Use this tag for questions related to the study of analytic functions of strictly more than one complex variables. For the single complex dimension case, use (complex-analysis).

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Definitions of complxe singularity exponent

If $\Omega\subset\mathbb C^n$ is an open subset of $\mathbb C^n$ and $\varphi$ is a plurisubharmonic function in $\Omega$, for a point $x\in \Omega$, we define the complex singularity exponent of ...
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Proof or Counterexample:Is every open connected set $D \subset \mathbb C$ is a domain of holomorphy?

Def: An open set $D \subset \mathbb C^n$ is called a domain of Holomorphy if there exists a holomorphic function $f$ on $D$ such that $f$ cannot be extended to a bigger set. Is every non empty open ...
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What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
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Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...
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Is it so easy to show a locally bounded function with holomorphic cross sections is holomorphic?

Let $f: U \rightarrow \mathbb{C}$ where $U$ is open in $\mathbb{C}^n$ and suppose every coordinate cross section of $f$ is holomorphic (I hope that's not too colloquial). I've heard it's a somewhat ...
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48 views

Finding an entire function $f$

Let $U\subset\mathbb{C}^n$ be a bounded domain. Give an example of an entire function $f:\mathbb{C}^n\longrightarrow\mathbb{C}$ such that: $$f[U]\subset D(0,1)$$ $$f[ext(U)]\subset ext[{D(0,1)}]$$ ...
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Convexity of the complex ellipsoid

Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I ...
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50 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
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Is any simply connected domain conformally equivalent to Cartesian product of unit disks?

By Riemann mapping theorem, any simply connected domain is conformally equivalent to the unit disk. Is any simply connected domain in the complex plane conformally equivalent to the Cartesian product ...
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31 views

Understanding a Wermer's counterexample.

I am reading some lecture notes on holomorphic functions of several complex variables, see page 105. The part I am struggling with is a proof by Wermer I have asked about runge domains, and ...
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51 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
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Complex Variables1

Let $f: G \to C$ be analytic and suppose that $G$ is bounded. fix $z_0$ in $\{G\}$ and suppose that $\lim _{z\to w}\sup|f(z)| \leq M$ for w in $\{G\}$, $w \neq z_0$. Show that if $\lim _{z\to ...
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Proof of Weierstrass Preparation Theorem

In Griffiths and Harris, Principles of Algebraic Geometry, on page 8, near the end of the proof of the Weierstrass Preparation Theorem, he states that $h(z,w)$ has only removable singularities in the ...
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16 views

Can a complete pluripolar set be a single point?

Let $f:\mathbb{C}^n\rightarrow\mathbb{R}\cup\{-\infty\}$ be a plurisubharmonic function which is not identically $-\infty$.The set $\mathcal{P}:=\{z\in\mathbb{C}^n:f(z)=-\infty\}$ is called a complete ...
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38 views

Calculating integral using Cauchy integral formula in two variables

I want to compute the integral: $\iint_{\partial_0P}\frac{1}{1-4zw}dzdw$ (or any similar integral) using Cauchy integral formula for two complex variables over polydiscs. The distinguished boundary is ...
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Hurwitz's theorem for a system of functions (corrected version)

This is the corrected version. First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$. We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times ...
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23 views

Multivariate Complex Function

Suppose $f(x,w)\not=0$ for all $x,w\in H^+\cup H^-$ (open upper and lower half planes) and $f$ is a multivariate entire function. Must there exist univariate entire functions $\phi_1$ and $\phi_2$ ...
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Notions of convergence for maps of complex manifolds

Suppose that $X_t$ and $Y_t$ are two families of complex manifolds, parametrized by maps $\sigma : \mathcal X \to T$ and $\tau : \mathcal Y \to T$. Suppose too I have holomorphic maps $f_t : X_t \to ...
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21 views

Analytic continuation in several variables

Suppose we have a function $f : U \to \mathbb{R}$, where $U = (0,1)^n \subset \mathbb{R}^n$ is the open box, and that $f(x_1,x_2,\cdots,x_n)$ is separately real analytic in each $x_i$. Does there ...
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Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
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sub mean value property of plurisubharmonic function

It is well known that a plurisubharmonic function $\varphi$ defined in a domain $\Omega\subset \mathbb C^n$ satisfies the sub mean value property. Now if $\varphi$ is defined on a complex manifold ...
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Holomorphic curve with unit norm

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \|\gamma(z) \| =1$? if the answer is yes, can the method of unit ...
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26 views

