Function of several complex variables

The theory of functions of several complex variables^{[clarification needed]} is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function ${\displaystyle f:(z_{1},z_{2},\ldots ,z_{n})\rightarrow f(z_{1},z_{2},\ldots ,z_{n})}$ is n-tuples of complex numbers, classically studied on the complex coordinate space ${\displaystyle \mathbb {C} ^{n}}$.

As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z_i. Equivalently, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, the every domain^{[note 1]}(${\displaystyle D\subset \mathbb {C} }$), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy.^{[ref 1][ref 2]} For several complex variables, this is not the case; there exist domains (${\displaystyle D\subset \mathbb {C} ^{n},\ n\geq 2}$) 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.^{[ref 1]} 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 (${\displaystyle \mathbb {CP} ^{n}}$) and has a different flavour to complex analytic geometry in ${\displaystyle \mathbb {C} ^{n}}$ or on Stein manifolds.

Historical perspective[]

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series. Naturally also 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, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function

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

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

After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory; while for every open connected set D in ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {C} ^{n}}$ 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^{[ref 3]} 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 ${\displaystyle \mathbb {C} ^{n}}$ is the Cartesian product of n copies of ${\displaystyle \mathbb {C} }$, and when ${\displaystyle \mathbb {C} ^{n}}$ is a domain of holomorphy, ${\displaystyle \mathbb {C} ^{n}}$ can be regarded as a Stein manifold. ${\displaystyle \mathbb {C} ^{n}}$ is also considered to be a complex projective variety, a Kähler manifold,^{[ref 4]} etc. It is also an n-dimensional vector space over the complex numbers, which gives its dimension 2n over ${\displaystyle \mathbb {R} }$.^{[note 3]} Hence, as a set and as a topological space, ${\displaystyle \mathbb {C} ^{n}}$ may be identified to the real coordinate space ${\displaystyle \mathbb {R} ^{2n}}$ 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

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

with determinant

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

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

Connected space[]

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

Compact[]

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

Holomorphic functions[]

Definition[]

This section may be confusing or unclear to readers. In particular, it is unclear whether this definition is equivalent with the definition given in the lead of holomorphic function. Please help clarify the section. There might be a discussion about this on the talk page. (August 2021) (Learn how and when to remove this template message)

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 ${\displaystyle z^{0}}$ on D;

Let ${\displaystyle |z-z^{0}|=|z_{1}-z_{1}^{0}|+|z_{2}-z_{2}^{0}|+\cdots +|z_{n}-z_{n}^{0}|,\varepsilon (z,z^{0})=f(z)-f(z^{0})-\sum _{\nu =1}^{n}\alpha _{\nu }(z_{\nu }-z_{\nu }^{0})\ (\alpha _{\nu }\in \mathbb {C} )}$,

${\displaystyle \lim _{z\to z^{0}}{\frac {\varepsilon (z,z^{0})}{|z-z^{0}|}}=0}$, since such ${\displaystyle \alpha _{1},\dots ,\alpha _{n}}$ are uniquely determined, they are called the partial differential coefficients of f, and each are written as ${\displaystyle {\frac {\partial f}{\partial {\overline {z}}_{1}}}(z^{0}),\dots ,{\frac {\partial f}{\partial {\overline {z}}_{n}}}(z^{0})}$

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

1. f is continuous^{[note 4]} on D^{[note 5]}
2. For each variable ${\displaystyle z_{\nu }}$, f is holomorphic, namely,

   ${\displaystyle {\frac {\partial f}{\partial {\overline {z}}_{\nu }}}=0}$

   these conditions are called Osgood's lemma.

