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

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6
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0answers
55 views

Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
-2
votes
1answer
153 views

Degree of a rational function [on hold]

I would like to have a simple proof for the following result: Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
1
vote
1answer
28 views

About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
8
votes
2answers
242 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of ...
3
votes
0answers
84 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
1
vote
0answers
133 views

About real roots of complex multivariable polynomials

Let $f(z,w_1,w_2,..,w_n)$ be a multivarible complex polynomial mapping $\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ and it has all real coefficients. Assume that this is "real-stable" i.e it has no roots ...
5
votes
1answer
360 views

Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
16
votes
1answer
474 views
+50

Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

There is a 50 point bounty on this question. Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define $$\pi_k(x)=\#\{n\leq x: ...
0
votes
0answers
73 views

Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
2
votes
0answers
79 views

Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...
2
votes
1answer
224 views

Triviality of holomorphic vector bundles over contractible Stein manifolds

If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
-1
votes
0answers
49 views

Other Ways for Riemann Zeta Analytic Continuation [migrated]

A well-known way for analytic continuing riemann zeta function is using from the functional equation between $\zeta$, $\theta$ and $\Gamma$ function. but I know that there is or there are other ways ...
1
vote
0answers
49 views

Define an entire function with zeros in a given set [closed]

how to define an entire function that it's zeros are from a given set. for example, define an entire function that it's zeros are prime numbers on real axis.
0
votes
0answers
134 views

Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
1
vote
1answer
81 views

Lifting quadratic forms on the cotangent bundle to higher level forms

Backround In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates. If $\alpha$ is a $(p,q+1)$ form on a domain ...
0
votes
0answers
45 views

Maximum modulus principle and restriction

Suppose I have a complex multivariate polynomial $f\in \mathbb{C}[z_1,\ldots,z_n]$ and a connected, bounded open set $U$. The maximum modulus principle says $f$ achieves its maximum value on the ...
5
votes
1answer
165 views

Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully). Suppose $g$ is holomorphic on ...
1
vote
1answer
116 views

How to find isothermal coordinates equivalent to circles in far limit?

I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...
3
votes
0answers
91 views

Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild". For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
5
votes
1answer
217 views

Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?

Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f_n$ is a sequence of holomorphic functions converging pointwise to 0 on ...
0
votes
0answers
20 views

A bound in Schramm-Loewner evolutions

Recently, I read a paper in SLE that claims the following (which I do not really understand why): Recall the differentiability estimate which says that $$|g_K(z) - z- \frac{\text{hcap}(K)}{z}| ...
3
votes
0answers
126 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: ...
2
votes
0answers
103 views

Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define $$f_{\mathrm{symm}}(x_1,\ldots,x_n) = 2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1} ...
0
votes
1answer
89 views

holomorphic continuation

consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$. This function can be continued holomorphically for all complex numbers with ...
1
vote
1answer
130 views

Sequence of smooth maps converging to the identity [closed]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
1
vote
1answer
67 views

Finding Laurent Series of a function [closed]

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
0
votes
0answers
103 views

Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...
10
votes
1answer
208 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
2
votes
1answer
122 views

Flatness of a morphism of complex analytic spaces

Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced. ...
3
votes
1answer
147 views

A weak analytic version of the valuative criterion of properness

EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that (a) $f$ is injective on points; (b) $f$ is local imbedding near each point $x\in ...
1
vote
0answers
74 views

Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
2
votes
1answer
144 views

polynomial inequality in complex variable (generalized)

Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number. What happens if ...
0
votes
1answer
154 views

existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
0
votes
0answers
50 views

Analytic functions space on Riemann surface

I have some questions about the analytic function space on Riemann surface and distinguished varieties: Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...
7
votes
1answer
169 views

Analytic diffeomorphisms of the circle from complex domains

Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms: $$\phi : (D^2,S^1) \to ...
1
vote
0answers
171 views

Estimating the kernel of Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for complex $z$

Let $f(z,a)$ be an analytic function on $C^+=\{\Re z>0\}$ for each fixed $a>0$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
2
votes
0answers
72 views

Is logarithmic convexity of the heat kernel with complex time a general fact?

Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
7
votes
0answers
140 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 ...
2
votes
1answer
121 views

Complex function for mapping a circle to a superellipse

I was wondering if anyone knows an analytic complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated! Thanks, Kayvan
3
votes
0answers
97 views

Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1) \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space $$ Z_n^d(\eta, ...
4
votes
2answers
241 views

Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps. I think it is interesting to ask whether we ...
0
votes
0answers
44 views

on the existence of holomorphic coordinates under bounded curvature

Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...
2
votes
0answers
69 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
1
vote
1answer
65 views

An optimization problem in complex space

Consider the following optimization problem $$ \min \| \textbf{Ax-B}\| $$ $$ s.t.|x_i|=1,i=1,...,n $$ where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...
7
votes
1answer
148 views

Conformal map of polygon with circle segments

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a ...
13
votes
3answers
397 views

Entire function bounded at every line

I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
0
votes
0answers
65 views

kostant partition function vs Haar measure

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant: $$ \Delta(\theta) = \prod_{i< j} ...
4
votes
2answers
124 views

Bound areas of disks with respect to a quadratic differential

Let $q = q(z)\,dz^2$ be a quadratic differential on the unit disk $D(1)$, normalized so that $\int_{D(1)}|q| = 1$. I have a fairly convoluted argument that for any smaller disk $D(r)$ with $r < 1$, ...
4
votes
1answer
109 views

Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
8
votes
0answers
90 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 ...