The theory offunctions of several complex variables is the branch ofmathematics dealing with functions defined onthe complex coordinate space${\displaystyle {\mathbb{C}}^n}$, that is,n-tuples ofcomplex numbers. The name of the field dealing with the properties of these functions is calledseveral complex variables (andanalytic space), which theMathematics Subject Classification has as a top-level heading.

As incomplex analysis of functions of one variable, which is the casen = 1, the functions studied areholomorphic orcomplex analytic so that, locally, they arepower series in the variablesz_i. Equivalently, they are locallyuniform limits ofpolynomials; or locallysquare-integrable solutions to then-dimensionalCauchy–Riemann equations.^[1][2][3] For one complex variable, everydomain^{[note 1]}(${\displaystyle D\subset \mathbb{C}}$), is thedomain 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 (${\displaystyle D\subset {\mathbb{C}}^n,n\ge 2}$) 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 ofmeromorphic 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 andcomplex projective varieties (${\displaystyle {\mathbb{C}\mathbb{P}}^n}$)^[6] and has a different flavour to complex analytic geometry in${\displaystyle {\mathbb{C}}^n}$ or onStein manifolds, these are much similar to study of algebraic varieties that is study of thealgebraic geometry than complex analytic geometry.

Many examples of such functions were familiar in nineteenth-century mathematics;abelian functions,theta functions, and somehypergeometric series, and also, as an example of an inverse problem; theJacobi inversion problem.^[7] Naturally also same function of one variable that depends on some complexparameter is a candidate. The theory, however, for many years didn't become a full-fledged field inmathematical analysis, since its characteristic phenomena weren't uncovered. TheWeierstrass preparation theorem would now be classed ascommutative algebra; it did justify the local picture,ramification, that addresses the generalization of thebranch points ofRiemann surface theory.

With work ofFriedrich Hartogs,Pierre Cousin [fr],E. E. Levi, and ofKiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time wereHeinrich Behnke,Peter Thullen,Karl Stein,Wilhelm Wirtinger andFrancesco Severi. Hartogs proved some basic results, such as everyisolated singularity isremovable, for every analytic function$${\displaystyle f:{\mathbb{C}}^n\to \mathbb{C}}$$whenevern > 1. Naturally the analogues ofcontour integrals will be harder to handle; whenn = 2 an integral surrounding a point should be over a three-dimensionalmanifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to adouble integral over a two-dimensional surface. This means that theresidue calculus will have to take a very different character.

After 1945 important work in France, in the seminar ofHenri Cartan, and Germany withHans Grauert andReinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that ofanalytic continuation. Here a major difference is evident from the one-variable theory; while for every open connected setD in${\displaystyle \mathbb{C}}$ we can find a function that will nowhere continue analytically over the boundary, that cannot be said forn > 1. In fact theD of that kind are rather special in nature (especially in complex coordinate spaces${\displaystyle {\mathbb{C}}^n}$ and Stein manifolds, satisfying a condition calledpseudoconvexity). The natural domains of definition of functions, continued to the limit, are calledStein manifolds and their nature was to makesheaf cohomology groups vanish, on the other hand, theGrauert–Riemenschneider vanishing theorem is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan.^[4] 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 foralgebraic geometry, in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied toanalytic geometry,^{[note 2]}automorphic forms of several variables, andpartial differential equations. Thedeformation theory of complex structures andcomplex manifolds was described in general terms byKunihiko Kodaira andD. C. Spencer. The celebrated paperGAGA ofSerre^[8] pinned down the crossover point fromgéometrie analytique togéometrie algébrique.

C. L. Siegel was heard to complain that the newtheory of functions of several complex variables had fewfunctions in it, meaning that thespecial function side of the theory was subordinated to sheaves. The interest fornumber theory, certainly, is in specific generalizations ofmodular forms. The classical candidates are theHilbert modular forms andSiegel modular forms. These days these are associated toalgebraic groups (respectively theWeil restriction from atotally real number field ofGL(2), and thesymplectic group), for which it happens thatautomorphic 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 thehyperfunction theory, and theedge-of-the-wedge theorem, both of which had some inspiration fromquantum field theory. There are a number of other fields, such asBanach algebra theory, that draw on several complex variables.