Cauchy–Riemann equations[]

For each index ν let

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

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

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

Let^{[clarification needed]}

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

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

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

therefore,

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

Cauchy's integral formula[]

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

${\displaystyle {\begin{aligned}f(z_{1},\ldots ,z_{n})&={\frac {1}{2\pi i}}\int _{\partial D_{1}}{\frac {f(\zeta _{1},z_{2},\ldots ,z_{n})}{\zeta _{1}-z_{1}}}\,d\zeta _{1}\\[6pt]&={\frac {1}{(2\pi i)^{2}}}\int _{\partial D_{2}}\,d\zeta _{2}\int _{\partial D_{1}}{\frac {f(\zeta _{1},\zeta _{2},z_{3},\ldots ,z_{n})}{(\zeta _{1}-z_{1})(\zeta _{2}-z_{2})}}\,d\zeta _{1}\\[6pt]&={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{n}}\,d\zeta _{n}\cdots \int _{\partial D_{2}}\,d\zeta _{2}\int _{\partial D_{1}}{\frac {f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{(\zeta _{1}-z_{1})(\zeta _{2}-z_{2})\cdots (\zeta _{n}-z_{n})}}\,d\zeta _{1}\end{aligned}}}$

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

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

(1)

Cauchy's evaluation formula[]

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

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

(2)

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

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

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

Therefore, Liouville's theorem hold.

Power series expansion of holomorphic functions on polydisc[]

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

${\displaystyle {\begin{aligned}&f(z)=\sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ ,\\&c_{k_{1}\cdots k_{n}}={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-a_{1})^{k_{1}+1}\cdots (\zeta _{n}-a_{n})^{k_{n}+1}}}\,d\zeta _{1}\cdots d\zeta _{n}\end{aligned}}}$

In addition, f 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 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 ${\displaystyle f_{1},\ldots ,f_{n}}$ 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 5]}

Radius of convergence of power series[]

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

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

${\displaystyle \omega (z)=\sum _{k=0}^{\infty }{\frac {1}{k!}}{\frac {1}{(2\pi i)^{n}}}\int _{|\zeta _{\nu }|=R_{\nu }}\cdots \int \omega (\zeta )\times \left[{\frac {d^{k}}{dz^{k}}}{\frac {1}{\zeta -z}}\right]_{z=0}df_{\zeta }\cdot z^{k}+\sum _{k=1}^{\infty }{\frac {1}{k!}}{\frac {1}{2\pi i}}\int _{|\zeta _{\nu }|=r_{\nu }}\cdots \int \omega (\zeta )\times \left(0,\cdots ,{\sqrt {\frac {k!}{\alpha _{1}!\cdots \alpha _{n}!}}}\cdot \zeta _{n}^{\alpha _{1}-1}\cdots \zeta _{n}^{\alpha _{n}-1},\cdots 0\right)df_{\zeta }\cdot {\frac {1}{z^{k}}}\ (\alpha _{1}+\cdots +\alpha _{n}=k)}$

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 ${\displaystyle r'_{\nu }<|z|<R'_{\nu }}$, where ${\displaystyle r'_{\nu }>r_{\nu }}$ and ${\displaystyle R'_{\nu }<R_{\nu }}$, and so it is possible to integrate term.^{[ref 6]}

Bochner–Martinelli formula[]

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 Bochner–Martinelli formula.

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

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

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

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

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

Identity theorem[]

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

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 ${\displaystyle \mathbb {C} ^{n}}$, the bijective holomorphic function ${\displaystyle \phi :V\to U}$ and the inverse mapping ${\displaystyle \phi ^{-1}:V\to U}$ is also holomorphic. At this time, ${\displaystyle \phi }$ 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 ${\displaystyle n>1}$, 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.^{[ref 7][ref 2]}

Analytic continuation[]

Let U, V be domain on ${\displaystyle \mathbb {C} ^{n}}$, ${\displaystyle f\in {\mathcal {O}}(U)}$ and ${\displaystyle g\in {\mathcal {O}}(V)}$. Assume that ${\displaystyle U,V,U\cap V\neq \varnothing }$ 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 arbitrarily. Several complex variables have restrictions on this domain, and depending on the shape of the domain , all analytic functions g belonging to V are connected to ${\displaystyle V\subset D}$, and there may be not exist function g with ${\displaystyle \partial V}$ as the natural boundary. In other words, V cannot be defined arbitrarily. 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 single-valued holomorphic function, but as a multivalued analytic function. This means that W is not unique and has different properties in the neighborhood of the branch point than in the case of one variable.

Reinhardt domain[]

In polydisks and the Reinhardt domain, Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but the unique radius of convergence is not defined for each variable. Also, since the Riemann mapping theorem does not hold, polydisks and unit spheres are not biholomorphic mapping. Therefore, the domain of convergence of the power series is not as simple as the case of one variable, but it satisfies the property called Logarithmically-convex. There are various convexity for the domain of convergence of several complex variables.

