mathbb {C} ^{n}$  is also considered to be a complex projective variety, a Kähler manifold, etc. It is also an n-dimensional vector space over the complex numbers, which gives its dimension 2n over $\mathbb {R}$ .[note 3] Hence, as a set and as a topological space, $\mathbb {C} ^{n}$  may be identified to the real coordinate space $\mathbb {R} ^{2n}$  and its topological dimension is thus 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

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

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 $\mathbb {C} ^{n}$  to $\mathbb {C} ^{n}$ .

### Connected space

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

### Compact

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

## Holomorphic functions

### Definition

When a function f defined on the domain D is complex-differentiable at each point on D, f is said to be holomorphic on D. When the function f defined on the domain D satisfies the following conditions, it is complex-differentiable at the point $z^{0}$  on D;

Let $|z-z^{0}|=|z_{1}-z_{1}^{0}|+|z_{2}-z_{2}^{0}|+\cdots +|z_{n}-z_{n}^{0}|,\varepsilon (z,z^{0})=f(z)-f(z^{0})-\sum _{\nu =1}^{n}\alpha _{\nu }(z_{\nu }-z_{\nu }^{0})\ (\alpha _{\nu }\in \mathbb {C} )$ ,
$\lim _{z\to z^{0}}{\frac {\varepsilon (z,z^{0})}{|z-z^{0}|}}=0$ , since such $\alpha _{1},\dots ,\alpha _{n}$  are uniquely determined, they are called the partial differential coefficients of f, and each are written as ${\frac {\partial f}{\partial z_{1}}}(z^{0}),\dots ,{\frac {\partial f}{\partial z_{n}}}(z^{0})$

Therefore, when a function f is holomorphic on the domain $D\subset \mathbb {C} ^{n}$ , then f satisfies the following two conditions.

1. f is continuous on D[note 4]
2. f is holomorphic in each variable separately, that is f is separate holomorphicity, namely,
${\frac {\partial f}{\partial {\overline {z}}_{\nu }}}=0$
On the converse, when these conditions are satisfied, the function f is holomorphic (as described later), and this condition is called Osgood's lemma. However, note that condition (B) depends on the properties of the domain (as described later).

### Cauchy–Riemann equations

For each index ν let

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

and[clarification needed]

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

Then as expected,

$\left({\frac {\partial }{\partial z_{\nu }}}\right)dz_{\lambda }=\delta _{\nu \lambda },\left({\frac {\partial }{\partial {\bar {z}}_{\nu }}}\right)dz_{\lambda }=0,\left({\frac {\partial }{\partial z_{\nu }}}\right)d{\bar {z}}_{\lambda }=0,\left({\frac {\partial }{\partial {\bar {z}}_{\nu }}}\right)d{\bar {z}}_{\lambda }=\delta _{\nu \lambda }$

through, let $\delta _{\nu \lambda }$  be the Kronecker delta, that is $\delta _{\nu \nu }=1$ , and $\delta _{\nu \lambda }=0$  if $\nu \neq \lambda$ .

When, ${\frac {\partial f}{\partial {\overline {z}}_{\nu }}}=0\ (\nu =1,\dots ,n)$

then,

${\frac {1}{2}}\left[\left({\frac {\partial }{\partial x_{\nu }}}-i{\frac {\partial }{\partial y_{\nu }}}\right)+\left({\frac {\partial }{\partial x_{\nu }}}+i{\frac {\partial }{\partial y_{\nu }}}\right)\right]=0\ (\nu =1,\dots ,n)$

therefore,

${\frac {\partial u}{\partial x_{\nu }}}={\frac {\partial v}{\partial y_{\nu }}},\ \ \ \ {\frac {\partial u}{\partial y_{\nu }}}=-{\frac {\partial v}{\partial x_{\nu }}}\ (\nu =1,\dots ,n).$

This satisfies the Cauchy–Riemann equation of one variable to each index ν, then f is a separate holomorphic.