Thecomplex coordinate space${\displaystyle {\mathbb{C}}^n}$ is theCartesian product ofn copies of${\displaystyle \mathbb{C}}$, and when${\displaystyle {\mathbb{C}}^n}$ is a domain of holomorphy,${\displaystyle {\mathbb{C}}^n}$ can be regarded as aStein manifold, and more generalized Stein space.${\displaystyle {\mathbb{C}}^n}$ is also considered to be acomplex projective variety, aKähler manifold,^[9] etc. It is also ann-dimensional vector space over thecomplex numbers, which gives its dimension2n over${\displaystyle \mathbb{R}}$.^{[note 3]} Hence, as a set and as atopological space,${\displaystyle {\mathbb{C}}^n}$ may be identified to thereal coordinate space${\displaystyle {\mathbb{R}}^{2n}}$ and itstopological dimension is thus2n.

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, wherea complex structure is specified by alinear operatorJ (such thatJ ² =−I) which definesmultiplication by theimaginary uniti.

Any such space, as a real space, isoriented. On thecomplex plane thought of as aCartesian plane,multiplication by a complex numberw =u +iv may be represented by the realmatrix

withdeterminant

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

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

A functionf defined on a domain${\displaystyle D\subset {\mathbb{C}}^n}$ and with values in${\displaystyle \mathbb{C}}$ is said to be holomorphic at a point${\displaystyle z\in D}$ if it is complex-differentiable at this point, in the sense that there exists a complex linear map${\displaystyle L:{\mathbb{C}}^n\to \mathbb{C}}$ such that

${\displaystyle f(z+h)=f(z)+L(h)+o(\Vert h\Vert )}$

The functionf is said to be holomorphic if it is holomorphic at all points of its domain of definitionD.

Iff is holomorphic, then all the partial maps :

${\displaystyle z\mapsto f(z_1,\dots ,z_{i-1},z,z_{i+1},\dots ,z_n)}$

are holomorphic as functions of one complex variable : we say thatf is holomorphic in each variable separately. Conversely, iff is holomorphic in each variable separately, thenf is in fact holomorphic : this is known asHartog's theorem, or asOsgood's lemma under the additional hypothesis thatf iscontinuous.

In one complex variable, a function${\displaystyle f:\mathbb{C}\to \mathbb{C}}$ defined on the plane is holomorphic at a point${\displaystyle p\in \mathbb{C}}$ if and only if its real part${\displaystyle u}$ and its imaginary part${\displaystyle v}$ satisfy the so-calledCauchy-Riemann equations at${\displaystyle p}$ :$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}(p)=\frac{\mathrm{\partial }v}{\mathrm{\partial }y}(p)\text{ and }\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(p)=-\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(p)}$$

In several variables, a function${\displaystyle f:{\mathbb{C}}^n\to \mathbb{C}}$ is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part${\displaystyle u}$ and the imaginary part${\displaystyle v}$ of${\displaystyle f}$ satisfy the Cauchy Riemann equations :$${\displaystyle \mathrm{\forall }i\in \{1,\dots ,n\},\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}=\frac{\mathrm{\partial }v}{\mathrm{\partial }{y}_i}\text{ and }\frac{\mathrm{\partial }u}{\mathrm{\partial }{y}_i}=-\frac{\mathrm{\partial }v}{\mathrm{\partial }{x}_i}}$$

Using the formalism ofWirtinger derivatives, this can be reformulated as :$${\displaystyle \mathrm{\forall }i\in \{1,\dots ,n\},\frac{\mathrm{\partial }f}{\mathrm{\partial }\overline{z_i}}=0,}$$or even more compactly using the formalism ofcomplex differential forms, as :$${\displaystyle \overline{\mathrm{\partial }}f=0.}$$

Prove the sufficiency of two conditions (A) and (B). Letf meets the conditions of being continuous and separately homorphic on domainD. Each disk has arectifiable curve${\displaystyle \gamma }$,${\displaystyle {\gamma }_{\nu }}$ is piecewisesmoothness, class${\displaystyle {\mathcal{C}}^1}$ Jordan closed curve. (${\displaystyle \nu =1,2,\dots ,n}$) 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${\displaystyle \overline{D_1}\times \overline{D_2}\times \cdots \times \overline{D_n}\in D}$. Also, take the closedpolydisc${\displaystyle \overline{\mathrm{\Delta }}}$ so that it becomes${\displaystyle \overline{\mathrm{\Delta }}\subset D_1\times D_2\times \cdots \times D_n}$.${\displaystyle \overline{\mathrm{\Delta }}(z,r)=\left\{\zeta =({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n)\in {\mathbb{C}}^n;\left|{\zeta }_{\nu }-z_{\nu }\right|\le r_{\nu }\text{ for all }\nu =1,\dots ,n\right\}}$ and let${\displaystyle \{z_{\nu }{\}}_{\nu =1}^n}$ be the center of each disk.) Using theCauchy's integral formula of one variable repeatedly,^{[note 4]}

