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

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7
votes
3answers
364 views

Summation of a series

I would like to sum the series $$ \sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} . $$ It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...
0
votes
0answers
212 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with $\tau$, $\lambda$, and ...
27
votes
1answer
440 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
2
votes
1answer
307 views

Definition of a complex space

In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...
0
votes
1answer
87 views

quotient of holomorphic functions at a point

Let $0\in U \subset \mathbb{C}$ be a small neighborhood of origin in the complex plane and $f_1,f_2\colon U\to \mathbb{C}$ be two complex valued functions such that $$f_1(0)=f_2(0)=0$$ ...
0
votes
1answer
424 views

automorphism groups of unit disk $\mathbf{D}^n $ and unit ball $ B^n $

How does one compute the group of biholomorphisms of $\mathbf{D}^n = \{(z_1, \ldots, z_n) \in \mathbb{C}^n: \forall_i \; |z_i| \leq 1\}$, i.e., the unit polydisk, and of the unit ball $B^n = \{(z_1, ...
3
votes
1answer
178 views

Local holomorphic equations for symplectic divisors

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ ...
3
votes
1answer
227 views

Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
12
votes
0answers
210 views

Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? ...
0
votes
0answers
90 views

Arakelian's approximation theorem

I have a difficulty in understanding how one gets relation (2) in the proof of the Theorem, in the nice paper [Jean-Pierre Rosay and Walter Rudin, Arakelian's Approximation Theorem, The American ...
3
votes
0answers
229 views

The Poisson-kernel in the plane and polynomials

Let \begin{align*} p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\ & = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j} \end{align*} be a non-constant complex polynomial with ...
1
vote
0answers
112 views

Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that $\sigma \colon S \mapsto S + S^2 + S^3$ $\sigma \colon X \mapsto X + S$. It is easy to see that the ideal $(S)$ is stable ...
3
votes
1answer
301 views

Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...
4
votes
0answers
77 views

status of Invariant subspace problem on Krein Space

What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
0
votes
1answer
121 views

smooth holomorphic functions are CR on the boundary? [closed]

Is this true that any holomorphic functions in a domain with smooth boundary, and which is smooth on the boundary is a CR function ?
2
votes
1answer
184 views

Can the natural boundary be part of the unit circle?

It is well-known that the function $f(z)=\sum_{n=0}^\infty z^{n!}$ is analytic in the open unit disk and it can not be extended analytically to any proper open superset of the unit disk, i.e., the ...
3
votes
2answers
344 views

The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...
1
vote
1answer
83 views

Which rate of growth of the Sobolev norms guarantees analyticity?

Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.) We define the semi-norm ...
5
votes
1answer
248 views

definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the transformation $ (T − \lambda)/(T + \bar \lambda\ ) $ with domain and range in the Hilbert space is contractive for ...
0
votes
0answers
48 views

Finding stable ideals of F_3[[X,S]] by group action.

Let k > 1 be a positive integer and define the action σ_k on F_3[[X,S]] by σ_k: X ---> X + S + X^k σ_k: S ---> S + S^3. Then, Conjecture: There exists a principal ideal (a) other than (S) such ...
0
votes
1answer
77 views

A subspace of the algebra of infinitesimal CR automorphisms

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR ...
6
votes
0answers
130 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
4
votes
1answer
176 views

Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set : $$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$ Question : If $K(f)$ ...
2
votes
1answer
163 views

Why do holomorphic maps increase extremal length?

Let $\mathcal{L}$ denote extremal length. The following theorem appears in http://arxiv.org/abs/math/0505191. Theorem Let $U$ and $V$ be Riemann surfaces, and let $f:U\rightarrow V$ be a holomorphic ...
2
votes
3answers
282 views

Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...
0
votes
1answer
95 views

Integral and conformal mappings II

Assume that $f_n$ is a sequence of conformal injective mappings of the unit disk $D$ onto the nested smooth Jordan domains $D_n\subset D$, such that $\cup_{n=1}^\infty D_n=D$ and $D_n$ are images of ...
2
votes
1answer
118 views

Uniform convergence of conformal mappings

Assume that $D_n\subset D$, where $D$ is the unit disk is an increasing sequence of Jordan domains with smooth boundaries such that $\cup_{n=1}^\infty D_n=D$ and let $f_n: D \to D_n$ be conformal ...
2
votes
2answers
277 views

Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. I am pretty sure that the answer to the ...
2
votes
0answers
73 views

Analytic differential equations with estimates

I recently came across the following theorem of Martineau (and others). Let $P$ be a polynomial of $n$ variables with complex coefficients. If we identify naturally powers of partial derivatives ...
2
votes
2answers
257 views

A question on Koebe theorem

Assume that $f$ is a conformal mapping of a bounded Jordan domain $\Omega$ onto the unit disk U such that $f(0)=0$. How to prove the following inequality $(1-|f(z)|)\le K \sqrt{dist(z,\partial ...
1
vote
1answer
111 views

Why there are so few conformal mappings in higher dimension? [duplicate]

I think the conformal mapping theory in the plane are quit interesting and useful in physics. I learned that there is very few conformal mappings in higher dimensions, is there any reason for that?
2
votes
1answer
217 views

hayman's result for $ A^2(D) $

Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. Thus for $|z|\leq 1$ we have $ ...
1
vote
1answer
160 views

ddbar lemma for positive closed (1,1)-currents

This is probably fairly elementary, but does someone know how to prove the following or know a reference. Let $X$ be a Kaehler manifold. Let $\theta$ be a closed $(1,1)$-form and $T$ be a closed ...
1
vote
0answers
95 views

Forcing a set of complex points to be closed under conjugation

I am a PhD student and I have to address the optimization of a real scalar function of complex variables $f(z_1,z_2,\ldots,z_n)$. The function is real valued in my case because each complex $z_i$ ...
0
votes
1answer
117 views

Chow lemma for complex spaces

Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of ...
1
vote
0answers
73 views

analytical behaviour of holomorphic functions on boundary

Is there any survey about criterions for a holomorphic function to be analytical on the boundary of a pseudo convex domain on which it is defined ?
19
votes
0answers
364 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
2
votes
2answers
293 views

Topological information via cohomology of sheaves

On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The ...
-1
votes
1answer
142 views

Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product." I try my best to understand Stone-Cech compactification theorem by ...
5
votes
1answer
58 views

An corona pair (a,b) satisfies (A_2)

Let $b$ be a non-extreme point in the unit ball of $H^\infty$. Let $(a,b)$ be a corona pair, that is $|a|+|b|$ is bounded away from zero in the unit disc. Also let $|a|^2$ satisfy Hunt-Muckenhoupt ...
2
votes
1answer
339 views

Finding the residue for a complex function defined using an infinite product

Suppose we define the infinite product $\displaystyle \prod_{n=1}^{\infty} (1+a^{-ns})^{a_n}$, where $a_n$ is some given sequence of positive integers. Is there a way, supposing there is a pole at ...
-2
votes
1answer
137 views

Holder class of analytic functions

Assume that $\lim_{(nt) |z|\to 1}|f(z)|(1-|z|)^p=0$, where $f$ is analytic in the unit disk and $p>0$,where $(nt)|z|\to 1$ nontangentially. Does this implies that $\lim_{|z|\to ...
3
votes
1answer
269 views

Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
2
votes
2answers
229 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...
3
votes
1answer
152 views

Weighted projective space with rational or real weights

The most common formulation of the weighted projective space is perhaps the global quotient $$ (\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast $$ with the $\mathbb{C}^\ast$ group action ...
4
votes
1answer
172 views

Integral and conformal mappings

Let $f$ be a conformal mapping of the unit disk $U$ into $C$. Is the following integral convergent $$\int_U \frac{dx dy}{|f'(z)|}?$$
5
votes
2answers
541 views

An extension of Morera's Theorem

Morera's Theorem states that If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$. I have two questions: ...
3
votes
1answer
303 views

A particular contour integral

Mathoverflow, I'd like to carry out the following integral, $$f(t) = \int_{- \infty}^{\infty}\frac{-i\Omega e^{i \Omega t}}{1-\sqrt{-i\Omega}\coth(\sqrt{-i\Omega})} d\Omega.$$ Here's what I've ...
17
votes
1answer
540 views

Almost linearly dependent functions

Everyone knows that analytic functions $f_1,\ldots,f_n$ are linearly dependent if and only if their Wronski determinant is identically equal to zero. There are several proofs of this, one in ...
0
votes
1answer
139 views

Inverse “Riemann mapping” [closed]

The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...