# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### One-parameter groups acting on dual Banach spaces

Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...

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**1**answer

728 views

### Plurisubharmonic exhaustion functions without critical points at infinity

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.
For example, Stein manifolds are weakly pseudoconvex (in this ...

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**2**answers

2k views

### Proper holomorphic map from unit disk to punctured unit disk

It is easy to see that there cannot be a proper holomorphic map from the punctured unit disk to the unit disk in the complex plane. What about the other direction; does there exist a proper ...

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**1**answer

503 views

### The identity $\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})$

As in the famous Euler product identity, the primes occur on
only one side of the following:
$\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})\ .$
My basic question: Does this ...

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**3**answers

721 views

### Is there a “Riemann mapping theorem” for a circle in C^2 ?

The Riemann mapping theorem says that if you have a simple closed curve in $\mathbb{C}$, then there is an essentially unique way to map a holomorphic disc to the interior. Is there any reasonable ...

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**1**answer

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### Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...

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**3**answers

639 views

### magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal

What can you say about a function defined on a square region of the complex plane, if the integral of the function along any horizontal, vertical or diagonal of the square is equal ? - an analytic ...

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**0**answers

279 views

### Question in complex analysis arising from large $N$ gauge theory

This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...

**25**

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**2**answers

4k views

### Which almost complex manifolds admit a complex structure?

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau'...

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260 views

### Finding representatives of PSL_2(Z) orbits

Given $\tau$ in the upper half plane, what is a good, systematic way to find a representative in the usual fundamental domain for the $PSL_2(Z)$-orbit of $\tau$? For example, let $\tau=\frac{2}{3} + \...

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645 views

### Pole data of meromorphic matrix function

Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a right pole of order $r$ ...

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207 views

### Picard Fuchs and Lefschetz trace

In Clemen's book "A Scrapbook of Complex Curve Theory", he discusses in Chapter 2 how the infinite sum giving the period of the Legendre curve matches (mod p) the sum giving the number of points over ...

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**3**answers

668 views

### Holomorphic function with a.e. vanishing radial boundary limits

Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...

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votes

**1**answer

2k views

### Relation between complex analysis and harmonic function theory [closed]

There are some theorems in harmonic function theory that resemble results in complex analysis, like:
Holomorphic functions and complex functions are analytic;
Cauchy's integral formula in complex ...

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2k views

### Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...

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270 views

### Universally open morphism with reduced fibers.

Hi.
I asked in the last post if, for a flat morphism $f:X\rightarrow S$ of complex spaces with reduced fibers and $S$ reduced, $X$ is reduced or not. In the algebraic setting, Liu said that the ...

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**1**answer

1k views

### Zeros of a holomorphic function

Suppose Ω is a bounded domain in the plane whose boundary consist of m+1 disjoint analytic simple closed curves.
Let f be holomorphic and nonconstant on a neighborhood of the closure of Ω such that
|...

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**2**answers

686 views

### Flat map with reduced fibers.

Hi.
Let $f:X\rightarrow S$ be a flat, surjective morphism of complex spaces with reduced fibers over $S$ reduced.
Q1: Is $X$ reduced too?
Q2: Is the property " reduced fiber" preserved by base ...

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votes

**1**answer

711 views

### Non-standard enlargements, $\zeta(s)$ and analytic continuation

Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...

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**1**answer

6k views

### A paper to the question, if the six dimensional sphere is a complex manifold [duplicate]

for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...

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**0**answers

218 views

### Dimension of pluripolar sets

Let $\Omega$ be an open set in $\mathbb C^n$, and let $A$ be a closed pluripolar set in $\Omega$. Is there a notion of dimension of $A$ such that the following theorem is true?
Theorem.
Let $\phi$ ...

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votes

**1**answer

767 views

### Irreducibility of Analytic Sets

How does one prove that an Analytic set $V$ in $C^n$ is irreducible if the set of regular points $V^*$ is connected?
Proceeding by contradiction, if we assume that $V$ is in fact reducible and if $...

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361 views

### Boundary behavior of Kähler cone with curvature restriction

Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.
A fundamental result is due to Demailly and Paun: they ...

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**1**answer

1k views

### Kolodziej's acta paper “the complex monge-ampere equation”——a detailed ploblem [closed]

Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.
In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}...