Because${\displaystyle \mathrm{\partial }D}$ is a rectifiable Jordanian closed curve^{[note 5]} andf is continuous, so the order of products and sums can be exchanged so theiterated integral can be calculated as amultiple integral. Therefore,

${\displaystyle f(z_1,\dots ,z_n)=\frac{1}{(2\pi i{)}^n}{\int }_{\mathrm{\partial }{D}_1}\cdots {\int }_{\mathrm{\partial }{D}_n}\frac{f({\zeta }_1,\dots ,{\zeta }_n)}{({\zeta }_1-z_1)\cdots ({\zeta }_n-z_n)}d{\zeta }_1\cdots d{\zeta }_n}$

1

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

${\displaystyle \frac{{\mathrm{\partial }}^{k_1+\cdots +k_n}f(z_1,z_2,\dots ,z_n)}{\mathrm{\partial }{z_1}^{k_1}\cdots \mathrm{\partial }{z_n}^{k_n}}=\frac{k_1!\cdots k_n!}{(2\pi i{)}^n}{\int }_{\mathrm{\partial }{D}_1}\cdots {\int }_{\mathrm{\partial }{D}_n}\frac{f({\zeta }_1,\dots ,{\zeta }_n)}{({\zeta }_1-z_1{)}^{k_1+1}\cdots ({\zeta }_n-z_n{)}^{k_n+1}}d{\zeta }_1\cdots d{\zeta }_n.}$

2

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

From (2), iff is holomorphic, on polydisc${\displaystyle \left\{\zeta =({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n)\in {\mathbb{C}}^n;|{\zeta }_{\nu }-z_{\nu }|\le r_{\nu },\text{ for all }\nu =1,\dots ,n\right\}}$ and${\displaystyle |f|\le M}$, the following evaluation equation is obtained.

${\displaystyle \left|\frac{{\mathrm{\partial }}^{k_1+\cdots +k_n}f({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n)}{{\mathrm{\partial }{z}_1}^{k_1}\cdots \mathrm{\partial }{z_n}^{k_n}}\right|\le \frac{M{k}_1!\cdots k_n!}{{r_1}^{k_1}\cdots {r_n}^{k_n}}}$

Therefore,Liouville's theorem hold.

If functionf is holomorphic, on polydisc, from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

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 onD :

${\displaystyle f(z)=\sum _{k_1,\dots ,k_n=0}^{\mathrm{\infty }}{c}_{k_1,\dots ,k_n}(z_1-a_1{)}^{k_1}\cdots (z_n-a_n{)}^{k_n},}$

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${\displaystyle f_1,\dots ,f_n}$ which converges uniformly on compacta inside a domainD, the limit functionf of${\displaystyle f_v}$ also uniformly on compacta inside a domainD. Also, respective partial derivative of${\displaystyle f_v}$ also compactly converges on domainD to the corresponding derivative off.

${\displaystyle \frac{{\mathrm{\partial }}^{k_1+\cdots +k_n}f}{\mathrm{\partial }{z_1}^{k_1}\cdots \mathrm{\partial }{z_n}^{k_n}}=\sum _{v=1}^{\mathrm{\infty }}\frac{{\mathrm{\partial }}^{k_1+\cdots +k_n}{f}_v}{\mathrm{\partial }{z_1}^{k_1}\cdots \mathrm{\partial }{z_n}^{k_n}}}$^[10]

It is possible to define a combination of positive real numbers${\displaystyle \{r_{\nu }(\nu =1,\dots ,n)\}}$ such that the power series$\sum _{k_1,\dots ,k_n=0}^{\mathrm{\infty }}{c}_{k_1,\dots ,k_n}(z_1-a_1{)}^{k_1}\cdots (z_n-a_n{)}^{k_n}$ converges uniformly at 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 6]} for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

