# Questions tagged [cv.complex-variables]

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

**5**

votes

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688 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 &...

**2**

votes

**2**answers

317 views

### Coefficients of holomorphic functions defined by Borel probability measures on the unit disc

Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\...

**4**

votes

**4**answers

2k views

### Minimizing the modulus of a polynomial around a circle

I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...

**4**

votes

**3**answers

1k views

### Coprimality and squarefree numbers

As observed on Mathworld, "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of ...

**24**

votes

**5**answers

2k views

### Continuous + holomorphic on a dense open => holomorphic?

Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$.
Let ...

**1**

vote

**2**answers

524 views

### Weierstrass Theorem [closed]

Hi--
Where can I find a proof of this theorem: For each $r \in \mathbb{Z}_{+}$,
there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but
$f(r+1)=f(r+2)=\cdots =0$,
i.e. $f(z) \in ...

**16**

votes

**5**answers

1k views

### What is the spectrum of the ring of entire functions?

Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$.
...

**10**

votes

**0**answers

479 views

### Adeles of Holomorphic Functions

In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...

**0**

votes

**1**answer

571 views

### Fiberwise torsion free and generically null sheaf for flat morphism

Hi.
Has some one an example of sheaf $A$ on flat morphism $f:X\rightarrow S$ of reduced complex spaces with fibers of constant positive dimension (or locally noetherian excellent schemes without ...

**15**

votes

**3**answers

1k views

### What is a reasonable finitary analogue of the statement that harmonic functions are smooth?

In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way ...

**3**

votes

**1**answer

559 views

### Kahler forms on Cohen Macaulay spaces

Hi.
Can anyone answer the two following questions:
For $n$-dimensional $X$ Cohen-Macaulay complex space, is it true that the sheaf of top degree homolorphic forms $\Omega^{n}_{X}$ has no ...

**6**

votes

**1**answer

889 views

### Simply-connected domain around a curve

In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this ...

**7**

votes

**0**answers

174 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 ...

**17**

votes

**1**answer

803 views

### Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not $\...

**13**

votes

**4**answers

1k views

### “Simple” Kahler manifolds

I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...

**36**

votes

**3**answers

5k views

### On linear independence of exponentials

Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...

**9**

votes

**3**answers

4k views

### Functions of several complex variables: book recommendations?

Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...

**15**

votes

**1**answer

2k views

### Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...

**0**

votes

**1**answer

3k views

### Is there an Isomorphism between R and C under addition? [duplicate]

Possible Duplicate:
AC in group isomorphism between R and R^2
Somewhere, I recall being told that there is an isomorphism between $\mathbb{R}$ and $\mathbb{C}$ under addition. However, despite a ...

**11**

votes

**1**answer

1k views

### Dirichlet series expansion of an analytic function

Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...

**0**

votes

**0**answers

491 views

### Motivation of proof of Riemann-Roch for elliptic curve and generalizations

Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...

**0**

votes

**1**answer

1k views

### pure dimensional and embedded components

Hi.
Let $X$ be a pure $n$-dimensional complex subspace of manifold $Z$. It is true that $X$ has no embedded components? (perhaps that is obvious with Weierstrass preparation theorem or Noether ...

**1**

vote

**1**answer

512 views

### Points at twice the distance from (-1, 0) that they are from (1, 0) in hyperbolic geometry [closed]

In answer to the question Demystifying complex numbers, Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex numbers....

**1**

vote

**1**answer

174 views

### real-valued functions on the modular surface

How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over ...

**70**

votes

**36**answers

15k views

### Demystifying complex numbers

At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...

**1**

vote

**1**answer

1k views

### Duality and isomorphism of functor

Hi.
First of all thanks to Zsolt for the answer to the question on "Cohen Macaulay morphism".
I want to show for which proper and flat morphisms $f:X\rightarrow S$ of complex spaces with $n$ pure ...

**1**

vote

**1**answer

380 views

### Cohen macaulay morphism

Hi.
I have a doubt about this fact:
Let f:XS be a flat, proper and surjective morphism of complex spaces (or locally noetherian, excellent schemes) with n-pure dimensional fibers. Then f is Cohen-...

**33**

votes

**2**answers

2k views

### Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \...

**4**

votes

**1**answer

922 views

### torsion freeness of tensor product continued

Hi.
Question 1: If $f:A\rightarrow B$ be a morphism of local noetherian rings with $B$ is $A$-flat. Let $M$ (resp. $N$) be a $B$ (resp. $A$-)-module of finite type (fin. generated). We assume that $...

**3**

votes

**0**answers

341 views

### torsion freeness of tensor product

Hi.
Let $f:A\rightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)\geq 2$ and $N$ is torsion free.
Then it is ...

**1**

vote

**0**answers

956 views

### Trigonometric identities and (several?) complex variables

I don't know anything about several complex variables nor whether that topic will answer my questions below, but in one complex variable one learns that since $\sin x$ and $\cos x$ are entire ...

**1**

vote

**2**answers

517 views

### Extension of harmonic function at infinity

Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property ...

**4**

votes

**0**answers

655 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 ...

**6**

votes

**1**answer

1k views

### Approximation by analytic functions

Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...

**4**

votes

**1**answer

444 views

### Is the following a sufficient condition for flatness?

Hi.
Let $f\rightarrow S$ be an open morphism of reduced finite dimensional complex spaces (or a universally open morphism of locally noetherian excellents without embedded components or reduced ...

**2**

votes

**1**answer

681 views

### finite tor dimension

Hi. Can, every one, give me an example of finite surjective morphism of finite tor dimension (but not flat!) between reduced schemes or complex analytic spaces... Thank you.

**10**

votes

**2**answers

2k views

### Picard-Fuchs equations for modular functions

Hello, MathOverflow community!
Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil ...

**14**

votes

**2**answers

722 views

### Highly connected, compact complex manifolds

Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:
If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...

**2**

votes

**2**answers

547 views

### L^2 space of holomorphic functions with given weight

Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product
$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^...

**6**

votes

**2**answers

2k views

### Relative canonical sheaf

Hi.
I want to know if for $f:X\to S$ a proper flat holomorphic map with n-dimensionnal fibers over reduced complex space S, the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing ...

**31**

votes

**7**answers

7k views

### Interpreting the Famous Five equation [closed]

$$e^{\pi i} + 1 = 0$$
I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us?
Best that I can figure out is that it just ...

**9**

votes

**1**answer

2k views

### When are entire functions surjective?

Is there some useful criterion to determine whether or not an entire function is surjective?

**19**

votes

**0**answers

530 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 ...

**3**

votes

**3**answers

724 views

### Analytic ODE with complex time

Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...

**12**

votes

**5**answers

5k views

### References for complex analytic geometry?

I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc. A ...

**4**

votes

**2**answers

509 views

### The link of a singular quintic hypersurface in CP^4

Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...

**15**

votes

**2**answers

870 views

### Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with $...

**11**

votes

**3**answers

1k views

### Analytic continuation of holomorphic functions

Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, ...

**-4**

votes

**1**answer

477 views

### Meaning of the Mobius transformations video [closed]

What is this video trying to tell us?
http://www.youtube.com/watch?v=JX3VmDgiFnY
The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic ...

**23**

votes

**4**answers

3k views

### Genealogy of the Lagrange inversion theorem

A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...