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

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

A Reinhardt domain D with ${\displaystyle a=0}$ is invariant under the transformations ${\displaystyle \left\{z^{0}\right\}\to \left\{z_{\nu }^{0}e^{i\theta _{\nu }}\right\}}$, ${\displaystyle 0\leq \theta _{\nu }<2\pi }$, ${\displaystyle \nu =1,\dots ,n}$. The Reinhardt domains constitute a subclass of the Hartogs domains ^{[ref 8]} and a subclass of the circular domains, which are defined by the following condition; Together with all points of ${\displaystyle z^{0}\in D}$, the domain contains the set

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

i.e. all points of the circle with center ${\displaystyle a}$ 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 ${\displaystyle a}$ and ${\displaystyle z^{0}}$.

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

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

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

Logarithmically-convex[]

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

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

under the mapping

${\displaystyle \lambda ;z\rightarrow \lambda (z)=(\ln |z_{1}|,\dots ,\ln |z_{n}|)}$

is a convex set in the real coordinate space ${\displaystyle \mathbb {R} ^{n}}$.

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

Some results[]

Hartogs's extension theorem and Hartogs's phenomenon[]

Look at the example on the Hartogs's phenomenon 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 ${\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2};|z_{1}|<\varepsilon \ \cup \ 1-\varepsilon <|z_{2}|\}\ (0<\varepsilon <1)}$

Hartogs's extension theorem (1906);^{[ref 9]} 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 ${\displaystyle \mathbb {C} ^{n}}$ (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.^{[ref 11][ref 10]}

It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for holomorphic functions of several complex variables. The nature of these singularities is also derived from Weierstrass preparation theorem. A generalization of this theorem using the same method as Hartogs was proved in 2007.^{[ref 12][ref 13]}

From Hartogs's extension theorem the domain of convergence extends from ${\displaystyle H_{\varepsilon }}$ to ${\displaystyle \Delta ^{2}}$. Looking at this from the perspective of the Reinhardt domain, ${\displaystyle H_{\varepsilon }}$ is the Reinhardt domain containing the center z = 0, and the domain of convergence of ${\displaystyle H_{\varepsilon }}$ has been extended to the smallest complete Reinhardt domain ${\displaystyle \Delta ^{2}}$ containing ${\displaystyle H_{\varepsilon }}$.^{[ref 14]}

Thullen's classic results[]

Thullen's^{[ref 15]} 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. ${\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|<1,~|w|<1\}}$ (polydisc);
2. ${\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{2}<1\}}$ (unit ball);
3. ${\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{\frac {2}{p}}<1\}(p>0,\neq 1)}$ (Thullen domain).

Sunada's results[]

Toshikazu Sunada (1978)^{[ref 16]} established a generalization of Thullen's result:

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

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 ${\displaystyle \mathbb {C} ^{n}}$ call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen.^{[ref 17]} Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for ${\displaystyle \mathbb {C} ^{2}}$,^{[ref 18]} later extended to ${\displaystyle \mathbb {C} ^{n}}$.^{[ref 19][ref 20]})^{[ref 21]} Kiyoshi Oka's^{[ref 22][ref 23]} notion of idéal de domaines indéterminés is interpreted sheaf cohomology by H. Cartan.^{[ref 24][note 12]} In sheaf cohomology,^{[ref 25]} the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.^{[ref 26]} The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.^{[ref 1]}

Domain of holomorphy[]

When a function f is holomorpic on the domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ and cannot directly connect to the domain outside D, including the point of the domain boundary ${\displaystyle \partial D}$, the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f. In other words, the domain of holomorphy D is the supremum of the domain where the holomorphic function f is holomorphic, and the domain D, which is holomorphic, cannot be extended any more. For several complex variables, i.e. domain ${\displaystyle D\subset \mathbb {C} ^{n}\ (n\geq 2)}$, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.