Let${\displaystyle \omega (z)}$ be holomorphic in theannulus and continuous on their circumference, then there exists the following expansion ;

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, where${\displaystyle r_{\nu }^{\prime }>r_{\nu }}$ and, and so it is possible to integrate term.^[11]

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

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

${\displaystyle \omega (\zeta ,z)=\frac{(n-1)!}{(2\pi i{)}^n}\frac{1}{|z-\zeta {|}^{2n}}\sum _{1\le j\le n}({\overline{\zeta }}_j-{\overline{z}}_j)d{\overline{\zeta }}_1\wedge d{\zeta }_1\wedge \cdots \wedge d{\zeta }_j\wedge \cdots \wedge d{\overline{\zeta }}_n\wedge d{\zeta }_n}$

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

In particular iff is holomorphic the second term vanishes, so

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

Holomorphic functions of several complex variables satisfy anidentity theorem, as in one variable : two holomorphic functions defined on the same connected open set${\displaystyle D\subset {\mathbb{C}}^n}$ and which coincide on an open subsetN ofD, are equal on the whole open setD. This result can be proven from the fact that holomorphics functions have power series extensions, and it can also be deduced from the one variable case. Contrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps${\displaystyle f(z_1,z_2)=0}$ and${\displaystyle g(z_1,z_2)=z_1}$coincide on the whole complex line of${\displaystyle {\mathbb{C}}^2}$ defined by the equation${\displaystyle z_1=0}$.

Themaximal principle,inverse function theorem, and implicit function theorems also hold. For a generalized version of the implicit function theorem to complex variables, see theWeierstrass preparation theorem.

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

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

When${\displaystyle n>1}$, open balls and open polydiscs arenot biholomorphically equivalent, that is, there is nobiholomorphic mapping between the two.^[12] This was proven byPoincaré in 1907 by showing that theirautomorphism groups have different dimensions asLie groups.^[5][13] 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.^[14]

LetU, 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 onU.) assume that${\displaystyle U,V,U\cap V\ne \varnothing }$ and${\displaystyle W}$ is aconnected component of${\displaystyle U\cap V}$. If${\displaystyle f{|}_W=g{|}_W}$ thenf is said to be connected toV, andg is said to be analytic continuation off. From the identity theorem, ifg exists, for each way of choosingW it is unique. When n > 2, the following phenomenon occurs depending on the shape of the boundary${\displaystyle \mathrm{\partial }U}$: there exists domainU,V, such that all holomorphic functions${\displaystyle f}$ over the domainU, have an analytic continuation${\displaystyle g\in \mathcal{O}(V)}$. In other words, there may not exist a function${\displaystyle f\in \mathcal{O}(U)}$ such that${\displaystyle \mathrm{\partial }U}$ as the natural boundary. This is called the Hartogs's phenomenon. Therefore, investigating when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. In addition, if${\displaystyle n\ge 2}$, it would be that the aboveV has an intersection part withU other thanW. This contributed to advancement of the notion of sheaf cohomology.

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

Let${\displaystyle D\subset {\mathbb{C}}^n}$ (${\displaystyle n\ge 1}$) to be a domain, with centre at a point${\displaystyle a=(a_1,\dots ,a_n)\in {\mathbb{C}}^n}$, such that, 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 domainD is called a Reinhardt domain if it satisfies the following conditions:^[15][16]

Let${\displaystyle {\theta }_{\nu }(\nu =1,\dots ,n)}$ is a arbitrary real numbers, a domainD is invariant under the rotation:${\displaystyle \left\{z^0-a_{\nu }\right\}\to \left\{e^{i{\theta }_{\nu }}(z_{\nu }^0-a_{\nu })\right\}}$.

The Reinhardt domains which are defined by the following condition; Together with all points of${\displaystyle z^0\in D}$, the domain contains the set

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

${\displaystyle \left\{z=(z_1,\dots ,z_n);\left|z_{\nu }-a_{\nu }\right|\le \left|z_{\nu }^0-a_{\nu }\right|,\nu =1,\dots ,n\right\}.}$

A complete Reinhardt domainD isstar-like with regard to its centrea. Therefore, the complete Reinhardt domain issimply connected, also when the complete Reinhardt domain is the boundary line, there is a way to prove theCauchy's integral theorem without using theJordan curve theorem.