### Cauchy's integral formula I (Polydisc version)

Prove the sufficiency of two conditions (A) and (B). Let f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve $\gamma$ , $\gamma _{\nu }$  is piecewise smoothness, class ${\mathcal {C}}^{1}$  Jordan closed curve. ($\nu =1,2,\ldots ,n$ ) Let $D_{\nu }$  be the domain surrounded by each $\gamma _{\nu }$ . Cartesian product closure ${\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}$  is ${\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}\in D$ . Also, take the closed polydisc ${\overline {\Delta }}$  so that it becomes ${\overline {\Delta }}\subset {D_{1}\times D_{2}\times \cdots \times D_{n}}$ . (${\overline {\Delta }}(z,r)=\left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in \mathbb {C} ^{n};\left|\zeta _{\nu }-z_{\nu }\right|\leq r_{\nu }{\text{ for all }}\nu =1,\dots ,n\right\}$  and let $\{z\}_{\nu =1}^{n}$  be the center of each disk.) Using the Cauchy's integral formula of one variable repeatedly, [note 5]

{\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 $\partial D$  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,

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

(1)

#### Cauchy's evaluation formula

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

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

(2)

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

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

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

Therefore, Liouville's theorem hold.

#### Power series expansion of holomorphic functions on polydisc

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

{\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}}

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

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

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

We have already explained that holomorphic functions 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 $f_{1},\ldots ,f_{n}$  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.
${\frac {\partial ^{k_{1}+\cdots +k_{n}}f}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}=\sum _{v=1}^{\infty }{\frac {\partial ^{k_{1}+\cdots +k_{n}}f_{v}}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}$ 

#### Radius of convergence of power series

It is possible to define a combination of positive real numbers $\{r_{\nu }\ (\nu =1,\dots ,n)\}$  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 $\left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};|z_{\nu }-a_{\nu }|  and does not converge uniformly at $\left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};|z_{\nu }-a_{\nu }|>r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}$ .

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.

#### Laurent series expansion

Let $\omega (z)$  be holomorphic in the annulus $\left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};r_{\nu }<|z|  and continuous on their circumference, then there exists the following expansion ;

{\begin{aligned}\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{aligned}}

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 $r'_{\nu }<|z| , where $r'_{\nu }>r_{\nu }$  and $R'_{\nu } , and so it is possible to integrate term.

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

The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many domain, so we introduce the Bochner–Martinelli formula.

Suppose that f is a continuously differentiable function on the closure of a domain D on $\mathbb {C} ^{n}$  with piecewise smooth boundary $\partial D$ , and let the symbol $\land$  denotes the exterior or wedge product of differential forms. Then the Bochner–Martinelli formula states that if z is in the domain D then, for $\zeta$ , z in $\mathbb {C} ^{n}$  the Bochner–Martinelli kernel $\omega (\zeta ,z)$  is a differential form in $\zeta$  of bidegree $(n,n-1)$ , defined by

$\omega (\zeta ,z)={\frac {(n-1)!}{(2\pi i)^{n}}}{\frac {1}{|z-\zeta |^{2n}}}\sum _{1\leq j\leq n}({\overline {\zeta }}_{j}-{\overline {z}}_{j})\,d{\overline {\zeta }}_{1}\land d\zeta _{1}\land \cdots \land d\zeta _{j}\land \cdots \land d{\overline {\zeta }}_{n}\land d\zeta _{n}$
$\displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z)-\int _{D}{\overline {\partial }}f(\zeta )\land \omega (\zeta ,z).$

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

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

### Identity theorem

When the function f,g is analytic in the 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 point of analytic. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold. For a generalized version of the implicit function theorem to complex variables, see the Weierstrass preparation theorem

### Biholomorphism

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

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

#### The Riemann mapping theorem does not hold

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

### Analytic continuation

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

## Reinhardt domain

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

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

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

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

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

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

A Reinhardt domain D is called a complete Reinhardt domain if together with all point $z^{0}\in D$  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 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 the Cauchy's integral theorem without using the Jordan curve theorem.

### Logarithmically-convex

A Reinhardt domain D is called logarithmically convex if the image $\lambda (D^{*})$  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 $\mathbb {R} ^{n}$ .

Every such domain in $\mathbb {C} ^{n}$  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 every power series in $z_{1},\dots ,z_{n}$  is a logarithmically-convex Reinhardt domain with centre $a=0$ . [note 10]

### Some results

#### Hartogs's extension theorem and Hartogs's phenomenon

When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the $\mathbb {C} ^{n}$  were all connected to larger domain.

On the polydisk consisting of two disks $\Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}$  when $0<\varepsilon <1$ .
Internal domain of $H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2};|z_{1}|<\varepsilon \ \cup \ 1-\varepsilon <|z_{2}|\}\ (0<\varepsilon <1)$
Hartogs's extension theorem (1906); Let f be a holomorphic function on a set G \ K, where G is a bounded (surrounded by a rectifiable closed Jordan curve) domain[note 11] on $\mathbb {C} ^{n}$  (n ≥ 2) and K is a compact subset of G. If the complement 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.
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.

