Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
969
questions with no upvoted or accepted answers
8
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146
views
Extremal length of graphs in surfaces
Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be
$$
\sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\...
8
votes
0
answers
139
views
What does this number tell me about a convex lattice polygon?
EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices).
Suppose I have a convex lattice polygon $P$, ...
8
votes
0
answers
285
views
Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?
Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i
\geq 0$ with $\sum ik_i = n$, bounded in terms ...
8
votes
0
answers
488
views
The natural generalization of Euler's derivation of the Basel sum
Euler proved that $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{{\pi}^2}{6}$$ by comparing the $z^3$ term in the power series expression of $\sin(z)$ given by
$$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{...
8
votes
0
answers
858
views
Etymology of the O-notation for algebras of holomorphic functions
The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
7
votes
0
answers
214
views
Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
7
votes
0
answers
210
views
Partitions, weights and polynomials with roots on the unit circle
Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
7
votes
0
answers
297
views
Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
7
votes
0
answers
201
views
Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
7
votes
0
answers
156
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
7
votes
0
answers
122
views
holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)
Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|...
7
votes
0
answers
196
views
Polynomials having all zeros in the closed left half plane
Let $$P(z) =\sum_{k=0}^n(\alpha_k+e^{i\gamma}\beta_k)z^k=P_1(z)+e^{i\gamma}P_2(z)$$ be a polynomial of degree $n$ with $\alpha_k, \beta_k\geq 0$ for $0\leq k\leq n, $ where $$P_1(z) =\sum_{k=0}^n\...
7
votes
0
answers
299
views
Zero derivative on a connected set
I apologize in advance for this (not really research level) question whose answer should be well known. Complex differentiability of a function $f:A\to\mathbb C$ for $A\subseteq \mathbb C$ without ...
7
votes
0
answers
159
views
Limiting behavior of a sequence of polynomials
Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line
$\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary
fact (equivalent to Lemma 9.13 here) that
if $u\in\mathbb{C}$ and $|u|=1$, ...
7
votes
0
answers
600
views
How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
7
votes
0
answers
317
views
If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?
I hope this question isn't too basic or ambiguous for this site.
The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$:
$$\left[\begin{matrix}p & ...
7
votes
0
answers
455
views
On a paper of Alain Connes entitled 'Around Wilson's Theorem '
A relatively recent paper Alain Connes - Around Wilson's theorem
introduced the function
$$
S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr).
$$
In the same paper, he proved that the ...
7
votes
0
answers
198
views
Does this bound on an average over character sums have a more direct proof?
A special case of a well known result of Ingham is that
$$\sum_{n\leq x} d(n)d(n+1)=\frac{6}{\pi^2}x(\log x)^2+O(x\log x)$$
where $d(n)$ is the number of divisors of $n$.
Ingham's results, which ...
7
votes
0
answers
428
views
Sufficient condition on coefficients for a complex power series to be bounded
Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form
$$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$
where $c_{2n}\geq 0$ for all $n$.
Is there a ...
7
votes
0
answers
201
views
Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
7
votes
0
answers
155
views
Finite covers in complex analytic geometry
Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
7
votes
0
answers
727
views
What function space does holomorphic functional calculus give us?
Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
7
votes
0
answers
453
views
Convergence at the radius of convergence
Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...
7
votes
0
answers
225
views
A zeta function using half of the primes
It is well known that the zeta function satisfies the Euler product formula. See this wikipedia article.
Enumerate all primes by $p_1, p_2, \ldots $ in ascending order.
Set $S$ to be the set of all $...
7
votes
0
answers
892
views
Is this Fourier integral well-known?
The following integral is a special case of one that arises in an economics problem:
$I(u_{1}, u_{2}) := \displaystyle \int_{z_{1}=-\infty}^{\infty} \int_{z_{2}=-\infty}^{\infty} \frac{ \displaystyle ...
7
votes
0
answers
491
views
planar mappings that preserve elliptic measure
Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...
7
votes
0
answers
189
views
When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?
If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when ...
6
votes
0
answers
573
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
6
votes
0
answers
189
views
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
votes
0
answers
140
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
114
views
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
168
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
365
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
305
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
votes
0
answers
74
views
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
votes
0
answers
139
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
218
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
6
votes
0
answers
294
views
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
votes
0
answers
279
views
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
votes
0
answers
214
views
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
votes
0
answers
234
views
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
votes
0
answers
159
views
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
votes
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
votes
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
votes
0
answers
142
views
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
votes
0
answers
132
views
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
views
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}...