When a some complete Reinhardt domain to be the domain of convergence of a power series, an additional condition is required, which is called logarithmically-convex.

A Reinhardt domainD is calledlogarithmically convex if the image${\displaystyle \lambda (D^{\ast })}$ of the set

${\displaystyle D^{\ast }=\{z=(z_1,\dots ,z_n)\in D;z_1,\dots ,z_n\ne 0\}}$

under the mapping

${\displaystyle \lambda ;z\to \lambda (z)=(\ln |z_1|,\dots ,\ln |z_n|)}$

is aconvex 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 of some power series in$\sum _{k_1,\dots ,k_n=0}^{\mathrm{\infty }}{c}_{k_1,\dots ,k_n}(z_1-a_1{)}^{k_1}\cdots (z_n-a_n{)}^{k_n}$, 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 7]} But, there is an example of a complete Reinhardt domain D which is not logarithmically convex.^[17]

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 when.

Internal domain of

Hartogs's extension theorem (1906);^[19] Letf be aholomorphic function on asetG \ K, whereG is a bounded (surrounded by a rectifiable closed Jordan curve) domain^{[note 8]} on${\displaystyle {\mathbb{C}}^n}$ (n ≥ 2) andK is a compact subset ofG. If thecomplementG \ K is connected, then every holomorphic functionf regardless of how it is chosen can be each extended to a unique holomorphic function onG.^[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 fromWeierstrass 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${\displaystyle H_{\epsilon }}$ to${\displaystyle {\mathrm{\Delta }}^2}$. Looking at this from the perspective of the Reinhardt domain,${\displaystyle H_{\epsilon }}$ is the Reinhardt domain containing the center z = 0, and the domain of convergence of${\displaystyle H_{\epsilon }}$ has been extended to the smallest complete Reinhardt domain${\displaystyle {\mathrm{\Delta }}^2}$ containing${\displaystyle H_{\epsilon }}$.^[24]

Thullen's^[25] classical result says that a 2-dimensional bounded Reinhard domain containing the origin isbiholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

1. (polydisc);
2. (unit ball);
3. (Thullen domain).

Toshikazu Sunada (1978)^[26] established a generalization of Thullen's result:

Twon-dimensional bounded Reinhardt domains${\displaystyle G_1}$ and${\displaystyle G_2}$ are mutually biholomorphic if and only if there exists a transformation${\displaystyle \phi :{\mathbb{C}}^n\to {\mathbb{C}}^n}$ given by${\displaystyle z_i\mapsto r_i{z}_{\sigma (i)}(r_i>0)}$,${\displaystyle \sigma }$ being a permutation of the indices), such that${\displaystyle \phi (G_1)=G_2}$.

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 ofH. 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 ofidéal de domaines indéterminés is interpreted theory ofsheaf cohomology byH. Cartan and more development Serre.^{[note 10][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]

When a functionf is holomorpic on the domain${\displaystyle D\subset {\mathbb{C}}^n}$ and cannot directly connect to the domain outsideD, including the point of the domain boundary${\displaystyle \mathrm{\partial }D}$, the domainD is called the domain of holomorphy off and the boundary is called the natural boundary off. In other words, the domain of holomorphyD is the supremum of the domain where the holomorphic functionf is holomorphic, and the domainD, which is holomorphic, cannot be extended any more. For several complex variables, i.e. domain${\displaystyle D\subset {\mathbb{C}}^n(n\ge 2)}$, 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 domainD in then-dimensional complex coordinate space${\displaystyle {\mathbb{C}}^n}$ is called adomain 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 functionf onD there exists a holomorphic functiong onV with${\displaystyle f=g}$ onU.

For the${\displaystyle n=1}$ case, every domain (${\displaystyle D\subset \mathbb{C}}$) is a domain of holomorphy; we can find a holomorphic function that is not identically 0, but whose zerosaccumulate everywhere on theboundary of the domain, which must then be anatural boundary for a domain of definition of its reciprocal.

• If${\displaystyle D_1,\dots ,D_n}$ are domains of holomorphy, then their intersection$D=\bigcap _{\nu =1}^n{D}_{\nu }$ 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$D=\bigcup _{n=1}^{\mathrm{\infty }}{D}_n$ is also a domain of holomorphy (seeBehnke–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 firstCousin problem is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for${\displaystyle n\ge 3}$.^[45] this is also true, with additional topological assumptions, for the second Cousin problem.

