Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
141 questions from the last 365 days
3
votes
1
answer
308
views
Holomorphic functions from the perspective of functional analysis
Fix an open set $U\subseteq \mathbb{C}$. Let $C(U)$ denote all continuous $f:U\to\mathbb{C}$ and $H(U)$ denote all holomorphic $f:U\to\mathbb{C}$. Equip $C(U)$ with compact-open topology, and note ...
- 3,083
13
votes
2
answers
556
views
Alternative proofs for the connectedness of the Mandelbrot set?
Douady and Hubbard proved that the Mandelbrot set is connected by constructing an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk....
2
votes
0
answers
139
views
Metrics on the Moduli / Teichmüller space of higher dimensional complex tori / abelian varieties
I am trying to study canonical metrics on the moduli / Teichmüller space of complex tori/abelian varieties. What would be a good reference for this?
Usually the classical references like Complex Tori ...
2
votes
1
answer
256
views
Paley-Wiener functions that concentrate on small interval
For $r> 0$ denote by $\mathcal{P}_r$ the space of entire functions $f$ that can be represented in the form
$$
f(z) = \int_0^r g(x)e^{izx}\, dx, \qquad g\in L^2([0, r]).
$$
Question. Do there exist ...
- 637
10
votes
5
answers
1k
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Formalisation of the notion that a complex analytic function $f(z)$ is "independent" of the variable $\overline{z}$
Suppose $U\subseteq\mathbb C$ is a connected open set, and $f:U\to\mathbb C$ is a function whose partial derivatives in the first and second components exist and are continuous on $U$. It is well-...
- 179
2
votes
0
answers
107
views
Transform a Maurer-Cartan formal solution into a convergent one
I am looking to the Bogomolov-Tian-Todorov theorem for Calabi-Yau manifolds. For any compact Kahler manifold $X$ with trivial canonical bundle ($K_X\simeq \mathcal{O}_X$) this theorem ensure that its ...
1
vote
0
answers
70
views
Split-signature wedge reflection positivity and Osterwalder-Schrader reconstruction
The Osterwalder-Schrader (OS) reconstruction theorem [1,2] establishes that Euclidean correlation functions satisfying reflection positivity (RP) in a codimension-1 hyperplane determine a Wightman QFT ...
6
votes
1
answer
572
views
Polynomial contractions acting as automorphisms of ${\Bbb C}^n$
Let $F:\; {\Bbb C}^n \to {\Bbb C}^n$ be a polynomial
automorphism. Assume that $0$ is the single attracting point, ${\Bbb C}^n$ is the attraction basin of $F$ and hence
$F$ is a holomorphic ...
- 10.3k
3
votes
1
answer
236
views
Bound on total radius of cover of level set of monic polynomial
I came across the following question: Let $P(z)$ be a monic polynomial with complex coefficients. Consider the set of points
$$E = \{z \in \mathbb{C} : |P(z)| \le 1\}.$$
Prove that $E$ can be covered ...
- 1,035
2
votes
0
answers
34
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On spaces of Laurent-type polynomials but involving fractional powers on a polygonal domain
I am interested in a polygonal domain $\Omega$ in the complex plane. For simplicity, we may assume that its closure $\overline{\Omega}$ is compact (so it has no edges going to infinity, for example). ...
- 5,497
2
votes
0
answers
203
views
Behavior of polynomial roots when removing a single term
This question was asked on MSE here
Let $n \in \mathbb{N}$ and let $C > 0$ be a fixed constant. Consider the class of monic polynomials of degree $n$ with complex coefficients bounded by $C$:
$$\...
- 783
1
vote
0
answers
105
views
Partial fraction decomposition generalization
Define two polynomials $P(x)$ and $Q(x) = (x-\alpha_1) \cdots (x-\alpha_n)$ for disinct $\alpha_i$. We have that $$\dfrac{P(x)}{Q(x)} = \sum_{i = 1}^n\dfrac{P(\alpha_i)}{Q'(\alpha_i)}\dfrac{1}{x-\...
- 579
2
votes
0
answers
83
views
Positivity of an analytic continuation of a Hermite-moment series
Let $\gamma$ be the standard Gaussian measure on $\mathbb R$ and let $(H_n)_{n\ge 0}$ be the orthonormal probabilists' Hermite polynomials:
$$
\int_{\mathbb R} H_n(x)H_m(x)\,d\gamma(x)=\delta_{nm},\...
- 73
4
votes
0
answers
169
views
Generators of ideals in $\mathbb{C}\{x,y\}$ and their differentials
Let $I$ be an ideal of $\mathbb{C}\{x,y\}$ (the ring of convergent power series in two variables) whose vanishing locus is $\{0\}$. Denote
$$
dI := \{ dg \mid g \in I \} \subset \Omega^1_{\mathbb{C}^2,...
- 315
1
vote
0
answers
73
views
Simplicity of complex zeros of a $\Gamma$-ratio / Bernstein function
Question. Fix $\beta>0$ and define
$$\phi_\beta(z)=\int_{0}^{\infty}(1-e^{-zx})(1-e^{-x})^{\beta}e^{-(\beta+2)x}\,dx,$$
interpreted by analytic continuation. With $t=e^{-x}$,
$$\phi_\beta(z)=\int_{...
- 11