Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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27
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3k views

A paper to the question, if the six dimensional sphere is a complex manifold

Hi, for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf Because I am not able to ...
23
votes
0answers
978 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
21
votes
0answers
410 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
20
votes
0answers
955 views

Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by $$\gamma(K) := \sup |f'(\infty)|$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
17
votes
0answers
415 views

Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
13
votes
0answers
735 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
12
votes
0answers
292 views

Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
12
votes
0answers
249 views

Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? ...
12
votes
0answers
288 views

Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of ...
11
votes
0answers
248 views

Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$ where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...
11
votes
0answers
328 views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
11
votes
0answers
866 views

Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb ...
9
votes
0answers
143 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 ...
9
votes
0answers
625 views

Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ...
9
votes
0answers
1k views

Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of $1$ complex variable, one ...
9
votes
0answers
404 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
8
votes
0answers
315 views

rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index. Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index. Let $M_0(\Gamma_i)$ denote the ring of modular ...
8
votes
0answers
150 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
8
votes
0answers
227 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...
8
votes
0answers
743 views

Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: ...
8
votes
0answers
236 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
0answers
354 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
7
votes
0answers
70 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} ...
7
votes
0answers
175 views

Complex structures on $\Bbb R^4$

Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...
7
votes
0answers
93 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 $P$, ...
7
votes
0answers
350 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
0answers
401 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!} + ...
7
votes
0answers
809 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
0answers
154 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 is ...
6
votes
0answers
180 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
0answers
254 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 ...
6
votes
0answers
190 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
0answers
172 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
6
votes
0answers
186 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 ...
6
votes
0answers
252 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
0answers
153 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 ...
5
votes
0answers
84 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 ...
5
votes
0answers
64 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 a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...
5
votes
0answers
192 views

Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
5
votes
0answers
118 views

Density of rational functions in open Stein

I repost here, after I tried here. Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
5
votes
0answers
187 views

proper mapping between Stein manifolds

My question is the following: Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set ...
5
votes
0answers
173 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 ...
5
votes
0answers
219 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 ...
5
votes
0answers
144 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 ...
5
votes
0answers
176 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 ...
5
votes
0answers
379 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 ...
5
votes
0answers
681 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 ...
5
votes
0answers
582 views

some questions about properties of harmonic measure

The original post The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...
5
votes
0answers
410 views

Lacunar series with an interesting (in-formula) symmetry.

So, I wrote out a table of functions like so: $\sum_{n=1}^{\infty} (-1)^{n+1}q^{n}=$ $+q^{1}$ $-q^{2}$ $+q^{3}$ $-q^{4}$ $+q^{5}$ + $\ldots$ $\sum_{n=1}^{\infty} (-1)^{n}q^{n^{2}}=$ $-q^{1}$ ...
5
votes
0answers
449 views

Two meromorphic functions with overlapping sets of poles

Assume that we have two meromorphic functions $f(z,w)$ and $g(z,w)$, where $z$ and $w$ are complex (we are interested only in behavior on compact sets). Fix $z$ and assume that the sets of poles of ...