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

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

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

The domain${\displaystyle G}$ is calledholomorphically convex if for every compact subset${\displaystyle K,{\hat{K}}_G}$ is also compact inG. Sometimes this is just abbreviated asholomorph-convex.

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

When${\displaystyle n\ge 1}$, iff satisfies the above holomorphic convexity onD it has the following properties.${\displaystyle \text{dist}(K,D^c)=\text{dist}({\hat{K}}_D,D^c)}$ for every compact subsetK inD, 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]

Hartogs showed that

Hartogs (1906):^[19] LetD be a Hartogs's domain on${\displaystyle \mathbb{C}}$ andR be a positive function onD such that the set${\displaystyle \mathrm{\Omega }}$ in${\displaystyle {\mathbb{C}}^2}$ defined by${\displaystyle z_1\in D}$ and is a domain of holomorphy. Then${\displaystyle -\log R(z_1)}$ is a subharmonic function onD.^[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 11]} Thesubharmonic function looks like a kind ofconvex function, so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.^[46]

A function

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

withdomain${\displaystyle D\subset {\mathbb{C}}^n}$

is calledplurisubharmonic if it isupper 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

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

Infull generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space${\displaystyle X}$ as follows. Anupper semi-continuous function

${\displaystyle f:X\to \mathbb{R}\cup \{-\mathrm{\infty }\}}$

is said to be plurisubharmonic if and only if for anyholomorphic map

${\displaystyle \phi :\mathrm{\Delta }\to X}$ the function

${\displaystyle f\circ \phi :\mathrm{\Delta }\to \mathbb{R}\cup \{-\mathrm{\infty }\}}$

is subharmonic, where${\displaystyle \mathrm{\Delta }\subset \mathbb{C}}$ 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 toz of one-variable complex function is subharmonic is${\displaystyle \mathrm{\Delta }=4\left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }z\mathrm{\partial }\overline{z}}\right)\ge 0}$. Therefore, if${\displaystyle u}$ is of class${\displaystyle {\mathcal{C}}^2}$, then${\displaystyle u}$ is plurisubharmonic if and only if thehermitian matrix${\displaystyle H_u=({\lambda }_{ij}),{\lambda }_{ij}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}_i\mathrm{\partial }{\overline{z}}_j}}$ is positive semidefinite.

Equivalently, a${\displaystyle {\mathcal{C}}^2}$-functionu is plurisubharmonic if and only if${\displaystyle \sqrt{-1}\mathrm{\partial }\overline{\mathrm{\partial }}f}$ is apositive (1,1)-form.^{[47]: 39–40}

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

Weak pseudoconvex is defined as : Let${\displaystyle X\subset {\mathbb{C}}^n}$ be a domain. One says thatX ispseudoconvex if there exists acontinuousplurisubharmonic function${\displaystyle \phi }$ onX such that the set${\displaystyle \{z\in X;\phi (z)\le supx\}}$ is arelatively compact subset ofX for all real numbersx.^{[note 12]} i.e. there exists a smooth plurisubharmonic exhaustion function${\displaystyle \psi \in \text{Psh}(X)\cap {\mathcal{C}}^{\mathrm{\infty }}(X)}$. Often, the definition of pseudoconvex is used here and is written as; LetX be a complexn-dimensional manifold. Then is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function${\displaystyle \psi \in \text{Psh}(X)\cap {\mathcal{C}}^{\mathrm{\infty }}(X)}$.^[47]: 49

LetX be a complexn-dimensional manifold.Strongly (or Strictly) pseudoconvex if there exists a smoothstrictly plurisubharmonic exhaustion function${\displaystyle \psi \in \text{Psh}(X)\cap {\mathcal{C}}^{\mathrm{\infty }}(X)}$, i.e.,${\displaystyle H\psi }$ is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.^[47]: 49 Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete^[48]) are often used interchangeably,^[49] see Lempert^[50] for the technical difference.