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

For the ${\displaystyle n=1}$ case, the every domain (${\displaystyle D\subset \mathbb {C} }$) 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 ${\displaystyle D_{1},\dots ,D_{n}}$ are domains of holomorphy, then their intersection ${\textstyle D=\bigcap _{\nu =1}^{n}D_{\nu }}$ is also a domain of holomorphy.
• If ${\displaystyle D_{1}\subseteq D_{2}\subseteq \cdots }$ is an 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 ${\displaystyle D_{1}}$ 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, also Cartan showed that the converse of this result was incorrect for ${\displaystyle n\geq 3}$.^{[ref 27]} this is also true, with additional topological assumptions, for the second Cousin problem.

Holomorphically convex hull[]

Let ${\displaystyle G\subset \mathbb {C} ^{n}}$ be a domain , 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

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

One obtains a narrower concept of polynomially convex hull by taking ${\displaystyle {\mathcal {O}}(G)}$ 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 ${\displaystyle n=1}$, every domain ${\displaystyle G}$ is holomorphically convex since then ${\displaystyle {\hat {K}}_{G}}$ is the union of K with the relatively compact components of ${\displaystyle G\setminus K\subset G}$.

When ${\displaystyle n\geq 1}$, if f satisfies the above holomorphically convexity on D it has the following properties. ${\displaystyle {\text{dist}}(K,D^{c})={\text{dist}}({\hat {K}}_{D},D^{c})}$ for every compact subset K in D, where ${\displaystyle {\text{dist}}(K,D^{c})}$ denotes the distance between K and ${\displaystyle D^{c}=\mathbb {C} ^{n}\setminus D}$. Also, at this time, D is a domain of holomorphy. Therefore, every convex domain ${\displaystyle (D\subset \mathbb {C} ^{n})}$ is domain of holomorphy.^{[ref 2]}

Levi convex (approximate from the inside on the analytic polyhedron domain)[]

${\textstyle \bigcup _{n=1}^{\infty }S_{n}\subseteq D}$ is union of increasing sequence of analytic compact surfaces with paracompact and Holomorphically convex properties such that ${\textstyle \bigcup _{n=1}^{\infty }S_{n}\rightarrow D,S_{n}\subset S_{n+1}}$. i.e. Approximate from the inside by analytic polyhedron. ^{[note 13]}

Pseudoconvex[]

Pseudoconvex Hartogs showed that ${\displaystyle -\log R}$ is subharmonic for the radius of convergence in the Hartogs series ${\displaystyle R(z)}$ when the Hartogs series is a one-variable ${\displaystyle f(z,\omega ),\ \omega =0}$.^{[ref 1]} 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.^{[note 14]} The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain. Pseudoconvex domain are important, as they allow for classification of domains of holomorphy.

Definition of plurisubharmonic function[]

A function

${\displaystyle f\colon D\to {\mathbb {R} }\cup \{-\infty \},}$

with domain ${\displaystyle D\subset {\mathbb {C} }^{n}}$

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

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

the function ${\displaystyle z\mapsto f(a+bz)}$ is a subharmonic function on the set

${\displaystyle \{z\in \mathbb {C} ;a+bz\in D\}.}$

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.

In one-complex variable, necessary and sufficient condition that the real-valued function ${\displaystyle u=u(z)}$, that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is ${\displaystyle \Delta =4\left({\frac {\partial ^{2}u}{\partial z\partial {\overline {z}}}}\right)\geq 0}$. There fore, if ${\displaystyle u}$ is of class ${\displaystyle {\mathcal {C}}^{2}}$, then ${\displaystyle u}$ is plurisubharmonic if and only if the hermitian matrix ${\displaystyle H_{u}=(\lambda _{ij}),\lambda _{ij}={\frac {\partial ^{2}u}{\partial z_{i}\partial {\bar {z}}_{j}}}}$ is positive semidefinite.

