Symmetric cone
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that
Hartogs's theorem on separate holomorphicity
In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuo
Ohsawa–Takegoshi L2 extension theorem
In several complex variables, the Ohsawa–Takegoshi L2 extension theorem is a fundamental result concerning the holomorphic extension of an -holomorphic function defined on a bounded Stein manifold (su
Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named
Biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose i
Plurisubharmonic function
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmoni
Andreotti–Norguet formula
The Andreotti–Norguet formula, first introduced by Aldo Andreotti and , is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely,
Complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the stud
Ushiki's theorem
In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of we
Tube domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of co
Cousin problems
In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in
Polar homology
In complex geometry, a polar homology is a group which captures holomorphic invariants of a complex manifold in a similar way to usual homology of a manifold in differential topology. Polar homology w
Behnke–Stein theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence (i.e., ) of domains of holomorphy is again a domain of holomorphy. It was p
Behnke–Stein theorem on Stein manifolds
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold. In other words, it states that there is
Positive current
In mathematics, more particularly in complex geometry,algebraic geometry and complex analysis, a positive currentis a positive (n-p,n-p)-form over an n-dimensional complex manifold,taking values in di
Bergman kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic func
Edge-of-the-wedge theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the
Suita conjecture
In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 exten
Domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which
Analytic polyhedron
In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cn of the form where D is a bounded connected open subset of Cn, are holomorphic on D and
Weierstrass preparation theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplicat
Bombieri norm
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or (there is also a version for non homogeneous univariate polynomials). This n
Poincaré residue
In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of s
Bochner–Martinelli formula
In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by Enzo Martinelli and Salomon Bochner.
Complex coordinate space
In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers. It is denoted , and is the n-fold Cartesian product of the comple
Bergman–Weil formula
In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by and .
Complex analytic variety
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence o
Skoda–El Mir theorem
The Skoda–El Mir theorem is a theorem of complex geometry, stated as follows: Theorem (Skoda, El Mir, Sibony). Let X be a complex manifold, and E a closed complete pluripolar set in X. Consider a clos
Hartogs's extension theorem
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that th
Polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs. More specifically, if we denote by the open disc of center z and radius r
Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several compl
Bochner's tube theorem
In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in can be extended to the convex hull of this domain. Theorem Let be a connect
Bateman transform
In the mathematical study of partial differential equations, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral o
Oka–Weil theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André
Theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann
Pseudoconvexity
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are im
Pluriharmonic function
In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of severa
Fatou–Bieberbach domain
In mathematics, a Fatou–Bieberbach domain is a proper subdomain of , biholomorphically equivalent to . That is, an open set is called a Fatou–Bieberbach domain if there exists a bijective holomorphic
Holomorphic separability
In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.
Malgrange–Zerner theorem
In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and ) shows that a function on allowing holomorphic extension in each variable separately can be extended, under certain condition
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 comple
Thin set (analysis)
In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the