# Tagged Questions

1answer
31 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 ...
2answers
59 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 ...
1answer
23 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 $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
0answers
51 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 ...
1answer
25 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 $f^{-1}$ ...
1answer
56 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 ...
0answers
18 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 ...
1answer
32 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 ...
0answers
24 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 ...
0answers
16 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 ...
2answers
42 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 ...
0answers
23 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 ...
1answer
26 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 ...
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. ...
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 ...
0answers
64 views

0answers
48 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 ...
2answers
475 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 ...
1answer
75 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 ...
2answers
63 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.
1answer
124 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 ...
1answer
53 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 ...
1answer
48 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 ...
0answers
71 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 $g_1(z)=g(z)$?
0answers
80 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 ...
0answers
87 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 ...
1answer
74 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 ...
1answer
132 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?
1answer
269 views

### The sum of the residues of a meromorphic function on a Riemann surface

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $\Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.
1answer
53 views

### complex vector fields - hard d vs. soft d?

I believe this is a computation I have done before, but now I can't write the symbols to convince myself: What is the connection between the "hard" complex differential operator d/dz and the "soft" ...
0answers
72 views

### Can every complex space be covered by a finite number of Stein spaces?

Can every complex space be covered by a finite number of Stein spaces?
0answers
61 views

### Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
2answers
67 views

### Help with unknown notation

In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere): "writing $z=x+iy$, we can verify that: $$x:y:-1=x_1:x_2:x_3-1,$$ and this means that the points ...