Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cv.complex-variables]

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

5
votes
3answers
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
2answers
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
4answers
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
3answers
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
5answers
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
2answers
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
5answers
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
0answers
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
1answer
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
4answers
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
3answers
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
36answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
2answers
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
2answers
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
2answers
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
7answers
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
1answer
2k views

When are entire functions surjective?

Is there some useful criterion to determine whether or not an entire function is surjective?
19
votes
0answers
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
3answers
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
5answers
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
2answers
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
2answers
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
3answers
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
1answer
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
4answers
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, ...