Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,153
questions
3
votes
2
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495
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Connected complement manifold
I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...
11
votes
1
answer
369
views
Continuous Weierstrass map
Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set
of all non-negative divisors in $\mathbb C$.
Consider the map $Z:H(\mathbb C)\to D(\...
18
votes
2
answers
920
views
Vanishing of Dolbeault cohomologies and Steinness
That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
1
vote
2
answers
487
views
Generalized Schwarz Lemma for near-zeros
In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma:
Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain $...
0
votes
1
answer
120
views
Dirichlet series without order term
is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum_{0<n<N}a_n/n^s$
The ...
1
vote
0
answers
448
views
Complex Hessian Signature
It' all, simply, about the signature of a matrix.
Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
1
vote
1
answer
364
views
Cohomology of lattice with coefficients in field of rational functions
In my research, I came across a 1-cocycle in the following group cohomology complex:
Let $\Lambda_\mathbb{Z}$ be a lattice (i.e. isomorphic to $\mathbb{Z}^n)$; let $\Lambda_\mathbb{C} = \Lambda_\...
3
votes
1
answer
415
views
Singularities of the Remmert reduction of a holomorphic convex manifold
Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...
1
vote
1
answer
175
views
About the upper bound on the roots of the matching polynomial
Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$.
Is there a modern exposition of ...
2
votes
1
answer
392
views
If a polynomial $p(z)$ omits a value, then $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)$ also omits that value
Suppose that a polynomial $p(z)$ of degree $n$ does not assume the value $w$ for $|z|<1$, that is $p(z)\neq w$ for $|z|<1.$ Show that $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)\neq w$ for $|z|&...
2
votes
1
answer
151
views
Prove or disprove an inequality concerning zeros of a polynomial
If a polynomial $p(z)$ of degree $n$ with zeros $z_1,z_2,\cdots,z_n$ assumes maximum at $w$ on $|z|=1.$ Prove or disprove that the Harmonic mean of $|z_k-w|,$ $k=1,2,\cdots,n$ is greater or equal to $...
3
votes
1
answer
209
views
Determining the Fourier transform
Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...
2
votes
1
answer
148
views
hyperbolic metrics
Let $D_1\subset D_2$ be simply connected domains in the complex plane. Let $\lambda_1$ and
$\lambda_2$ be the corresponding hyperbolic (Poincare) metrics. It seems intuitive to me that
$\lambda_2$ is ...
6
votes
1
answer
278
views
Is the Poincaré metric continuous with respect to the domain?
Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ ...
3
votes
1
answer
223
views
A conjecture regarding the integral of the square of an entire function
Can some help me prove or disprove the following assertion which I encountered in research? Thanks!
Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some $...
6
votes
1
answer
532
views
Bound on the sum of arguments
Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...
0
votes
1
answer
2k
views
Holomorphic Function on Disk
Let $f$ be holomorphic function on unit disk and it is continuous on boundary of the disk.
It is known that $f$ is constant and equal to zero if $f$ is vanishing on sub-arc of boundary (Maximum ...
3
votes
1
answer
365
views
If $X$ fails to be holomorphic, what is $\mathcal{L}_X \bar\partial - \bar\partial \mathcal{L}_X$?
I present a simplified version of the problem, and if that's easy to answer then someone can try answering the more complicated version.
Simplified version
Suppose $X$ is a tangent vector field on a ...
2
votes
1
answer
312
views
Exact reference for Liouville theorem
It seems hard for me to find that the solution of the following equation
$$
\Delta u+e^u=0
$$
defined on a simply-connected domain $D\subset R^2$ must be of form
$$
u(z)=\log\frac{4|f'|^2}{(1+|f|^2)^2}...
18
votes
1
answer
834
views
Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?
Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$
and consider the generating functions
\...
2
votes
1
answer
1k
views
Irreducible non-singular M-matrices and complex numbers
It is well known that a non-singular M-matrix that is irreducible has a strictly positive inverse (all entries $>0$).
An M-matrix is a matrix that has eigenvalues with positive real part, and the ...
11
votes
5
answers
4k
views
Applications of Liouville's theorem
I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...
-2
votes
1
answer
985
views
Degree of a rational function [closed]
I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
8
votes
2
answers
837
views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of $...
37
votes
1
answer
3k
views
Circles and rational functions
Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere,
and there exist two rational functions $f$ and $g$ such that
$f$ maps $\gamma$ into a circle, and $g$ maps a circle into $\gamma$...
1
vote
1
answer
126
views
About convex combinations of real-stable multivariable complex polynomials
Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
5
votes
1
answer
589
views
Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?
Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on $\mathbb{C}^n$...
5
votes
1
answer
453
views
Structure of the automorphism group of a Riemann surface
I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
0
votes
0
answers
252
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
1
vote
1
answer
820
views
Triviality of holomorphic vector bundles over contractible Stein manifolds
If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
2
votes
0
answers
181
views
Real-rooted polynomials and higher rank matrices
For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + tB)...
2
votes
1
answer
981
views
A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane
I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...
1
vote
1
answer
162
views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain $\...
3
votes
0
answers
320
views
Implicit function theorem for singularities
I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
5
votes
1
answer
734
views
Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?
Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f_n$ is a sequence of holomorphic functions converging pointwise to 0 on $\...
4
votes
0
answers
661
views
A difficult integral which the Risch algorithm shows is not elementary
For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
$$\int_{\delta}^...
2
votes
0
answers
173
views
Does this symmetrization operator have a name? Any theory?
Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define
$$f_{\mathrm{symm}}(x_1,\ldots,x_n) =
2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1}
f(\varepsilon_1x_1,\ldots,\...
0
votes
1
answer
265
views
holomorphic continuation
consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.
This function can be continued holomorphically for all complex numbers with ...
12
votes
0
answers
286
views
How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?
One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
2
votes
1
answer
251
views
polynomial inequality in complex variable (generalized)
Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number.
What happens if ...
1
vote
1
answer
188
views
Sequence of smooth maps converging to the identity [closed]
Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
1
vote
1
answer
495
views
Finding Laurent Series of a function [closed]
I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
1
vote
0
answers
215
views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
10
votes
1
answer
338
views
Is there a solvable point on any variety over the field of complex rational functions?
Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...
2
votes
1
answer
306
views
Flatness of a morphism of complex analytic spaces
Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...
5
votes
1
answer
352
views
A weak analytic version of the valuative criterion of properness
EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...
1
vote
0
answers
406
views
Possible generalizations of Hadamard's three line lemma
Let $f$ be an analytic function on a sector
$$
S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\}
$$
with opening angle $\gamma$ at the origin. Suppose $f$ is ...
1
vote
1
answer
364
views
existence of positive curved line bundles on a compact Riemann surface
Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
6
votes
1
answer
642
views
Analytic diffeomorphisms of the circle from complex domains
Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to (\...
36
votes
1
answer
3k
views
Is S^2 x S^4 a complex manifold?
As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...