# Tagged Questions

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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### Any convex Reinhardt domain is logarithmically convex

I have the following question in Shabat p.59: Prove that any convex Reinhardt domain is logarithmically convex. I think I have a good idea about how to show this, but need to be clear on the ...
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### Domain of convergence is complete Reinhardt

Let $W\subset \mathbb C^n$ be a domain such that $W$ is the domain of convergence for a certain power series at the origin. How to show that the interior of $W$ is a complete Reinhardt domain?
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### 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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### Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
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### 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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### 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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### 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 $f:\mathbb C^2\to\mathbb C$...
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### On the hypothesis of the Additive Cousin Problem

The Additive Cousin Problem is the following: Assume that $D$ is a region (open, connected) of $\mathbb{C}^n$. Assume that the Dolbeault Cohomology Group of $D$, $H^1_{\bar{\partial}}(D)$ is equal ...
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### 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 ...
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### Why are there no discrete zero sets of a polynomial in two complex variables?

Why is the zero set in $\mathbb{C}$ of a polynomial $f(x,y)$ in two complex variables always non-discrete (no zero of $f$ is isolated)?
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### 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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### 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 theorem,...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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