1
vote
1answer
24 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
0
votes
0answers
16 views

If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
0
votes
1answer
42 views

Question about a paragraph in the book complex analysis by Ahlfors.

By $C_1$, we denote family of circles passing through $a,b$ and by $C_2$ we denote family of Appolonius circles with limit point $a,b$. In section $3.5$ entitled Families of circles, in one paragraph ...
0
votes
0answers
33 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
0
votes
0answers
26 views

is this definition of meromorphic function missing a requirement?

Looking at Definition 1.1.33 in the book Complex geometry by Huybrechts: Def. Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere ...
1
vote
1answer
40 views

complement of zero set of holomorphic function is connected

I'm stuck with the following part of exercise 1.1.8 in Hubrechts book Complex geometry: Prove that, if $U \subset \mathbb C^n$ is open connected, then $U \setminus Z(f)$, the complement of zero set ...
5
votes
2answers
65 views

Physical or geometric meaning of complex derivative?

As in, the real derivative of a function at a point is a slope of a function at that point. What is the physical or geometric meaning of complex derivative of a function at a point? Any help is ...
0
votes
1answer
30 views

truncate power series to approximate holomorphic function by polynomial

Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
0
votes
0answers
65 views

Zero moment of arc length measure

Suppose $\gamma$ is a simple smooth closed curve and is not a circle. Does there exist a monomial $z^n$ so that $\int_{\gamma}z^n ds(z)=0$ for some positive integer $n$? (In here, $ds$ is the arc ...
0
votes
1answer
28 views

Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
1
vote
1answer
60 views

Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
1
vote
0answers
23 views

Does the conformal class of a complex projective curve contain the Fubini-Study metric?

Let $X \subset \mathbb CP^2$ be a complex curve with metric $g$ induced by the Fubini-Study metric on $\mathbb CP^2$. Since in the case of two-dimensional real manifolds a complex structure is ...
0
votes
1answer
38 views

Intuition for the limit of complex functions

We have intuition and somehow geometrical point of view about the limit in the Real functions.I mean we can think of the limit on the graph of the Real function and imagine how close we can get on the ...
3
votes
0answers
26 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
0
votes
0answers
17 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
2
votes
2answers
43 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
1
vote
0answers
27 views

Complex structures on punctured disks.

Let $X$ be a smooth surface diffeomorphic to the punctured unit disk $\{(x,y)\in \mathbb{R}^2 \ | \ 0<x^2+y^2<1\}$ in the plane. It admits a lot of non equivalent complex structures, for example ...
0
votes
1answer
27 views

Isometries of the plane and fixed lines

I am given that for all reflections $g$ there are infinitely many lines $L$ satisfying $g(L) = L$ which makes perfect sense (just take lines perpendicular to the axis of reflection). I am asked to ...
0
votes
0answers
19 views

External map of a polynomial like mappings

I'm having trouble understanding the method used in the following paper on page 297 in the case that the filled Julia set is not connected to construct an external map. ...
1
vote
0answers
35 views

Explicit Weierstrass Preparation Theorem decomposition

Consider the function $f:\mathbb{C}^2\to\mathbb{C}$ given by $(z_1,z_2)\to z_1^3z_2+z_1z_2+z_1^2z_2^2+z_2^2+z_1z_2^3$. Find an explicit decomposition $f=h\cdot g_w$ as per the WPT, i.e. $h$ is ...
0
votes
0answers
72 views

Univalent functions whitch are not starlike or convex

$S$ denotes the univalent function class, $S=\{f \in \mathcal{A}:f \in H_{u}(\mathrm{U})\}$. $S^{\ast}=\left\{f\in\mathcal{A}:\operatorname{Re}\frac{zf^{\prime}(z)}{f(z)}>0, ...
1
vote
0answers
57 views

What will be the integration region?

Where $\Omega_s$ is new integration region, due to change in geometry integration region will also change. Also note that $\Omega_l$ is $\Omega$ with $|x|<l$ is the integration region for ...
1
vote
1answer
88 views

Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
5
votes
2answers
121 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
3
votes
1answer
28 views

Is every analytic hypersurface in $\mathbb{C}^n$ cut out by one holomorphic function?