Radii problem in a power series

I was studying some basic matters of several complex variables (here $\Omega\subseteq\Bbb C^n$, open): After this, before the proof, the author pointed what follows: So I'm going to tell you ...
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39 views

Cauchy formula in Polydisks

I don't understand a remark after the proof. Here's the theorem: The proof is done by induction on $n$; starting from $n=1$ on the unitary disk in $\Bbb C$, which is the well known Cauchy formula. ...
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32 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
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103 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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58 views

$f$ is holomorphic iff $df$ is $\Bbb C$-linear

Let $\Omega\subseteq\Bbb C^n$ open connected, $f:\Omega\to\Bbb C$ differentiable in the real sense. We know that $f$ is holomorphic iff $\partial_{\bar z_j}f=0\;\;\forall j=1,\dots,n$ . We know also ...
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35 views

Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to ...
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Commutative Operators from QM

In Theoretical Chemistry, there seems to be a lot of assumptions about mathematics that are incorporated without justification. One example that I found questionable is this: $$\int \Psi_1^*\ ...
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How can I prove that $\operatorname{hess}r(iu,iu)\neq-\operatorname{hess}r(u,u)$?

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know that $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as ...
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54 views

How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
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28 views

What are the real and imaginary parts of the complex function?

So, it is asked to find the real and imaginary parts of the specific complex function: $f(z)=sin(z)+i(3z+2) $ So I use $z$ as $z=x+iy$ everything seemed clear till I met Mr. Sinus: $u+iv= ...
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54 views

How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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1answer
64 views

Does in this case exist necessarely an eigenvalue equal to $0$?

I pasted more than I refer, hoping to be more clear. Look at the claim of the theorem: it states we can change coordinates untill we reach a "good" form for the equation of $r$, which defines the ...
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What is the definition of the conformal class of a bilinear form?

In the fourth line you can see. It talks about conformal class of a Levi form. I know what a conformal map is, but I can't deduce what a conformal class is from this. Can somebody help me? Many ...
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26 views

Is this the hessian bilinear form or $1/2$ of it?

Look at: the author writes "full hessian", but it's clearly $\frac12\operatorname{hess}r$. Or not? Other question: why does the author mean by "real harmonic polynomial"? And why it should be ...
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85 views

Chain rule in several complex variables: Wirtinger derivatives

Let $\Omega,\Omega'\subseteq\Bbb C^n$ open, $F:\Omega\to\Omega'$ holomorphic invertible function; it's a variable change, so let's call $F(z)=\tilde z$. Let $r:\Omega\to\Bbb R$ twice differentiable, ...
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Local normal form of a (several complex variable) holomorphic map at a point?

Suppose $F\colon\Omega\subseteq\mathbb C^m\to\mathbb C^n$ is a holomorphic map. WLOG, $0\in\Omega$ and $F(0)=0$. I want to determine the local normal form of $F$, i.e. classifying $F$ up to local ...
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52 views

Proving that a holomorphic function is constant

I am attempting to prove the following: Let $X$ be a connected complex manifold, and $f\in \mathcal{O}(X)$. For any $x\in X$, there is a complex submanifold of $X$ which is biholomorphic to ...
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Relation between being holomorphic in $\Delta\times\Delta$ and in every relatively compact polydisk.

I asked to some professors of my university and no one was able to help me (the one who held the course is abroad for a period, otherwise I'd ask him, obviously). My problem is that simply I don't ...
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A necessary condition for a multi-complex-variable holomorphic function. [closed]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ ...
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Detail in proof of Hartog Theorem

I am stuck in the middle of what you can see below: when the book says "we can repeat the same construction [...] horizontal strip arbitrary close to $z_2=0$": We define different sets $E_l$ on any ...
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Is there a name for these sets of functions of several complex variables (most not analytic)?

For each $n \in \mathbb{N}$, I came up with the following sets that I found interesting; at least I've never seen them in the literature before. $S_n = $span{$z_1z_2 \cdots z_n, \bar{z}_1z_2\cdots ...
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The conditions that partial derivatives commute

State the conditions that partial derivatives commute, namely, $D_1D_2f = D_2D_1f$. I understand how to prove that these partial derivatives are equal but I don't understand what commute means. ...
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Interpretation of the Weierstrass Preparation Theorem

I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition). The ...
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Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
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About multivariable quadratic polynomials

Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one ...
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Can a product of a Stein manifold and a compact manifold be again Stein?

A Stein manifold is a manifold which is holomorphically separable and convex. It is well known that a product of two holomorphically convex (resp. Stein) manifolds is again holomorphically convex ...
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Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$ \sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha} $$ ...