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922 views

### Pedagogical question concerning $\Gamma(z)$

Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...

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312 views

### Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$

Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for ...

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**2**answers

442 views

### Analytic functions with isotopic x-rays

Following Arias-De-Reyna, the x-ray of an analytic function $f$ means markings on the complex plane, with one color showing the real locus of $f$ and another color the purely imaginary locus.
...

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4k views

### If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...

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votes

**1**answer

1k views

### Surgery in complex geometry

I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...

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186 views

### plurisubharmonic sublevel sets

Let $X$ be a complex manifold, let $\Omega \subseteq {\bf C} \times X$ be defined by
$\Omega = \{ (z,p) \in {\bf C} \times X : a(p) < Im z < - b(p) \} $ where $a$ and $b$ are plurisubharmonic ...

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**1**answer

7k views

### Behaviour of power series on their circle of convergence

I asked myself the following question while preparing a course on power series for 2nd year students. Let F be the set of power series with convergence radius equal to 1. What subsets S of the unit ...

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707 views

### Landau's constant

(Hi. This is my first question here.)
A well known result in complex analysis says that there is an $\varepsilon\gt 0$ such that if $f$ is holomorphic in (a neighborhood of) the closed disk ${\mathbb ...

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**1**answer

662 views

### on common fixed points of commuting polynomials (and rational functions)

By the Ritt's classification, for any pair of commuting polynomials (i.e. $f(g(z))=g(f(z))$) over $\mathbb C$ there is a common fixed point of them. My questions are:
Is that true that this can be ...

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766 views

### Relationships between the roots of an entire function and the roots of its derivative

Hey everyone,
I would like to know if anybody could help me find references for the following.
Take a suitably well defined entire function $f(x)$ and it's derivative $\tilde{f}(x)$ to which the ...

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**1**answer

896 views

### Harmonic forms on Ricci-flat Kahler manifolds

Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian.
...

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705 views

### Compactness properties of plurisubharmonic functions

I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\...

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**1**answer

347 views

### Conjugate Groups of (quasi) Fuchsian Groups

I apologize in advance if this question is so trivial or too low level.
Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...

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**1**answer

702 views

### Holomorphic functions in almost-complex geometry

Maximum principle implies that every holomorphic function on a compact complex manifold is constant. Is this still true if the manifold is only almost complex?

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**1**answer

517 views

### How manifold-like is Aut(C^n) in the holomorphic category?

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just $\...

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4k views

### Specializing in Complex Analysis [closed]

May someone kindly provide a useful list of books on complex analysis that would be appropriate for a graduate student intending to specialize in that area. Thanks, your help is appreciated!
William

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**1**answer

491 views

### monodromy of plane curve singularities

Are there two IRREDUCIBLE plane curve singularities having different equisingular type with the same monodromy (linear action on the first homology group of the (regular) Milnor fibre)?

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696 views

### What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $...

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votes

**1**answer

329 views

### Do proper polynomial mappings have a path-lifting property?

Suppose $f: \mathbb{C}^n \to \mathbb{C}^n$ is a proper polynomial mapping and $\gamma: [0,1] \to \mathbb{C}^n$ is a continuous path. Further, suppose $z_0 \in \mathbb{C}^n$ satisfies $f(z_0)=\gamma(0)...

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124 views

### Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function)

2 random fields $b$ and $c$ are derived from random field $a$ by
$b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a $
and
$c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$.
(...

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vote

**3**answers

785 views

### A simple ordinary differential equation

Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function
$$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and $g(0)=...

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1k views

### Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...

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**1**answer

2k views

### Wick rotation and the Riemann zeta function

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now ...

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2k views

### Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...

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510 views

### Does the tangent bundle of this fiber product split?

Let $\mathcal X \to S$ be the local universal family of an elliptic curve, and let $E \to S$ be a vector bundle over $S$. Then we can form the fiber product $\mathcal Y = \mathcal X \times_S E$, which ...

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147 views

### Holomorphic automorphism of strictly psudo-convex domain smooth on boundary

I am wondering if anything is known about this. I couldn't find anything in the literature.
In '74 C. Fefferman published a solution to the following problem.
Let $\sigma:D\rightarrow D$ be an ...