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

#### Thullen's classic results

Thullen's 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. $\{(z,w)\in \mathbb {C} ^{2};~|z|<1,~|w|<1\}$  (polydisc);
2. $\{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{2}<1\}$  (unit ball);
3. $\{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{\frac {2}{p}}<1\}\,(p>0,\neq 1)$  (Thullen domain).

Toshikazu Sunada (1978) 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 $\varphi :\mathbb {C} ^{n}\to \mathbb {C} ^{n}$  given by $z_{i}\mapsto r_{i}z_{\sigma (i)}(r_{i}>0)$ , $\sigma$  being a permutation of the indices), such that $\varphi (G_{1})=G_{2}$ .

## Natural domain of the holomorphic function (domain of holomorphy)

When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of the domain are natural boundaries (In the complex coordinate space $\mathbb {C} ^{n}$  call the domain of holomorphy), the first result of 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. (First for $\mathbb {C} ^{2}$ , later extended to $\mathbb {C} ^{n}$ .) Kiyoshi Oka's notion of idéal de domaines indéterminés is interpreted theory of sheaf cohomology by H. Cartan and more development Serre.[note 13] In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds. The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.

### Domain of holomorphy

The sets in the definition. Note: On this section, replace $\Omega$  in the figure with D

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

Formally, a domain D in the n-dimensional complex coordinate space $\mathbb {C} ^{n}$  is called a domain of holomorphy if there do not exist non-empty domain $U\subset D$  and $V\subset \mathbb {C} ^{n}$ , $V\not \subset D$  and $U\subset D\cap V$  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, the every domain ($D\subset \mathbb {C}$ ) was the 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.

#### Properties of the domain of holomorphy

• If $D_{1},\dots ,D_{n}$  are domains of holomorphy, then their intersection ${\textstyle D=\bigcap _{\nu =1}^{n}D_{\nu }}$  is also a domain of holomorphy.
• If $D_{1}\subseteq D_{2}\subseteq \cdots$  is an increasing 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 $D_{1}$  and $D_{2}$  are domains of holomorphy, then $D_{1}\times D_{2}$  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 $n\geq 3$ . this is also true, with additional topological assumptions, for the second Cousin problem.

### Holomorphically convex hull

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

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

One obtains a narrower concept of polynomially convex hull by taking ${\mathcal {O}}(G)$  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 $K,{\hat {K}}_{G}$  is also compact in G. Sometimes this is just abbreviated as holomorph-convex.

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

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

### Pseudoconvexity

Hartogs showed that

Hartogs (1906): Let D be a Hartogs's domain on $\mathbb {C}$  and R be a positive function on D such that the set $\Omega$  in $\mathbb {C} ^{2}$  defined by $z_{1}\in D$  and $|z_{2}|  is a domain of holomorphy. Then $-\log {R}(z_{1})$  is a subharmonic function on D.

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.[note 14] 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 are important, as they allow for classification of domains of holomorphy.

#### Definition of plurisubharmonic function

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

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

$\{a+bz;z\in \mathbb {C} \}\subset \mathbb {C} ^{n}$  with $a,b\in \mathbb {C} ^{n}$
the function $z\mapsto f(a+bz)$  is a subharmonic function on the set
$\{z\in \mathbb {C} ;a+bz\in D\}.$
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
$f\colon X\to \mathbb {R} \cup \{-\infty \}$
is said to be plurisubharmonic if and only if for any holomorphic map

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

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

is subharmonic, where $\Delta \subset \mathbb {C}$  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 $\Delta =4\left({\frac {\partial ^{2}u}{\partial z\,\partial {\overline {z}}}}\right)\geq 0$ . There fore, if $u$  is of class ${\mathcal {C}}^{2}$ , then $u$  is plurisubharmonic if and only if the hermitian matrix $H_{u}=(\lambda _{ij}),\lambda _{ij}={\frac {\partial ^{2}u}{\partial z_{i}\,\partial {\bar {z}}_{j}}}$  is positive semidefinite.

