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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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1answer
45 views

On extensions of holomorphic mappings with image in a projective algebraic variety

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that: Let $N$ be a complex manifold, $S\...
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0answers
50 views

How to fix multi-valued function on contour?

I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function. I want to calculate the Fourier transformation of a muti-valued ...
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1answer
126 views

Poles of equivariant meromorphic functions on Riemann surfaces

Let $p:\Sigma\to \mathbb{P}^1$ be the cyclic cover of $\mathbb{P}^1$ with Galois group $\Gamma$. Let $\Gamma\cdot p$ be a free $\Gamma$-orbit on $\Sigma$. Given any character $\chi$ of $\Gamma$, does ...
16
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1answer
376 views

Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "...
3
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1answer
155 views

Is a factorial scheme with Noetherian stalks locally Noetherian?

Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...
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1answer
180 views

Cauchy path integral as a linear operator: kernel and image?

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
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0answers
73 views

Meromorphic mappings between complex projective spaces

Let $n>2$ and $\phi: \mathbb{P}_{\mathbb{C}}^n \setminus S \rightarrow \mathbb{P}_{\mathbb{C}}^n$ be a holomorphic map and $S$ a closed analytic subset of $\mathbb{P}_{\mathbb{C}}^n$ with ...
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1answer
231 views

A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that $$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$. where $\zeta$ ...
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3answers
213 views

Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

Denote by $\zeta$ the Riemann zeta function. It is known that $$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$ But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
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1answer
174 views

Exponential type of a product of entire functions

Let $\{a_n\}_{n=1}^\infty$ and $\{b_m\}_{m=1}^\infty$ be two sequences of points in $\mathbb{C}$ such that $$ f(z)=\prod_{n=1}^\infty\left(1-\frac{z}{a_n}\right)\quad\mbox{and}\quad g(z)=\prod_{m=1}^\...
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0answers
85 views

Understanding proof of q-theta function expression

In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed $$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
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2answers
220 views

Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
3
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1answer
104 views

Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type $$\int_C f(z) M(\lambda g(z)) dz$$ for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
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1answer
141 views

Complex structures on topological surfaces

I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...
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0answers
50 views

Equations needed to define a normal complex surface singularity

This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...
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0answers
31 views

Smoothings over a real interval

I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear. Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
2
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1answer
79 views

Smoothings of isolated non-irreducible surface singularities

Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing. Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
4
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1answer
140 views

Julia set containing smooth curve

I have two realted questions. Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
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1answer
112 views

Computing the convex hull of a region of $\mathbb{C}^2$

Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When ...
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0answers
47 views

Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
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63 views

The use of concavity of $\log\det\left(u_{j\overline{k}}\right)$

Let $F:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}$ be the function $$ F\left(a_{1\overline{1}},a_{1\overline{2}},\ldots,a_{n\overline{n}}\right):=\log\det\left(\begin{array}{ccc} a_{1\overline{1}} &...
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1answer
116 views

Analytic continuation of 2 variable function

Consider a function $F(x, y)$ of two complex variables. For $\Re(y)>0$, we know the analytic structure of the function. In that case, the function is meromorphic, with simple poles in $x$ at ...
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0answers
54 views

Global interior estimate complex Monge-Ampere equation

Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
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1answer
63 views

Riesz measure of a smooth subharmonic function on a ball

Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...
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3answers
702 views

How to find a conformal map of the unit disk on a given simply-connected domain

By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
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0answers
154 views

Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
3
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0answers
58 views

Is a domain of a holomorphic flow pseudoconvex?

Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
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0answers
190 views

Does the Riemann Xi function possess the universality property?

Here is the question.   Does the Riemann Xi function possess the universality property,  or something similar to Voronin 's universality property?  This is the reason why the answer to this question ...
3
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1answer
111 views

A question about distribution of fractional part of $2^k\alpha$

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$. The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
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1answer
252 views

Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
6
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1answer
260 views

Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...
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0answers
86 views

Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\...
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0answers
119 views

Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$. Is ...
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1answer
445 views

Are there enough meromorphic functions on a compact analytic manifold?

Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
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0answers
87 views

Is there a probabilistic proof/interpretation of Mergelyan Theorem

I came across Mergelyan's Theorem:- Let K be a compact subset of the complex plane C such that C∖K is connected. Then, every continuous function $f : K \to C$, such that the restriction f to int(K) ...
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0answers
232 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
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1answer
286 views

A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$ Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$ The answer is ...
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2answers
125 views

Lelong numbers and integrability of psh functions

Let $\varphi$ be a plurisubharmonic function in the unit ball $B_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that ...
16
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1answer
1k views

A conjecture of Littlewood

The following is a conjecture due to Littlewood. For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality $$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds....
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1answer
192 views

An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given. Let $f(z)$ be an analytic function on a subdomain $...
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1answer
92 views

Algebraic independence of certain values implies algebraic independence of functions?

It is quite general and elementary question. Is it possible that some holomorphic functions $f_1,\cdots,f_m $ on a region $\Omega $ of $\mathbb C$ satisfies: Whenever $(f_1(z), \cdots, f_m (z)) $ is ...
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2answers
138 views

Simple Closed Hyperbolic Geodesics on Punctured Spheres

Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
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0answers
417 views

Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&...
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0answers
133 views

Analytic Aspects of Rational Maps

I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
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0answers
90 views

Analytic continuation of an NLS soliton

The attractive nonlinear Schroedinger equation $i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi$ in the $H^1$-subcritical case $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...
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3answers
224 views

about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
2
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2answers
398 views

Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ...
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0answers
43 views

What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?

Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...
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4answers
185 views

PDE with Laplacian and squared of the gradient

Let $u$ be a real function in $\mathbb{R}^2$. Does anybody know that the following PDE $$\Delta u+|\nabla u|^2=0$$ has any non-constant general solution or not? It would be appreciated if any one ...
1
vote
1answer
190 views

Roots of modular functions

Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$...