Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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67 views

Riemann Mapping Theorem [on hold]

It is well known that the Riemann mapping theorem asserts that for any open simply connected $G\subset \mathbb{C}$ and $z_{0}\in G$, there exists a unique bijective analytic function $f:G\to ...
5
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0answers
97 views

Complex structures on $\Bbb R^4$

Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...
3
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0answers
82 views

Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1) \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space $$ Z_n^d(\eta, ...
-1
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0answers
58 views

Residue theorem and essential singularity [closed]

Can residue theorem be applied in the case of essential singularity? If it can, how to calculate the residue?Thank you.
4
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2answers
193 views

Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps. I think it is interesting to ask whether we ...
0
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0answers
34 views

on the existence of holomorphic coordinates under bounded curvature

Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...
-4
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0answers
45 views

How can i be distinguished from -i? [migrated]

Mathematicians designate one solution to x^2 = -1 as i and the other as -i. Would anybody notice if we switched their identities? Any polynomial p(x) with a complex root will also have its conjugate ...
1
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1answer
52 views

An optimization problem in complex space

Consider the following optimization problem $$ \min \| \textbf{Ax-B}\| $$ $$ s.t.|x_i|=1,i=1,...,n $$ where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...
7
votes
1answer
129 views

Conformal map of polygon with circle segments

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a ...
12
votes
2answers
303 views

Entire function bounded at every line

I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
0
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0answers
59 views

kostant partition function vs Haar measure

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant: $$ \Delta(\theta) = \prod_{i< j} ...
4
votes
2answers
119 views

Bound areas of disks with respect to a quadratic differential

Let $q = q(z)\,dz^2$ be a quadratic differential on the unit disk $D(1)$, normalized so that $\int_{D(1)}|q| = 1$. I have a fairly convoluted argument that for any smaller disk $D(r)$ with $r < 1$, ...
2
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0answers
114 views

weak form of Sendov conjecture

Suppose $p$ is a polynomial of degree $n$ and all roots $z_1,\cdots,z_n $ of $p$ are inside the unit disk. Then how to show that every disk of radius $\sqrt{2}$ and centered at $z_k$ for ...
5
votes
1answer
157 views

Starshapeness of polynomial tracts with respect to the (entire collection of) critical points contained in the tract

I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a connected component of a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be ...
0
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0answers
106 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
31
votes
1answer
1k views

$\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that $$\pi\cot(\pi ...
5
votes
0answers
50 views

Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...
0
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0answers
74 views

Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum: $$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$ as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...
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0answers
108 views

Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
1
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2answers
74 views

A question on deficient values of entire functions

Recently I come cross a question about deficient values of entire functions. I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse ...
0
votes
1answer
136 views

When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map $$Mor_{Sch}(S,X)\to ...
3
votes
0answers
111 views

Series of swapping terms

Suppose $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$ is defined in the unit disk, $\mathbb{D}$, and $\|f\|_{\infty}\leq 1.$ Lets form a series $f_k$ by interchanging $a_1$ and $a_k$ i.e. ...
0
votes
0answers
40 views

Lifting $SHFC$ to non singular foliations

In this question we would like to complexify the idea in the following post. We would like to lift a "singular holomorphic foliation by curves", briefly "SHFC", of $\mathbb{C}P^{2}$ to a ...
2
votes
1answer
108 views

Norm of swapped power series in the unit disk

Suppose $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$ is defined in the unit disk and $\|f\|_{\infty}\leq 1.$ Lets form another series $g$ by interchanging $a_1$ and $a_k$ i.e. ...
15
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3answers
354 views

Can all $L^2$ holomorphic functions on a domain vanish at a particular point?

Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space ...
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0answers
76 views

Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?

Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
0
votes
1answer
159 views

A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...
4
votes
2answers
610 views

What does analyticity imply in complex analysis? [closed]

In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not ...
6
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1answer
287 views

Applications of the Small and Great Theorems of Picard

I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications. List of applications (rather ...
0
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0answers
75 views

Non-trivial global solution for Dirichlet eigenvalue problem

Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is smooth everywhere except a set of measure zero.(i.e. A set of area zero) and satisfies the equation $\Delta f=\lambda f$ for some constant $\lambda$ off this ...
6
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170 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...
1
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1answer
164 views

meromorphic extension of a function

Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in ...
2
votes
1answer
126 views

Meromorphic functions with finitely prescribed zeros and poles on annuli

I hope that this (probably) naive question will not bother those experts. Anyway, please allow me to ask this question here: We set $$\mathbb{A}(1/2, 2) = \Big\{z \in \mathbb{C}: 1/2 < |z| < ...
2
votes
3answers
111 views

closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2

Let $$ (\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0, $$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$. Q: ...
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2answers
161 views

“Convolution” for Multiplying Random Variables

The following situation arises frequently in probability. Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...
7
votes
5answers
452 views

Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum: $$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$ as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...
1
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0answers
130 views

Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
1
vote
2answers
177 views

Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...
9
votes
3answers
399 views

How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch. EDIT: This is an edited version. Before I asked about roots ...
3
votes
1answer
66 views

Conditions conformal mapping to be expansive

Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that ...
5
votes
1answer
96 views

Special Kähler normal coordinates around a point

Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G_{p}$ of biholomorpisms fixing $p$. Let ...
5
votes
1answer
180 views

How to prove an elementary functional equation for polylogarithms?

Let $Li_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li_{-n}(z)=(-1)^{n-1}Li_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing ...
5
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0answers
171 views

Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
7
votes
3answers
288 views

A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability. A quick google search gave a lot of references on SLE ...
4
votes
2answers
469 views

Power series with funny behavior at the boundary

Consider a power series $$ \sum_{n=0}^{\infty}a_nz^n $$ where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...
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0answers
107 views

Existence of zero-free strip of a Laplace transform (edited ..)

Problem Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is ...
6
votes
1answer
257 views

Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...
0
votes
2answers
111 views

Metric properties of a quadratic differential at an essential singularity

Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| ...
6
votes
0answers
192 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...
12
votes
2answers
584 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...