Equivalently, a ${\mathcal {C}}^{2}$ -function u is plurisubharmonic if and only if ${\sqrt {-1}}\partial {\bar {\partial }}f$  is a positive (1,1)-form.: 39–40

##### Strictly plurisubharmonic function

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

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

Weak pseudoconvex is defined as : Let $X\subset {\mathbb {C} }^{n}$  be a domain. One says that X is pseudoconvex if there exists a continuous plurisubharmonic function $\varphi$  on X such that the set $\{z\in X;\varphi (z)\leq \sup x\}$  is a relatively compact subset of X for all real numbers x. [note 15] i.e. there exists a smooth plurisubharmonic exhaustion function $\psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)$ . 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 $\psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)$ .: 49

#### Strongly (Strictly) pseudoconvex

Let X be a complex n-dimensional manifold. Strongly pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function $\psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)$ ,i.e., $H\psi$  is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.: 49  The strong Levi pseudoconvex domain is simply called strong pseudoconvex and is often called strictly pseudoconvex to make it clear that it has a strictly plurisubharmonic exhaustion function in relation to the fact that it may not have a strictly plurisubharmonic exhaustion function.

#### (Weakly) Levi(–Krzoska) pseudoconvexity

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

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

For arbitrary complex manifold, Levi (–Krzoska) pseudoconvexity does not always have an plurisubharmonic exhaustion function, i.e. it does not necessarily have a (p-)pseudoconvex domain.

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

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

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

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

##### Strongly Levi (–Krzoska) pseudoconvex (Strongly pseudoconvex)

When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly pseudoconvex.

##### Levi total pseudoconvex

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

#### Oka pseudoconvex

##### Family of Oka's disk

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

##### Definition

When $Q(0)\subset D$  holds on any family of Oka's disk, D is called Oka pseudoconvex. Oka's proof of Levi's problem was that when the unramified Riemann domain over $\mathbb {C} ^{n}$  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.

#### Locally pseudoconvex (locally Stein, Cartan pseudoconvex, local Levi property)

For every point $x\in \partial D$  there exist a neighbourhood U of x and f holomorphic. ( i.e. $U\cap D$  be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. i.e., let $\psi :X\to Y$  be a holomorphic map, if every point $y\in Y$  has a neighborhood U such that $\psi ^{-1}(U)$  admits a ${\mathcal {C}}^{\infty }$ -plurisubharmonic exhaustion function (weakly 1-complete), 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 $\mathbb {C} ^{n}$  the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.

### Conditions equivalent to domain of holomorphy

For a domain $D\subset \mathbb {C} ^{n}$  the following conditions are equivalent.[note 16]:

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 $1\Leftrightarrow 2\Leftrightarrow 3$ ,[note 17] $1\Rightarrow 4$ ,[note 18] and $4\Rightarrow 5$  are standard results. Proving $5\Rightarrow 1$ , 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 solved the this ploblem for unramified Riemann domains over $\mathbb {C} ^{n}$  by Kiyoshi Oka,[note 19] but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\bar {\partial }}$ -problem(equation) with a L2 methods).

## Sheaf

### Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))

Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains". Specifically, it is a set $(I)$  of pairs $(f,\delta )$ , $f$  holomorphic on a non-empty open set $\delta$ , such that

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

The origin of indeterminate domains comes from the fact that domains change depending on the pair $(f,\delta )$ . Cartan translated this notion into the notion of the coherent (sheaf) (Especially, coherent analytic sheaf) in sheaf cohomology. This name comes from H. Cartan. Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf. The notion of coherent (coherent sheaf cohomology) helped solve the problems in several complex variables.

### Coherent sheaf

#### Definition

The definition of the coherent sheaf is as follows.: 83–89  A quasi-coherent sheaf on a ringed space $(X,{\mathcal {O}}_{X})$  is a sheaf ${\mathcal {F}}$  of ${\mathcal {O}}_{X}$ -modules which has a local presentation, that is, every point in $X$  has an open neighborhood $U$  in which there is an exact sequence

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

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

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

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

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

Also, Jean-Pierre Serre (1955) proves that

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

#### (Oka–Cartan) coherent theorem

(Oka–Cartan) coherent theorem says that each sheaf that meets the following conditions is a coherent.

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

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

### Cousin problem

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and Weierstrass factorization theorem was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). However, these theorems do not hold because the singularities of analytic function in several complex variables is not isolated points, 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 Oka who showed[note 20] the conditions for solving first Cousin problem for the domain of holomorphy[note 21] on the complex coordinate space, and 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 propertie a pure topological, and Serre called this the Oka principle. 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, that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.)

