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

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101 views

Comparing two hyperbolic structures on a surface

Let S be a compact hyperbolic surface (a compact riemann surface of genus ≥2). Let S′=S∖P where P⊂S is a finite subset of points. Then S′ is hyperbolic too. Question: how is related the hyperbolic ...
8
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1answer
294 views

Local polynomial form of holomorphic functions

It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...
2
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1answer
74 views

How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems: Let $\Omega$ be an open connected subset ...
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1answer
373 views

Recomendation of Complex variables book [closed]

I'd like to ask for a book of complex variables that includes a "large" discussion about the Dirichlet problem, Neumann problem, and problems like that, I have now read "basic complex analysis - ...
3
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1answer
192 views

If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic?

Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$ and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, ...
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0answers
249 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
1
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1answer
85 views

Normal family and arithmetic progression

It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}_{n=1}^{\infty}$, is normal $\forall m\geq 1$. ...
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0answers
47 views

A criterion or algorithm for polynomial which admits Markov partition on its Julia set

For a given polynomials $P(z)$, whether there exists general algorithm to check it admits a Markov partition on its Julia sets. (in finite computation time.) May be it is more difficult for the ...
3
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0answers
140 views

elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result: for a transcendental entire function $f$, the second itarate $f^{2}$ has at least has one fixed point. (Using the technique of Picard theorem) This result ...
5
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1answer
340 views

Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
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2answers
98 views

general variational principle for the Julia sets of mermorphic function?

I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge? I want to ...
4
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2answers
412 views

When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?

Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (non-compact) complex manifold $X$. If $X$ and $Y$ are two (non-compact) complex manifolds, then $\mathrm{Aut}(X)$ and ...
1
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1answer
80 views

Removing a hyperplane from flag manifolds

It should be known that if we remove a compact complex codimension one submanifold $X$ (hyperplane) of a flag manifold $Z=G/P$, then $Z\setminus X$ is a Stein manifold. I was wondering if anyone can ...
1
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1answer
235 views

Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
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0answers
57 views

irreducible analytic decomposition of sets invariant under a group action

Let $U$ be a complex analytic space with an action of a finitely generated group $\Gamma$. Under what assumptions is the following true: Every $\Gamma$-invariant closed analytic subset of $U$ ...
19
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5answers
815 views

Topology on the set of analytic functions

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on ...
3
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1answer
176 views

Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...
21
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4answers
1k views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
5
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1answer
140 views

Ubiquity/scarcity of non-analytically continuable functions

Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of ...
2
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2answers
162 views

Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?

Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...
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1answer
139 views

Forster's Theorem

I am looking for a detailed account, in English or French, of the main result and its proof from this paper by O. Forster Zur Theorie der Steinschen Algebren und Moduln, Mathematische Zeitschrift 97, ...
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0answers
135 views

Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
4
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4answers
2k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
0
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2answers
177 views

A Functional Equation concerning analytic functions

Let $P$ be a polynomial and suppose $f : \Bbb{C}\longrightarrow \Bbb{C}$ is a non-constant analytic function such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such ...
3
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1answer
153 views

The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral $$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge ...\wedge d\phi_N \prod_{i<N} (-\phi_i) ...
1
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1answer
250 views

Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?

I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is ...
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0answers
123 views

topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$, call a subset $Z\subseteq U$ $\Gamma$-closed iff it is a closed analytic subset and each of its irreducible components is an ...
5
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1answer
188 views

Function transformation of exponentials

I came across the following function transformation: $$ \sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)} $$ where $ j ...
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0answers
82 views

How can we describe explicitly the “infinitely complex differentiable” complex-valued local martingales?

Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly ...
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0answers
55 views

Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
0
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0answers
243 views

Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...
1
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1answer
77 views

Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
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2answers
159 views

Comparing two Delaunay tessellations on a hyperbolic surface

Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb ...
2
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0answers
89 views

A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...
2
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2answers
392 views

What is the closure of space of polynomials in a dense subspace along with a marked point equal to?

EDIT Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most $d$ in two variables. So an element of this space is essentially $$ f:=f_{00} + f_{10} x + f_{01} y + \ldots ...
4
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0answers
144 views

semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
12
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3answers
734 views

Do all the roots of the polynomial lie in the unit disk?

How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see ...
28
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1answer
1k views

Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh: Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros ...
2
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1answer
153 views

Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k ...
5
votes
1answer
223 views

Zariski's main theorem in the complex analytic category

Hello, I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex ...
3
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1answer
212 views

Which real analytic functions of two variables locally are magnitudes of complex-analytic functions [closed]

Assume we have a real-analytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complex-analytic function $w(z)$ such that $|w(z)|=f(x,y)$ for $z=x+iy$. One necessary ...
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2answers
386 views

Constructing Riemann maps using Brownian motion?

There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...
2
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2answers
379 views

Mellin Transform

What is the inverse Mellin transform of (s-1/2)^k on the vertical line Re(s)=a where 0 < a <1 and k is a natural number?
2
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2answers
158 views

What is the moduli space of germs of one-sided complex structures near the circle?

Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( $U$ contains $S$) on the plane with a choice of smooth complex stricture $\tau$ on $U$. By smoothness of $\tau$ on $U$ ...
2
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2answers
301 views

What is known about this product?

I bet the product $$ \prod_{n=2}^\infty\frac 1 {1-n^{-s}}, $$ which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If ...
5
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4answers
574 views

Visualizing functions with a number of independant variables

I need to graph real valued functions ( for exposition and analysis) the issue is the independent variables are more so that the conventional graphing method cant be used and further i don't want to ...
7
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1answer
323 views

Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras

Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f ...
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2answers
446 views

On the existence of a holomorphic logarithm

Hi, The following is probably well-known, but I couldn't find anything in the literature. Any reference would be nice. Let $\Omega$ be a domain in the complex plane, and let $f$ be holomorphic and ...
5
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0answers
168 views

proper mapping between Stein manifolds

My question is the following: Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set ...
0
votes
1answer
261 views

Riemann mapping

Let in the complex plane be a bounded Jordan region T (that is a bounded and simply connected set with the boundary a Jordan curve), containing the origin, with its Riemann mapping onto the open unit ...