Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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3
votes
2answers
238 views

j-invariant duplication, triplication and quintuplication formulae… how?

I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s ...
6
votes
2answers
494 views

Reason for studying coherent sheaves on complex manifolds.

Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it ...
2
votes
0answers
153 views

Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients? Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$, $f'(0)>0$. ...
1
vote
0answers
270 views

The relation between the weak Lefschetz theorem and the strong Lefschetz theorem

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective. The Hard Lefschetz Theorem states that for a compact Kahler manifold ...
3
votes
2answers
127 views

Weierstrass factorization with $L^2$ estimates?

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? ...
1
vote
2answers
126 views

Solution of an infinite differential system

Let $r\in \mathbb N$ and $f$ be an entire function on $\mathbb C$. One assumes that for every $R\in\mathbb C[z]$ there exists polynomials $P_{i,R}\in\mathbb C[z]$ ($0\le i\le r$) not all zero such ...
3
votes
3answers
764 views

Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$ Is it true or false that a combination of two (or more, in general) of the ...
1
vote
0answers
69 views

Volume of complex submanifolds

Let $X$ be a projective algebraic manifold and let $w$ be any Kahler metric on $X$. Then the volume of any complex submanifold of $X$ w.r.t $w$ is bounded below by a uniform positive constant. My ...
2
votes
9answers
744 views

functions of one complex variable: geometric theory

Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English? When I studied complex analysis, I used two textbooks: An ...
16
votes
2answers
417 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
1
vote
1answer
229 views

A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$. Stable polynomials with all real ...
2
votes
1answer
114 views

Entire functions of exponential type with small $L^1$ norm outside a finite real interval

I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following $$\int_{-\infty}^{\infty} |f(x)|dx=1,$$ whose norm is as small as possible outside a range ...
0
votes
0answers
180 views

Bounded mappings on unbounded domains

Ahlfors and Beurling (Ahlfors, Lars; Beurling, Arne Conformal invariants and function-theoretic null-sets. Acta Math. 83, (1950). 101–129.) provided an explicit example of a domain in $\mathbb C$ ...
7
votes
3answers
544 views

Injectivity bounds for complex analytic functions

Let $$f(z) = z - \sum^\infty_{n=2} a_nz^n.$$ What is the largest ball around $0$ where $f$ is injective? If we restrict to the case where $a_n \geq 0,$ it seems the radius should be given exactly by ...
1
vote
0answers
125 views

The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
2
votes
1answer
294 views

Where i can find the proof of Ostrowski theorem

Where i can find any proof of next theorem? Theorem of Ostrowski: Let $\sum_{n=0}^{\infty}a_nz^n$ be a power series with radius of convergence 1, which is analytically continuable beyond unit disk. ...
1
vote
1answer
142 views

On the set of zero radial limits of bounded analytic functions

Hi, Let $f$ be a non-identically zero bounded analytic function in the open unit disk $\mathbb{D}$. It is well-known that $f$ has radial limits almost everywhere on the unit circle $\mathbb{T}$. Let ...
1
vote
0answers
55 views

Are morphisms of intersection graphs of circle packings harmonic?

Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...
0
votes
0answers
97 views

Boundary behavior of Harmonic functions

Assume that $f$ is harmonic in the unit disk $|z|<1$, with boundary function of bounded variation, such that $$\lim_{r\to 1}f(re^{it})= 0$$ for $t\in[0,\pi]\setminus \mathbf{Q}$, where $\mathbf{Q}$ ...
2
votes
2answers
348 views

Bolza curve admits no anticonformal fixedpointfree involution

The Bolza curve B double covers the Riemann sphere with branching at the vertices of a regular octahedron. An affine model is given by the locus of $y^2=x^5-x$. How does one show that B does not ...
5
votes
1answer
321 views

Branched Regular Cover over 4-times punctured sphere

This is probably trivial but has been bothering me all day. Suppose $f:\Sigma_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma_g$ a connected genus $g$ surface and $f$ having ...
3
votes
1answer
253 views

The right conformal map to make a certain picture

This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin: Fitting a mesh to a density function I am trying to come up with a way to make a picture of an ...
8
votes
1answer
499 views

Polynomial with all zeros on a circle and many real coefficients

On a circle (or a line) $\Omega$ in the complex plane that is not symmetric w.r.t. the real axis, choose $n\ge5$ distinct points $z_1,...,z_n$ and consider the polynomial ...
3
votes
0answers
131 views

Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane. We call $$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$ the dispersion function of $g$. Now, one can ...
0
votes
1answer
102 views

Counting complex solutions on a disk.

