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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22 views

Describe set of $z^2$ as z moves over 2nd quadrant and show it is open and connected

Problem: Describe the set of points $z^2$ as $z$ varies over the second quadrant: {z = x + iy; x < 0 and y > 0}. Show this is an open connected set. (Hint: use the polar representation of z.) The ...
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1answer
170 views
+50

Geometric intuition for the Stein factorization theorem?

What is the intuition behind the Stein Factorization Theorem? I understand that it was originally a theorem in several complex variables, so I was wondering if there's some geometric explanation that ...
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0answers
14 views

Uniqueness set for analytic functions of several variables

Is there a simple (and not so restrictive) condition for a set to be an uniqueness set for the space of holomorphic functions defined on some open subset $U \subseteq \mathbb{C}^n$? By uniqueness set ...
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1answer
32 views

On the proof of Riemann extension theorem in Huybrechts

In the proof of the generalized Riemann extension theorem in Daniel Huybrechts's Complex Geometry, which is: Proposition 1.1.7 (Riemann extension theorem) Let $f$ be a holomorphic function on an ...
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2answers
33 views

Real-analytic function of two complex variables, holomorphic in first and anti-holo in second, which vanishes on the diagonal is identically zero.

The following theorem is stated as being a well-known result of the theory of several complex variables in a book I am reading (on a more or less unrelated subject): Let $f:\mathbb C^2\to\mathbb ...
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3answers
41 views

Questions on Quaternion Algebra (introductory stuff)

I am a relatively new Mathematics student who understands about complex numbers and how they work. I am currently trying to create a 3D computer graphics engine and I heard that quaternion algebra may ...
3
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1answer
48 views

When is a quasiprojective variety Kobayashi hyperbolic?

I am looking for some (simple and well-known) sufficient conditions on a quasiprojective variety to be Kobayashi hyperbolic. I realize that in this generality it may be a complicated (maybe even ...
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0answers
33 views

Every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy

I am trying to prove that every convex domain in $\mathbb{C^n}$ is a weak domain of Holomorphy. Let $G$ be a convex domain. I pick a point $p \in \partial{G}$. Then By Hahn-Banach Separation ...
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1answer
38 views

Riemann Mapping theorem doesn't hold true in Several Complex Variables

I need to show that Riemann Mapping Theorem is not true in general for $\mathbb{C^n}$. I know Cartan's Uniqueness Theorem and $Aut(B)$ acts transitively on $B$. But I am unable to deduce the result ...
4
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1answer
32 views

Gluing together Riemann surfaces, don't see why $Z$ is union of two compact sets.

Consider Lemma 1.7 from page 60-61 of Miranda's Algebraic Curves and Riemann Surfaces. For a link to the book, see here. Lemma 1.7. With the above construction, $Z$ is a compact surface of genus ...
4
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1answer
53 views

Automorphisms of $\mathbb{C}[x_1, \dots, x_n]$

Are the linear transformations, and the automorphisms of the form $\sigma(x_1, \dots, x_n) = (x_1 -f(x_2, \dots, x_n), x_2, \dots, x_n)$, where $f$ is a polynomial, generators of the group of ...
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0answers
32 views

Every Holomorphic function on a Hartog's figure can be extended holomorphically to the while of $P^n$

Let $f$ be a holomorphic function on the Euclidean Hartog's figure that is $$H=\{(z,w)\in P^2 : 1 \gt|z| \gt q_1 \text{or} |w| \lt q_2 \}$$ where $0 \lt q_i \lt 1$. I need to show that it has a ...
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0answers
21 views

global nullstellensatz

Let J be an ideal of analytic functions in several variables on the open unit ball. If Z(J), the analytic set of J (common zero set in the ball) is a compact subset of the ball does a global ...
1
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1answer
36 views

Proving a complex polynomial is identically $0$.