Is every analytic hypersurface in $\mathbb{C}^n$ cut out by one holomorphic function?
0
votes
0answers
29 views

Complex Analysis Questions Compilation

All of my questions are in relation to Gamelin's Complex Analysis. How was the parametric form of the line from the North Pole on the unit sphere through a point P come to be? It is $$ N + t (P-N) ...
1
vote
3answers
86 views

A question on differential topology

Let $\mathbb{C}P(1)$ denote the complex projective line. I am attempting to show that there does not exist a nonzero holomorphic differential $1$-form on $\mathbb{C}P(1)$. My intuition is as ...
10
votes
1answer
169 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
1
vote
0answers
28 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
2
votes
0answers
34 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
0
votes
0answers
38 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
6
votes
1answer
99 views

Level sets of holomorphic functions

It is a somewhat well known fact that any closed set (say in the plane) can be realized as the level set of a smooth ($C^\infty$) function, so level sets of smooth functions are as general as they can ...
1
vote
0answers
17 views

Maximal immerged disk on a hermitian Riemann surface

Let $S$ be a Riemann surface equipped with a hyperbolic metric, and $z_0 \in S$. Is there a way to estimate the maximal radius $r>0$ such that there is a holomorphic embedding $\psi : \Delta_r ...
3
votes
2answers
61 views

Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
1
vote
0answers
19 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
vote
2answers
74 views

A subset of $\mathbb C\times\mathbb C$

I'm trying to think if the space $\{(z,\,i\overline{z})\,:\,z\in\mathbb{C}\}$, where $\overline{z}$ is the complex conjugate of $z$ and $i$ is the imaginary number, is topologically equivalent to ...
1
vote
0answers
34 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
3
votes
2answers
156 views

Continuous complex functions

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq ...
0
votes
0answers
50 views

Question on zero locus of holomorphic function.

I am trying to figure out a statement in Girffiths and Harris' book "Principles of Algebraic Geometry: Given $f:U\rightarrow V$ a holomorphic map of open sets in $\mathbb{C}^{n}$, and let ...
1
vote
2answers
531 views

Show that the function $f(z)=1/z$ transforms a circle centered at the origin in the $xy$ plane in a circle centered at the origin in the $uv$ plane

Show that the function $f (z) = 1 / z$ transforms a circle centered at the origin in the $xy$ plane in a circle centered at the origin in the $uv$ plane. Someone can give me a clue to start the ...
1
vote
1answer
85 views

Singular points of an analytic variety

I am struggling with a question that was posed here before, about why a reducible analytic variety $V=V_1\cup V_2$ must be singular in $V_1\cap V_2$. I must say I didn't really figure out the ...
2
votes
2answers
67 views

Residue of a complex function at some pole.

How can one visualize residue of a complex valued function at some given pole? I know how to find it. but I want to know its significance and its geometric nature. Why do we study it? thank you.
0
votes
1answer
127 views

Mapping unit disc onto upper half plane

How can I map the unit disc onto the upper half plane? I tried mapping $(1,i,-1)\rightarrow(1,0,\infty)$ using cross-ratio: $z\rightarrow \frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, but didn't give ...
1
vote
1answer
54 views

Finding local normal form of a holomorphic function

So I'm trying to find local coordinates to compute the local normal form of a holomorphic function. I have $f : \mathbb{P}^1 \to \mathbb{P}^1$ given by $f(z) = \frac{z}{(z-1)^2}$. Now we have a nice ...
4
votes
1answer
49 views

Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
0
votes
0answers
73 views

$\bar{\partial}$-Poincare' lemma in one variable

my question regards the proof, as it is described in Principles of Algebraic Geometry by Griffiths and Harris, on pages 5-6: why does last equality in last equation hold true, namely $g_1(z)=g(z)$?
3
votes
0answers
82 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
1
vote
0answers
89 views

The Wirtinger theorem proof

Let $M$ be a complex hermitian manifold with symplectic form $\Omega$ and $N \subseteq M$ be its smooth $2n$-dimensional (real dimensions, $2n \geq 2$) compact submanifold. I want to show the ...
3
votes
1answer
75 views

Show that a complex expression is smaller than one

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains ...
4
votes
1answer
136 views

Why holomorphic injection on $C^n$must be biholomorphic?

This result is certainly right in the 1-dim'l case. But I don't know how to show the general case by induction. Can anyone tell me the detail please?