Function of several complex variables

The theory of functions of several complex variables 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 locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.^[1][2][3] For one complex variable, 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.^[4][5] 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.^[4] 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 (${\displaystyle \mathbb {CP} ^{n}}$)^[6] and has a different flavour to complex analytic geometry in ${\displaystyle \mathbb {C} ^{n}}$ 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.^[7] 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

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

 

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^[8] 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, and more generalized Stein space. ${\displaystyle \mathbb {C} ^{n}}$  is also considered to be a complex projective variety, a Kähler manifold,^[9] 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 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 ${\displaystyle z^{0}}$  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 ${\displaystyle {\frac {\partial f}{\partial z_{1}}}(z^{0}),\dots ,{\frac {\partial f}{\partial z_{n}}}(z^{0})}$ 

Therefore, when a function f is holomorphic on the domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ , 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,

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

   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

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

and^{[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)

Then as expected,

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

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

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

then,

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

therefore,

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

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 ${\displaystyle \gamma }$ , ${\displaystyle \gamma _{\nu }}$  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 closed 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 the Cauchy's integral formula of one variable repeatedly, ^{[note 5]}

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

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 the 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.

${\displaystyle {\frac {\partial ^{k_{1}+\cdots +k_{n}}f}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}=\sum _{v=1}^{\infty }{\frac {\partial ^{k_{1}+\cdots +k_{n}}f_{v}}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}}$ ^[10]

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 ;

${\displaystyle {\begin{aligned}\omega (z)&=\sum _{k=0}^{\infty }{\frac {1}{k!}}{\frac {1}{(2\pi i)^{n}}}\int _{|\zeta _{\nu }|=R_{\nu }}\cdots \int \omega (\zeta )\times \left[{\frac {d^{k}}{dz^{k}}}{\frac {1}{\zeta -z}}\right]_{z=0}df_{\zeta }\cdot z^{k}\\[6pt]&+\sum _{k=1}^{\infty }{\frac {1}{k!}}{\frac {1}{2\pi i}}\int _{|\zeta _{\nu }|=r_{\nu }}\cdots \int \omega (\zeta )\times \left(0,\cdots ,{\sqrt {\frac {k!}{\alpha _{1}!\cdots \alpha _{n}!}}}\cdot \zeta _{n}^{\alpha _{1}-1}\cdots \zeta _{n}^{\alpha _{n}-1},\cdots 0\right)df_{\zeta }\cdot {\frac {1}{z^{k}}}\ (\alpha _{1}+\cdots +\alpha _{n}=k)\end{aligned}}}$ 

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 ${\displaystyle r'_{\nu }<|z|<R'_{\nu }}$ , where ${\displaystyle r'_{\nu }>r_{\nu }}$  and ${\displaystyle R'_{\nu }<R_{\nu }}$ , and so it is possible to integrate term.^[11]

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 ${\displaystyle \mathbb {C} ^{n}}$  with piecewise smooth boundary ${\displaystyle \partial D}$ , and let the symbol ${\displaystyle \land }$  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 ${\displaystyle \zeta }$ , 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 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 ${\displaystyle \mathbb {C} ^{n}}$ , 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.^[13] This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.^[5][14] 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. ^[15]

Analytic continuation

Let U, V be domain on ${\displaystyle \mathbb {C} ^{n}}$ , such that ${\displaystyle f\in {\mathcal {O}}(U)}$  and ${\displaystyle g\in {\mathcal {O}}(V)}$ , (${\displaystyle {\mathcal {O}}(U)}$  is the set/ring of holomorphic functions on U.) 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. When n > 2, the following phenomenon occurs depending on the shape of the boundary ${\displaystyle \partial U}$ : there exists V, W such that arbitrary holomorphic functions ${\displaystyle f}$  over the domain U have an analytic continuation ${\displaystyle g\in {\mathcal {O}}(V)}$ . In other words, there may be not exist function ${\displaystyle f\in {\mathcal {O}}(U)}$  such that ${\displaystyle \partial U}$  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.

Reinhardt domain

In polydisks, the 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 open unit balls are not biholomorphic mapping, 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 ${\displaystyle \mathbb {C} ^{n}}$ , ${\displaystyle n\geq 1}$ , 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^[16] 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 ^[17] 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 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 ${\displaystyle \lambda (D^{*})}$  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

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 ${\displaystyle \mathbb {C} ^{n}}$  were all connected to larger domain.^[18]

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);^[19] 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.^[21][20]

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.^[22][23]

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 ${\displaystyle H_{\varepsilon }}$ .^[24]

Thullen's classic results

Thullen's^[25] 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)^[26] 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.^[27] Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for ${\displaystyle \mathbb {C} ^{2}}$ ,^[28] later extended to ${\displaystyle \mathbb {C} ^{n}}$ .^[29][30])^[31] Kiyoshi Oka's^[34][35] notion of idéal de domaines indéterminés is interpreted theory of sheaf cohomology by H. Cartan and more development Serre.^{[note 13][36][37][38][39][40][41][6]} In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.^[42] The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.^[4]

Domain of holomorphy

 

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

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.^[43]

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).^[44]
• 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 ${\displaystyle n\geq 3}$ .^[45] 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 holomorphic 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.^[5]

Pseudoconvexity

Hartogs showed that

Hartogs (1906):^[19] Let D be a Hartogs's domain on ${\displaystyle \mathbb {C} }$  and R be a positive function on D such that the set ${\displaystyle \Omega }$  in ${\displaystyle \mathbb {C} ^{2}}$  defined by ${\displaystyle z_{1}\in D}$  and ${\displaystyle |z_{2}|<R(z_{1})}$  is a domain of holomorphy. Then ${\displaystyle -\log {R}(z_{1})}$  is a subharmonic function on D.^[4]

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.^[46]

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 ${\displaystyle {\mathcal {C}}^{2}}$ -function u is plurisubharmonic if and only if ${\displaystyle {\sqrt {-1}}\partial {\bar {\partial }}f}$  is a positive (1,1)-form.^{[47]: 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 ${\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)}$ .^[47]: 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.^[47]: 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.^[48]

(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

${\displaystyle H(\rho )=\sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\,\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}$ ^[5]

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.^[48]

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) pseudoconvex (Strongly pseudoconvex)

When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly pseudoconvex.^[5]

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.^[49]

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.^[49][50]

Definition

When ${\displaystyle Q(0)\subset D}$  holds on any family of Oka's disk, D is called Oka pseudoconvex.^[49] Oka's proof of Levi's problem was that when the unramified Riemann domain over ${\displaystyle \mathbb {C} ^{n}}$ ^[51] 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.^[29]

March 12, 2023