Equivalently, a ${\displaystyle C^{2}}$-function u is plurisubharmonic if and only if ${\displaystyle {\sqrt {-1}}\partial {\bar {\partial }}f}$ is a positive (1,1)-form.^{[ref 28]: 39–40}

Strictly plurisubharmonic function[]

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

(Weakly) pseudoconvex (p-pseudoconvex)[]

Weak pseudoconvex is defined as : Let ${\displaystyle X\subset {\mathbb {C} }^{n}}$ be a domain. One says that X is pseudoconvex if there exists a continuous plurisubharmonic function ${\displaystyle \varphi }$ on X such that the set ${\displaystyle \{z\in X;\varphi (z)\leq \sup x\}}$ is a relatively compact subset of X for all real numbers x. ^{[note 15]} i.e. there exists a smooth plurisubharmonic exhaustion function ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$. 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 ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$.^{[ref 28]: 49}

Strongly (Strictly) pseudoconvex[]

Let X be a complex n-dimensional manifold. 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 28]: 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.^{[ref 29]}

(Weakly) Levi(–Krzoska) pseudoconvexity[]

If ${\displaystyle {\mathcal {C}}^{2}}$ boundary , 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 ${\displaystyle {\mathcal {C}}^{2}}$ 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

${\displaystyle H(\rho )=\sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\,\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}$^{[ref 2]}

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.^{[ref 29]}

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

Proposition 1 If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains ${\displaystyle D_{k}\subset D}$ with class ${\displaystyle {\mathcal {C}}^{\infty }}$-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 ${\displaystyle {\mathcal {C}}^{\infty }}$ exhaustion function.

Strongly Levi (–Krzoska) pseudoconvexity (Strongly pseudoconvex)[]

When the Levi (–Krzoska) form is positive-definite, it is called Strongly Levi (–Krzoska) pseudoconvex or often called simply Strongly pseudoconvex.^{[ref 2]}

Levi total pseudoconvex[]

If for every boundary point ${\displaystyle \rho }$ of D, there exists an analytic variety ${\displaystyle {\mathcal {B}}}$ passing ${\displaystyle \rho }$ which lies entirely outside D in some neighborhood around ${\displaystyle \rho }$, except the point ${\displaystyle \rho }$ itself. Domain D that satisfies these conditions is called Levi total pseudoconvex.^{[ref 30]}

Oka pseudoconvex[]

Family of Oka's disk[]

Let n-functions ${\displaystyle \varphi :z_{j}=\varphi _{j}(u,t)}$ 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 ${\displaystyle Q(t):=\{Z_{j}=\varphi _{j}(u,t);|u|\leq 1\}}$ is called an analytic disc de-pending on a parameter t, and ${\displaystyle B(t):=\{Z_{j}=\varphi _{j}(u,t);|u|=1\}}$ 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 30][ref 31]}

Definition[]

When ${\displaystyle Q(0)\subset D}$ holds on any Family of Oka's disk, D is called Oka pseudoconvex.^{[ref 30]} Oka's proof of Levi's problem was that when the unramified Riemann domain^{[ref 32]} 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.^{[ref 19][ref 31]}

Locally pseudoconvex (locally Stein,Cartan pseudoconvex,Local Levi property)[]

For every point ${\displaystyle x\in \partial D}$ there exist a neighbourhood U of x and f holomorphic. ( i.e. ${\displaystyle U\cap D}$ be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. i.e. A holomorphic map ${\displaystyle \psi :X\to Y}$ will be said to be locally pseudoconvex if every point ${\displaystyle y\in Y}$ has a neighborhood U such that ${\displaystyle \psi ^{-1}(U)}$ is Stein (weakly 1-complete). In this situation, we shall also say that X is locally pseudoconvex over Y. This was also called locally Stein and was classically called Cartan Pseudoconvex. In ${\displaystyle \mathbb {C} ^{n}}$ the Clocally pseudoconvexdomain is itself a pseudoconvex domain and is a domain of holomorphy.^{[ref 33][ref 30]}

Conditions equivalent to domain of holomorphy[]

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

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

The implications ${\displaystyle 1\Leftrightarrow 2\Leftrightarrow 3}$,^{[note 17]} ${\displaystyle 1\Rightarrow 4}$,^{[note 18]} and ${\displaystyle 4\Rightarrow 5}$ are standard results. Proving ${\displaystyle 5\Rightarrow 1}$, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka,^{[note 19]} and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\displaystyle {\bar {\partial }}}$-problem).

