Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,168
questions
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
6
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0
answers
211
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
6
votes
0
answers
141
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
6
votes
0
answers
115
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Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$
I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
6
votes
0
answers
169
views
Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
6
votes
0
answers
374
views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...
6
votes
0
answers
306
views
Are the two-valued homogeneous harmonic functions classified?
Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$?
For reference, multi-valued functions are familiar objects in ...
6
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0
answers
74
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Implications of combinatorial results towards discrete function theory on circle packings
Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex ...
6
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0
answers
141
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
6
votes
0
answers
736
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
6
votes
0
answers
219
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
6
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0
answers
298
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Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free
I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
6
votes
0
answers
323
views
Cheap bound on $\zeta'(s)/\zeta(s)$ or $L'(s,\chi)/L(s,\chi)$?
Say you are proving an explicit formula for $L(s,\chi)$ and/or the prime number theorem (in arithmetic progressions or not) in the usual way -- that is, shifting a line of integration from $\Re(s) = 1^...
6
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0
answers
279
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Complex factorization of the angular part of the Laplacian
Some time ago some research led me to the following equality:
\begin{equation}
\frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
6
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214
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Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
6
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234
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Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
6
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0
answers
159
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Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
6
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0
answers
151
views
Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?
Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
6
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0
answers
127
views
How big may the maximum set of entire function be?
Let us consider an entire function of several complex variables $f(z_1,\dots,z_n)$ and its modulus maximum
$$M(r,f):=\max \{ |f(z_1,\dots,z_n)|: |z_1|\le r,\dots,|z_n| \le r \} $$ with $r\ge 0$. How ...
6
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0
answers
142
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Evaluate $\sum_{\sigma} (2\pi i)^{-n}\oint \frac{f_{\sigma(1)}(u)\dots f_{\sigma_n(1)}(u)}{(u_2 - u_1)\dots (u_n - u_{n-1})}du_1\dots du_n$
In a probability theory paper I found this rather pleasant result:
Theorem 4.1 Let $n \geq 2$ and $f_1, \dots, f_n : \mathbb{C} \to \mathbb{C}$ be meromorphic with possible poles at $\{ \mathfrak{p}...
6
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132
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Is the map taking a matrix to its semisimple part algebraic (or at least holomorphic)?
Let $\text{Mat}_n(\mathbb{C})$ be the set of $n \times n$ complex matrices. Let $\sigma\colon \text{Mat}_n(\mathbb{C}) \rightarrow \text{Mat}_n(\mathbb{C})$ be the map that takes a matrix to its ...
6
votes
0
answers
1k
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Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$
For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist:
$$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
6
votes
0
answers
262
views
roots of a polynomial linked to mock theta function?
The following polynomial (after harmless factors dropped) is found in the paper entitled Mock theta functions and quantum modular forms by Folsom-Ono-Rhoades (see Theorem 1.1)
$$Q_k(z)=\sum_{n=0}^{k-1}...
6
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0
answers
321
views
Exact determinant of a Cauchy-like matrix
Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries
$$
A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n,
$$
where $z_i$ ...
6
votes
0
answers
147
views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
6
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0
answers
280
views
Is the space of holomorphic maps a manifold
To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
6
votes
0
answers
3k
views
What is the Beltrami differential?
Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$.
Local ...
6
votes
0
answers
329
views
Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
6
votes
0
answers
197
views
Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
6
votes
0
answers
210
views
Non-trivial bounds for polynomials at a fixed point
Let $f$ be a polynomial of degree $d$. Of course $|f(z)|\sim C|z|^d$ as $|z|\rightarrow\infty$ but also, since any polynomial is completely determined by its values at any $d+1$ points, we may ask how ...
6
votes
0
answers
126
views
Homogenous polynomially convex hull of $[0,1]^n$
I would like to calculate the set of $z\in \mathbb{C}^d$ such that there exists a constant $C >0$ such that for every homogeneous polynomial $p$ in $d$ variables $$|p(z)|\leq C\sup_{x\in [0,1]^d} |...
6
votes
0
answers
153
views
Grunsky-Motzkin-Schoenberg formula
I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:
Suppose that $...
6
votes
0
answers
392
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 ...
6
votes
0
answers
276
views
the "three-point" characterization of holomorphy
I want to know the source of the following "folkloric" fact about holomorphic functions.
It seems well described by the phrase:
The three-point characterization of holomorphy.
If F is a self-...
6
votes
0
answers
300
views
Invariant curves of rational functions
Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function
of degree at least 2 which
maps $\gamma$ onto itself homeomorphically. The following examples of such ...
6
votes
0
answers
175
views
Perturbations of zero-dimensional algebraic varieties
Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...
6
votes
0
answers
223
views
Complex manifold with non-finitely generated canonical ring
P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have ...
6
votes
0
answers
456
views
Jet differentials and hyperbolicity: possible mistake in the literature?
I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...
6
votes
0
answers
282
views
What is the status of the subadditivity problem for analytic capacity?
Hi,
Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by
$$\gamma(K):=\sup|f'(\infty)|,$$
where the supremum is taken ...
6
votes
0
answers
1k
views
Computing the Chern class for a flat line bundle using the holonomy group?
Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From Chern-...
6
votes
0
answers
161
views
Multiplicity of zero (higher dimensional analog)
Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold.
I want to associate to it some ...
6
votes
0
answers
486
views
Lacunar series with an interesting (in-formula) symmetry.
So, I wrote out a table of functions like so:
$\sum_{n=1}^{\infty} (-1)^{n+1}q^{n}=$ $+q^{1}$ $-q^{2}$ $+q^{3}$ $-q^{4}$ $+q^{5}$ + $\ldots$
$\sum_{n=1}^{\infty} (-1)^{n}q^{n^{2}}=$ $-q^{1}$ $+q^{4}...
5
votes
2
answers
1k
views
Is this a "folk theorem" about analytic functions of a complex variable?
In a comment on question 110345 I made a claim that might be incorrect. I claimed that if
f(z) is a non-constant analytic function defined by a power series whose circle of convergence C
has a ...
5
votes
1
answer
667
views
Integer polynomials mapping the unit disk into itself
Is there a complete characterization of those integer polynomials, that is $P\in{\mathbb Z}[X]$, such that $P(D)\subset D$, where $D$ is the unit disk ? At least, $P(X)=\pm X^k$ works, when $k\in\...
5
votes
3
answers
309
views
fixed points of quadratic iteration
Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period ...
5
votes
1
answer
695
views
Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?
Can anyone give me a reference for the following theorem on the Riemann zeta function?
If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...
5
votes
3
answers
779
views
Is there a "Riemann mapping theorem" for a circle in C^2 ?
The Riemann mapping theorem says that if you have a simple closed curve in $\mathbb{C}$, then there is an essentially unique way to map a holomorphic disc to the interior. Is there any reasonable ...
5
votes
3
answers
1k
views
Product of sine
For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...
5
votes
3
answers
898
views
Square of an elliptic curve and projective plane
Let's assume one takes $E = \mathbb{C}^* / \langle p \rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 \pi i \tau}$ where $1, \tau$ are the 2 periods in additive notation; $\Im \tau &...
5
votes
1
answer
820
views
A statement on complex polynomials
I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...