I am looking for an example of an analytic map $f$ on the complex domain for which there exist $r>0$ such that for $\gamma=\{z:|z|=r\}$, a.s. for $\theta \in [0,2\pi)$ we have that ...
0
votes
1answer
188 views

Asymptotic behavior of entire functions

Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{-x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{-x}/x \right)/f \left(x \right) \rightarrow 1$.
4
votes
0answers
204 views

What is the spectrum of a ring of holomorphic rational power series?

Let $R_\infty$ be the ring of power series in a single variable with rational coefficients that converge in the whole complex plane. Let $R_\rho$ be the subring of $R_\infty$ that defines holomorphic ...
2
votes
2answers
552 views

The integral inequality

Let $f$ be an entire function of exponential type. Does the inequality $|f(a)| \le C \int_{a-1/2}^{a+1/2}|f(x)|\,dx$ hold for every $a \in R$ with an absolute constant $C$? At most, the constant ...
8
votes
0answers
216 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 ...
6
votes
0answers
174 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 ...
1
vote
1answer
201 views

How to solve a linear algebraic complex equation in one function evaluated at different arguments?

Hello, I am trying to solve an equation of the form $C_1 f(k_1 z) + C_2 f(k_2 z) + C_3 f(k_3 z) + C_4 f(k_4 z) = C_5 z^2$ for $f(z)$. Everything is complex. The $C_i$'s and $k_i$'s depend on some ...
1
vote
0answers
196 views

bivariate polynomial

Hello, Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex. If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...
10
votes
4answers
884 views

Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature): Let $f$ be a holomorphic function mapping the ...
3
votes
0answers
101 views

How did Nochka find weights in his proof of Cartan's conjecture?

Good evening, I have just read Nochka's proof of Cartan's conjecture (Second Main Theorem of Nevanlinna Theory for linearly degenerate meromorphic curves in $\mathbb{CP}^n$). To prove the conjecture, ...
5
votes
2answers
409 views

Another proof of the bidisc and the ball are biholomorphically inequivalent?

Does this outline of a proof work? Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...
1
vote
0answers
107 views

Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?

Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good ...
9
votes
1answer
362 views

When does continuity imply holomorphy?

I was studying the construction of the modular lambda function and I started thinking about the following question. Suppose that $\Omega\subset \mathbb{C}$ is an open connected set and $f:\Omega\to ...
0
votes
1answer
356 views

The real and imaginary parts of the Incomplete Gamma function for second argument being purely imaginary

Dear all, I am looking for explicit (at least more explicit than the original expression) for 1) Re$(\Gamma(a, i\omega))$ as well as 2) Im$(\Gamma(a, i\omega)),$ where i Re and Im denote the real ...
1
vote
1answer
148 views

Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

Given a complex algebraic, affine variety $X$ and a subvariety $V$, let $I_V$ indicate the ideal of polynomials on $X$ with vanish on $V$. Given an open subset $U$ of $X$, is it true that the ideal ...
3
votes
1answer
455 views

Complete intersections in complex and algebraic geometry

I'm wondering why (and therefore also if) the notions of "a projective variety/submanifold of projective space is a complete intersection" as used in algebraic geometry and the theory of, say, Riemann ...
0
votes
2answers
136 views

Cohomology of Complements by an analytic subset?

Good moring, Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? ...
15
votes
2answers
1k views

On the Universality of the Riemann zeta-function

Hi, I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference. First, recall Voronin's remarkable theorem ...
12
votes
1answer
453 views

Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...
2
votes
1answer
121 views

on the density of hypersurfaces in complex projective spaces

Good morning, Let $H$ be a hypersurface in a complex projective space $\mathbb{CP}^N.$ Let $d$ be the distance de Fubini-Study on $\mathbb{CP}^N.$ Let $x = [x_0: \ldots :x_N]$ and ...
11
votes
2answers
532 views

Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...
2
votes
2answers
251 views

Factorization of antisymmetric bounded holomorphic functions

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
6
votes
2answers
444 views

A question about $L^p$ integral of an entire function on $\mathbb{C}$

Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1)$. I guess that $f\equiv 0$ ...
2
votes
1answer
141 views

The distribution of roots of elliptic polynomial

If $p(x)$ is an $n$ variables polynomial of even degree with complex coefficients which satisfies the strong elliptic condition, that is, Re$p(x) \ge C|x|^{2m}$ for any $x \in \mathbb R^n$ where $2m$ ...
5
votes
0answers
199 views

Automorphisms of Compact Riemann Surfaces

I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has for the Jacobian $J(C)$ of the curve $C$: $$ Aut (J(C))\sim Aut C$$ when $C$ is hyperelliptic and ...
11
votes
2answers
318 views

Effective vanishing of the Schwarzian Derivative

Recall for any complex analytic function $f:\mathbb{D}\to \mathbb{C}$ the Schwarzian derivative of $f$ is $$ S(f)=\frac{f'''}{f'}-\frac{3}{2} \left( \frac{f''}{f'}\right)^2. $$ It's well known that ...