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".^{[ref 22][ref 23]} Specifically, it is a set ${\displaystyle (I)}$ of pairs ${\displaystyle (f,\delta )}$, ${\displaystyle f}$ holomorphic on a non-empty open set ${\displaystyle \delta }$, such that

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

The origin of indeterminate domains comes from the fact that domains change depending on the pair ${\displaystyle (f,\delta )}$. Cartan^{[ref 39][ref 24]} translated this notion into the notion of the coherent (sheaf) in sheaf cohomology.^{[ref 40][ref 41]} This name comes from H. Cartan.^{[ref 42]} The notion of coherent helped solve the problems in several complex variables.

Coherent sheaf[]

Definition[]

The definition of the coherent sheaf is as follows.^{[ref 43]}

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 arbitrary open set ${\displaystyle U\subseteq X}$, arbitrary natural number ${\displaystyle n}$, and arbitrary 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 43]} proves that

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

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

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

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

Oka's coherent theorem[]

Oka's coherent theorem^{[ref 22]} says that each sheaf that meets the following conditions is a coherent.^{[ref 44]}

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

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

Ideal sheaf[]

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

Cousin problem[]

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given pole, and Weierstrass factorization theorem was able to create a global meromorphic function from a given zero. 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.^{[ref 45]} It was Kiyoshi Oka who showed the conditions for solving First Cousin problem regard the domain of holomorphy on the complex coordinate space, and also Regarding the Second Cousin problem, if it is a domain of holomorphy on the complex coordinate space, the condition for solving it is purely topological, and J.P., Serre called this the Oka principle.^{[ref 46][ref 47][ref 48]} 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. ^{[ref 49]}

First Cousin problem[]

Definition without Sheaf cohomology words[]

Each difference ${\displaystyle f_{i}-f_{j}}$ is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that ${\displaystyle f-f_{i}}$ is holomorphic on U_i; 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.

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

By the long exact cohomology sequence,

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

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

Second Cousin problem[]

Definition without Sheaf cohomology words[]

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

Definition using Sheaf cohomology words[]

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

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

The long exact sheaf cohomology sequence associated to the quotient is

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

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

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

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

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

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

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

Manifolds with several complex variables[]

Stein manifold (Non-compact manifold)[]

Since an open Riemann surface^{[ref 50]} always has a non-constant single-valued holomorphic function,^{[ref 51]} and satisfies the second axiom of countability, the open Riemann surface can be thought of complex manifold to have a holomorphic embedding into a one-dimensional complex plane ${\displaystyle \mathbb {C} }$. 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 arbitrary compact connected complex manifold X: every 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 holomorphic embedded into ${\displaystyle \mathbb {C} ^{n}}$ are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.^{[note 20]}

A Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951).^{[ref 52]} A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. If the univalent domain on ${\displaystyle \mathbb {C} ^{n}}$ is connection to a manifold, can be regarded as a complex manifold and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.

Definition[]

Suppose X is a paracompact complex manifolds of complex dimension ${\displaystyle n}$ and let ${\displaystyle {\mathcal {O}}(X)}$ denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold:^{[ref 53]}

• X is holomorphically convex, i.e. for every compact subset ${\displaystyle K\subset X}$, the so-called holomorphically convex hull,

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

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 ${\displaystyle f(x)\neq f(y).}$^{[note 21]}
• The open neighborhood of every point on the manifold has a holomorphic Chart to the ${\displaystyle {\mathcal {O}}(X)}$.

Non-compact (open) Riemann surfaces are Stein[]

Let X be a connected, non-compact (open) Riemann surface. A deep theorem (1939)^{[ref 54]} of Heinrich Behnke and Stein (1948)^{[ref 51]} asserts that X is a Stein manifold.

Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0}$. The exponential sheaf sequence leads to the following exact sequence:

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

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

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

Levi problems[]

Cartan extended Levi's problem to Stein manifolds.^{[ref 55]}

If the relative compact open subset ${\displaystyle D\subset X}$ 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.^{[ref 56]}

This was proved by Bremermann^{[ref 57]} by embedding it in a sufficiently high dimensional ${\displaystyle \mathbb {C} ^{m}}$, and reducing it to the result of Oka.^{[ref 19]}

Also, Grauert proved for arbitrary complex manifolds M.^{[note 22][ref 60][ref 21][ref 58]}

If the relative compact subset ${\displaystyle D\subset M}$ 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^{[ref 61][ref 62]} 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.^{[ref 1][ref 63]}

Levi's problem remains unresolved in the following cases;