#### First Cousin problem

##### Definition without sheaf cohomology words

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

##### Definition using sheaf cohomology 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.

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

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 cohomology words

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

##### Definition using sheaf cohomology words

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

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

The long exact sheaf cohomology sequence associated to the quotient is

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

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

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

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

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

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

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

## Manifolds and analytic varieties with several complex variables

### Stein manifold (non-compact complex manifold)

Since a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, 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 $\mathbb {C}$ . (In fact, Gunning and Narasimhan have shown (1967) that every non-compact Riemann surface actually has a holomorphic immersion into the complex plane. In other words, there is a holomorphic mapping into the complex plane whose derivative never vanishes.) The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of $\mathbb {R} ^{2n}$ , whereas it is "rare" for a complex manifold to have a holomorphic embedding into $\mathbb {C} ^{n}$ . 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 $\mathbb {C} ^{n}$ , then the coordinate functions of $\mathbb {C} ^{n}$  would restrict to nonconstant holomorphic functions on X, contradicting compactness, except in the case that X is just a point. Complex manifolds that can be holomorphic embedded into $\mathbb {C} ^{n}$  are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.

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). 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 $\mathbb {C} ^{n}$  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 $n$  and let ${\mathcal {O}}(X)$  denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold:

1. X is holomorphically convex, i.e. for every compact subset $K\subset X$ , the so-called holomorphically convex hull,
${\bar {K}}=\left\{z\in X;|f(z)|\leq \sup _{w\in K}|f(w)|\ \forall f\in {\mathcal {O}}(X)\right\},$
is also a compact subset of X.
2. X is holomorphically separable,[note 22] i.e. if $x\neq y$  are two points in X, then there exists $f\in {\mathcal {O}}(X)$  such that $f(x)\neq f(y).$
3. The open neighborhood of every point on the manifold has a holomorphic chart to the ${\mathcal {O}}(X)$ .

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

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

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

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

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

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

#### Levi problems

Cartan extended Levi's problem to Stein manifolds.

If the relative compact open subset $D\subset X$  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.

This was proved by Bremermann by embedding it in a sufficiently high dimensional $\mathbb {C} ^{n}$ , and reducing it to the result of Oka.

Also, Grauert proved for arbitrary complex manifolds M.[note 23]

If the relative compact subset $D\subset M$  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 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.

Levi's problem remains unresolved in the following cases;

Suppose that X is a singular Stein space,[note 24] $D\subset \subset X$  . Suppose that for all $p\in \partial D$  there is an open neighborhood $U(p)$  so that $U\cap D$  is Stein space. Is D itself Stein?

more generalized

Suppose that N be a Stein space and f an injective, and also $f:M\to N$  a Riemann unbranched domain, such that map f is a locally pseudoconvex map (i.e. Stein morphism). Then M is itself Stein ?: 109

and also,

Suppose that X be a Stein space and $D=\bigcup _{n\in \mathbb {N} }D_{n}$  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. 

##### K-complete

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

Let X is complex manifold, X is K-complete if, to each point $x_{0}\in X$ , there exist finitely many holomorphic map $f_{1},\dots ,f_{k}$  of X into $\mathbb {C} ^{p}$ , $p=p(x_{0})$ , such that $x_{0}$  is an isolated point of the set $A=\{x\in X;f^{-1}f(x_{0})\ (v=1,\dots ,k)\}$ . This concept also applies to complex analytic space.

#### Properties and examples of Stein manifolds

• The standard[note 25] complex space $\mathbb {C} ^{n}$  is a Stein manifold.
• Every domain of holomorphy in $\mathbb {C} ^{n}$  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 n can be embedded into $\mathbb {C} ^{2n+1}$  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,[note 26] due to Behnke and Stein.
• Every Stein manifold X is holomorphically spreadable, i.e. for every point $x\in X$ , 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, i.e. a smooth real function $\psi$  on X (which can be assumed to be a Morse function) with $i\partial {\bar {\partial }}\psi >0$ , such that the subsets $\{z\in X\mid \psi (z)\leq c\}$  are compact in X for every real number c. This is a solution to the so-called Levi problem, named after E. E. Levi (1911). The function $\psi$  invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domain. A Stein domain is the preimage $\{z\mid -\infty \leq \psi (z)\leq c\}$ . 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 Xc agreeing with the usual orientation as the boundary of $f^{-1}(-\infty ,c).$  That is, $f^{-1}(-\infty ,c)$  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".