Let $f:\mathbb{C}^n\to\mathbb{C}$ be a complex polynomial of $n$ complex variables. Let $T^n:=\{(e^{i\theta_1},\dots,e^{i\theta_n}),\theta_j\in\mathbb{R}\}$ be the $n$-dimensional torus and $\sigma_n$ ...
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0answers
44 views

Zero set of an analytic function

If $f$ is a holomorphic function on $C^n$ which vanishes on $R^n$, it is easy to see that it vanishes everywhere. But if the zero set of $f$ is contained in $R^n$, can I deduce that $f$ vanishes ...
4
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1answer
69 views

Is every sheaf a subsheaf of a flasque sheaf?

Call a sheaf flasque if for all open sets $U \subset V$, the restriction map$$\mathcal{F}(V) \to \mathcal{F}(U)$$is surjective. Is every sheaf a subsheaf of a flasque sheaf?
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1answer
39 views

To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of ...
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0answers
28 views

To show $\bar{D}u=f$ where $u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge d\bar{\lambda}$,

Let $\Omega \subset \mathbb{C}$ be a bounded open set. Suppose $f \in C^{1 }(\Omega)$, $f$ is bounded and $$u(z)=\frac{1}{2\pi i} \int_{\Omega} \frac{f(\lambda)}{\lambda-z} d\lambda \wedge ...
3
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2answers
78 views

Determine the Automorphism group of the unit bidisc.

I'm currently reading through Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range, and I'm stuck on Exercise E.2.4, which states Let $\Delta^2$ be ...
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0answers
77 views

vanishing theorem in algebraic geometry

This is a general question: As we know there are lot of vanishing theorem like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for ...
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0answers
48 views

entension of a holomorphic map on projective algebraic manifolds

I'm reading a paper.It says if $M$ is a projective algebraic variety and $f:N-S\rightarrow M$ is a holomorphic map,where $N$ is a connected complex manifold and $S\subset N$ is a proper analytic ...
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1answer
34 views

Boundary of the image of a holomorphic map in several complex variables

Suppose we have a bounded open set $U \subset \mathbb{C}^n$ and a surjective holomorhpic map $f: U \rightarrow V$, where $V$ is open in $\mathbb{C}^n$ and not necessarily bounded. Also suppose we have ...
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0answers
38 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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1answer
45 views

what is the Jacobian for transformation from complex coordinate to real coordinate?

I'm trying to figure out what is the Jacobian when you do this simple transformation: $dzdz^* \rightarrow dxdy$ where $z=x+iy$ and $z^*=x-iy$. Follow the formula we have $$J(x,y)= \begin{vmatrix} ...
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0answers
50 views

Understanding Cartan's Theorem (Small hiccups in the proof)

I was going through the theorem and the Proof of "Cartan's Theorem". The theorem says: $$\text{Suppose $\Omega$ is a bounded region in $\mathbb{C^n}$, $F:\Omega \to \Omega$ is holomorphic, for some ...
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2answers
35 views

Are the Elementary Binary Operations Analytic?

Are the elementary binary operations of addition, multiplication, and exponentiation -- taken as multivariate functions over the real numbers -- analytic? That is, $f(a, b) = a + b$ and so forth. Does ...
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0answers
19 views

$(P,H)$ Euclidean Hartogs figure

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$ and $f:H\to \mathbb{C}^n$ a holomorphic injective map, then we know that $f$ extend holomorphically to polidisc $P$ (i.e. there is a ...
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1answer
27 views

Complex Hessian outputs don't match

Let $\Omega:=\{(x_1,y_1,z_2,\bar z_2)\;:\;\underbrace{y_1-f(z_2,\bar z_2)+g(z_2,\bar z_2)}_{=:r(x_1,y_1,z_2,\bar z_2)}<0\}\subset\Bbb C^2$ (we confuse consciously $\Bbb C$ with $\Bbb R^2$) where ...
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0answers
12 views

$r_1 + r_2 =$ constant (satisfied by these distances) implies the relation$ T. grad(r1 + r2) = 0$,