Suppose that X is a singular Stein space, ${\displaystyle D\subset \subset X}$ . Suppose that for all ${\displaystyle p\in \partial D}$ there is an open neighborhood ${\displaystyle U(p)}$ so that ${\displaystyle U\cap D}$ is Stein space. Is D itself Stein?^{[ref 1][ref 64][ref 65]}

more generalized

Suppose that N be a Stein space and f an injective, and also ${\displaystyle f:M\to N}$ a Riemann unbranched domain, such that map f is a locally pseudoconvex map (i.e. Stein morphism). Then M is itself Stein ?^{[ref 65][ref 66]: 109}

and also,

Suppose that X be a Stein space and ${\displaystyle D=\cup _{n\in \mathbb {N} }D_{n}}$ 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. ^{[ref 65]}

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 ${\displaystyle x_{0}\in X}$, there exist finitely many holomorphic map ${\displaystyle f_{1},\dots ,f_{k}}$ of X into ${\displaystyle \mathbb {C} ^{p}}$,${\displaystyle p=p(x_{0})}$, such that ${\displaystyle x_{0}}$ is an isolated point of the set ${\displaystyle A=\{x\in X;f^{-1}f(x_{0})\ (v=1,\dots ,k)\}}$.^{[ref 60]} This concept also applies to complex analytic space.^{[ref 67]}

Properties and examples of Stein manifolds[]

• The standard^{[note 23]} 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 n can be embedded into ${\displaystyle \mathbb {C} ^{2n+1}}$ by a biholomorphic proper map.^{[ref 68][ref 69]}

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^{[ref 70]} for Riemann surfaces,^{[note 24]} due to Behnke and Stein.^{[ref 51]}
• Every Stein manifold X is holomorphically spreadable, i.e. for every point ${\displaystyle x\in X}$, 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,^{[ref 60]} i.e. a smooth real function ${\displaystyle \psi }$ on 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 c. This is a solution to the so-called Levi problem,^{[ref 71]} 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 domain.^{[ref 72]} A Stein domain is the preimage ${\displaystyle \{z\mid -\infty \leq \psi (z)\leq c\}}$. 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 ${\displaystyle X_{c}=f^{-1}(c)}$ 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".

Complex projective varieties (Compact manifold)[]

Meromorphic function in one-variable complex function were studied in the closed Riemann surface, because the compact Riemann surface had a non-constant single-valued meromorphic function.^{[ref 50]} The compact one-dimensional complex manifold was the Riemann sphere ${\displaystyle {\hat {\mathbb {C} }}\cong \mathbb {CP} ^{1}}$. However, for high-dimensional (several complex variables) compact complex manifolds, the existence of meromorphic functions cannot be easily indicated because the singularity is not an isolated point. In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.^{[ref 33]} Consider expanding the closed (compact) Riemann surface to a higher dimension, such as, embedding close Complex submanifold M into the complex projective space ${\displaystyle \mathbb {CP} ^{n}}$, in particular, Complex projective varieties In high-dimensional complex manifolds, the phenomenon that the sheaf cohomology group disappears occurs, and it is Kodaira vanishing theorem and its generalization Nakano vanishing theorem etc. that gives the condition for this phenomenon to occur. Kodaira embedding theorem^{[ref 73]} gives complex Kähler manifold M, with a Hodge metric large enough dimension N into complex projective space, and also Chow's theorem^{[ref 74]} shows that the analytic subspace of a closed complex projective space is an algebraic manifold, and when combined with Kodaira's result, M embeds as an algebraic manifold. These give an example embeddings in manifolds with meromorphic functions. Similarities in Levi problems on the complex projective space ${\displaystyle \mathbb {CP} ^{n}}$, have been proved in some patterns, for example by Takeuchi.^{[ref 1][ref 75][ref 76][ref 77]}

See also[]

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

Annotation[]

References[]

Inline citations[]

Textbooks[]

Encyclopedia of Mathematics[]

PlanetMath[]

Further reading[]

External links[]

• Tasty Bits of Several Complex Variables open source book by Jiří Lebl
• Complex Analytic and Differential Geometry (OpenContent book See B2)
• Victor Guillemin. 18.117 Topics in Several Complex Variables. Spring 2005. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.