### Complex projective varieties (compact complex manifold)

Meromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the Riemann-Roch theorem (Riemann's inequality) holds for compact Riemann surfaces (Therefore the theory of compact Riemann surface can be regarded as the theory of (smooth projective) algebraic curve over $\mathbb {C}$ ). In fact, compact Riemann surface had a non-constant single-valued meromorphic function, and also a compact Riemann surface had enough meromorphic functions. A compact one-dimensional complex manifold was a Riemann sphere ${\widehat {\mathbb {C} }}\cong \mathbb {CP} ^{1}$ . However, the abstract notion of a compact Riemann surface is always algebraizable (The Riemann's existence theorem, Kodaira embedding theorem.),[note 27] but it is not easy to verify which compact complex analytic spaces are algebraizable. In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions. However, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic.  The generalization of the Riemann-Roch theorem to several complex variables was first extended by Kodaira to compact analytic surfaces, and then to three-dimensional, and then n-dimensional Kähler varieties. Serre formulated the Riemann-Roch theorem as a problem of dimension of coherent sheaf cohomology, and also Serre proved Serre duality. Cartan-Serre proved the following property: the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M. Hirzebruch generalized the theorem to compact complex manifolds in 1994 (The Hirzebruch–Riemann–Roch theorem) and Grothendieck more generalized it (The Grothendieck–Hirzebruch–Riemann–Roch theorem).  Next consider example of expanding the notion of closed (compact) Riemann surface to a higher dimension ,that is, consider that compactification of $\mathbb {C} ^{n}$ , specifically, consider the conditions that when embedding of compact complex submanifold X into the complex projective space $\mathbb {CP} ^{n}$ . [note 28] i.e., gives the conditions when a compact complex manifold is projective. The Kodaira vanishing theorem and its generalization Nakano vanishing theorem etc. gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of positivity. As an example given by this theorem, Kodaira embedding theorem 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. Chow's theorem 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. 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 gives an example of a complex manifold with enough meromorphic functions. Similarities in the Levi problems on the complex projective space $\mathbb {CP} ^{n}$ , have been proved in some patterns, for example by Takeuchi. 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.

## Annotation

1. ^ That is an open connected subset.
2. ^ 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.)
3. ^ The field of complex numbers is a 2-dimensional vector space over real numbers.
4. ^ Using Hartogs's theorem on separate holomorphicity, If condition (B) 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.
5. ^ Note that this formula only holds for polydisc. See §Bochner–Martinelli formula for the Cauchy's integral formula on the more general domain.
6. ^ According to the Jordan curve theorem, domain D is bounded closed set, that is, each domain $D_{\nu }$  is compact.
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 each domain is not always the natural boundary, so depending on how the domain is taken, there may not be a analytic 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 or Weierstrass preparation theorem , the zeros of analytic functions of several variables cannot have isolated singularities. Therefore, for several variables it is not enough that $f=g$  is satisfied at the accumulation point.
10. ^ 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.
11. ^ 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.
12. ^ Oka says that the contents of these two papers are different.
13. ^ The idea of the sheaf itself is by Jean Leray.
14. ^ In fact, this was proved by Kiyoshi Oka with respect to $\mathbb {C} ^{n}$  domain.See Oka's lemma.
15. ^ 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.
16. ^ 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 on the complex analytic space (when n = 2, the result by Hirzebruch, when n = 3 the result by Zariski 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. 
17. ^ This relation is called the Cartan–Thullen theorem.
18. ^ See Oka's lemma
19. ^ Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.
20. ^ This is called the classic Cousin problem.
21. ^ There are some counterexample in the domain of holomorphicity regarding second Cousin problem.
22. ^ From this condition, we can see that the Stein manifold is not compact.
23. ^ Levi problem is not true for domains in arbitrary manifolds.
24. ^ In the case of Stein space with isolated singularities, it has already been positively solved by Narasimhan.
25. ^ $\mathbb {C} ^{n}\times \mathbb {P} _{m}$  ($\mathbb {P} _{m}$  is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.
26. ^ The proof method uses an approximation by the polyhedral domain, as in Oka-Weil theorem.
27. ^ 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.
28. ^ This is the standard method for compactification of $\mathbb {C} ^{n}$ , but not the only method like the Riemann sphere that was compactification of $\mathbb {C}$ .