If $r_1$ and $r_2$ denote the distances from a point $(x, y)$ on an ellipse to its foci, show that the equation $r_1 + r_2 =$ constant (satisfied by these distances) implies the relation ...
1
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1answer
69 views

Riemann extension theorem for dimension greater than 1 case

This is a theorem in P.Griffiths Principles of Algebraic Geometry. I do not get a condition and the relation between two functions. $C^n$ below is the complex n-dimensional space. Theorem: Suppose ...
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0answers
8 views

Coordinate change for a real hypersurface

Let $\Omega\subseteq\Bbb C^n$ be a domain, $0_{\Bbb C^n}\in\Omega$, $r\in\mathcal{C}^2(\Omega,\Bbb R)$, $r(0)=0$. Call $M$ the real hypersurface described by $r=0$ around $0\in\Omega$. By implicit ...
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0answers
33 views

GCD for real analytic functions

In the theory of real analytic functions of several variables, is there a concept of greatest common divisor. If so, does it also hold true that if the gcd of a collection of functions is $1$, then ...
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1answer
26 views

Question about analytic polyhedra

Let $\Pi\subset\subset U\subset\mathbb{C}^n$ be an analytic polyhedron $$\Pi=\{z\in U:|f_j(z)|<1,1\le j\le m\}$$ where $f_1,\ldots,f_m\in H(U)$, the following equality holds? ...
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0answers
23 views

Power Series in Poly Discs (Assuming Cauchy Integral Formula)

Suppose that $f \in H(\Omega)$ and suppose that $\Omega$ contains the closure of some polydiscs $D(p;r)$. By polydiscs I means $D(p;r)=\{z=(z_1,z_2,..,z_n) \in \mathbb{C^n}| |z_i-p_i| \lt r_i \}$ for ...
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1answer
37 views

Connectedness of $\hat{K}_U$.

Let $U\subset\mathbb{C}^n$ be a domain and $K\subset U$ compact. If $K$ is connected then $\hat{K}_U$ is also connected? $\hat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \forall f\in ...
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1answer
36 views

Domain of holomorphy: finding a holomorphic function.

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy. The following proposition is true? For each $a\in\partial U$, there is a holomorphic function $f\in H(U)$, such that $\displaystyle\sup_{j\ge 1} ...
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0answers
65 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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0answers
15 views

$U:$ domain of holomorphy, $d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$.

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, then we know that $$d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$$ for each compact subset $K\subset U$, also that ...
4
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2answers
60 views

Definition of a complex fiber

We define a real hypersurface as a subset $M\subset\Bbb C^n$ which is locally defined as the zero-locus of some $r\in\mathcal C^2(\Omega,\Bbb R)$ ($\Omega\subseteq\Bbb C^n$ open). Then let $z_0\in M$. ...
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0answers
78 views

The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as an ...
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0answers
26 views

change of complex variables

Suppose that $z_1 , z_2,\cdots ,z_k,\cdots ,z_d$ are complex variable numbers. Locally, and suppose $f_1, f_2 \cdots, f_k $ are $k$ holomorphic functions on $z_1, z_2 \cdots,z_d$. At ...
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0answers
22 views

holomorphic range of a zero variety

Let D be the unit disk, and $d\ge 2$. $F=(f_1,...f_d)$ is a holomorphic map over the closure of $D^d$ to $C^d$. What information can we get about the image of $\{0\}$x $D^{d-1}$ under $F$? Hope some ...
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3answers
122 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
6
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1answer
161 views

Global Residue Theorem in $\mathbb{CP}^2$.

Consider the following meromorphic form defined on $\mathbb{CP}^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
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1answer
142 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
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0answers
14 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...
3
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3answers
68 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
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0answers
67 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
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0answers
33 views

Questions on Levi pseudoconvex domain

Here are some of the exercise questions which I am stuck: Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining ...
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1answer
44 views

Domain of holomorphy characterization

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy. The following proposition is true? There is a holomorphic function $f\in H(U)$ such that for all $a\in\partial U$, $\lim_{z\to a}f(z)=\infty$ ...