If${\displaystyle {\mathcal{C}}^2}$ boundary , it can be shown thatD 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, and${\displaystyle \mathrm{\partial }D=\{\rho =0\}}$. Now,D is pseudoconvex iff for every${\displaystyle p\in \mathrm{\partial }D}$ and${\displaystyle w}$ in the complex tangent space at p, that is,

${\displaystyle \mathrm{\nabla }\rho (p)w=\sum _{i=1}^n\frac{\mathrm{\partial }\rho (p)}{\mathrm{\partial }{z}_j}{w}_j=0}$, we have

${\displaystyle H(\rho )=\sum _{i,j=1}^n\frac{{\mathrm{\partial }}^2\rho (p)}{\mathrm{\partial }{z}_i\mathrm{\partial }\overline{z_j}}{w}_i\overline{w_j}\ge 0.}$^[5][51]

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

Proposition 1IfD is pseudoconvex, then there existbounded, strongly Levi pseudoconvex domains${\displaystyle D_k\subset D}$ with class${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}}$-boundary which are relatively compact inD, such that

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

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

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

If for every boundary point${\displaystyle \rho }$ ofD, there exists ananalytic variety${\displaystyle \mathcal{B}}$ passing${\displaystyle \rho }$ which lies entirely outsideD in some neighborhood around${\displaystyle \rho }$, except the point${\displaystyle \rho }$ itself. DomainD that satisfies these conditions is called Levi total pseudoconvex.^[52]

Letn-functions${\displaystyle \phi :z_j={\phi }_j(u,t)}$ be continuous on${\displaystyle \mathrm{\Delta }:|U|\le 1,0\le t\le 1}$, holomorphic in when the parametert is fixed in [0, 1], and assume that${\displaystyle \frac{\mathrm{\partial }{\phi }_j}{\mathrm{\partial }u}}$ are not all zero at any point on${\displaystyle \mathrm{\Delta }}$. Then the set${\displaystyle Q(t):=\{Z_j={\phi }_j(u,t);|u|\le 1\}}$ is called an analytic disc de-pending on a parametert, and${\displaystyle B(t):=\{Z_j={\phi }_j(u,t);|u|=1\}}$ is called its shell. If and${\displaystyle B(0)\subset D}$,Q(t) is called Family of Oka's disk.^[52][53]

When${\displaystyle Q(0)\subset D}$ holds on any family of Oka's disk,D is called Oka pseudoconvex.^[52] Oka's proof of Levi's problem was that when theunramified Riemann domain over${\displaystyle {\mathbb{C}}^n}$^[54] 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][53]

For every point${\displaystyle x\in \mathrm{\partial }D}$ there exist a neighbourhoodU ofx andf holomorphic. ( i.e.${\displaystyle U\cap D}$ be holomorphically convex.) such thatf cannot be extended to any neighbourhood ofx. i.e., let${\displaystyle \psi :X\to Y}$ be a holomorphic map, if every point${\displaystyle y\in Y}$ has a neighborhood U such that${\displaystyle {\psi }^{-1}(U)}$ admits a${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}}$-plurisubharmonic exhaustion function (weakly 1-complete^[55]), in this situation, we call thatX is locally pseudoconvex (or locally Stein) overY. As an old name, it is also called Cartan pseudoconvex. In${\displaystyle {\mathbb{C}}^n}$ the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.^[56][52] For example, Diederich–Fornæss^[57] found local pseudoconvex bounded domains${\displaystyle \mathrm{\Omega }}$ with smooth boundary on non-Kähler manifolds such that${\displaystyle \mathrm{\Omega }}$ is not weakly 1-complete.^{[58][note 13]}

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

1. D is a domain of holomorphy.
2. D is holomorphically convex.
3. D is the union of an increasing sequence ofanalytic polyhedrons inD.
4. D is pseudoconvex.
5. D is Locally pseudoconvex.

The implications${\displaystyle 1\iff 2\iff 3}$,^{[note 15]}${\displaystyle 1\Rightarrow 4}$,^{[note 16]} and${\displaystyle 4\Rightarrow 5}$ are standard results. Proving${\displaystyle 5\Rightarrow 1}$, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called theLevi problem (afterE. E. Levi) and was solved for unramified Riemann domains over${\displaystyle {\mathbb{C}}^n}$ by Kiyoshi Oka,^{[note 17]} but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,^[66] and then byLars Hörmander using methods from functional analysis and partial differential equations (a consequence of${\displaystyle \overline{\mathrm{\partial }}}$-problem(equation) with aL² methods).^{[1][43][3][67]}

The introduction ofsheaves into several complex variables allowed the reformulation of and solution to several important problems in the field.

Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains".^[34][35] 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