Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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Let $ f(w)=\frac{w(1-i)-(i-1)}{w-1} $, where $w$ is the left hand plane. What is the image of this map?

Let $$ f(w)=\frac{w(1-i)-(i-1)}{w-1} $$, where $w$ is the left hand plane. What is the image of this map? The answer should be $|z|^2<2$ if I did everything before correctly. This is a show ...
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24 views

Solving inequality in complex plane

I have to graphically represent the following subset in the complex plane being z a complex number: $A={1<|z|<2}$ However after trying to do it on WolframAlpha it says that "inequalities are ...
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90 views

How to interpret the cotangent bundle of a complex manifold?

Let $X$ be a complex manifold. I am not sure what people mean when they talk about the cotangent bundle $T^*X$ of $X$. I have two interpretations: At each point $x\in X$, $T_x^*X$ is the complex ...
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46 views

Is the complementary of a 2 codimensional set in a complex space simply connected?

It is well-known that if X is a complex manifold and $A \subset X$ an analytic set of codimension at least $2$ then $\pi_1(X)=\pi_1(X\setminus A)$. Can we generalize this equality when $X$ is a ...
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35 views

Verlinde Formula in geometric quantization?

I think I have a fair grasp on the $\rm{SU}(2)$ Verlinde Formula from the algebraic geometry perspective. I'm hoping to understand better how exactly this relates to the geometric quantization of a ...
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22 views

Relation between moduli space of flat connections and moduli space of bundles on curves

I would like to let $X$ be a genus $g$ curve, with $M_{g}(r,d)$ the moduli space of bundles on the curve such that $(r,d)=1$. Alternatively, we can pick a group $G$, consider principal $G$-bundles $...
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39 views

Dimension of Moduli Space of Bundles on Curves

I think I'm getting conflicting results for the dimension of the moduli space of rank $r$, degree $d$ stable vector bundles on a curve $X$ of genus $g$. I'm happy to look only at the nice case of $r$ ...
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58 views

Topological Degree of Map of Effective Divisors

Let $\Sigma$ be a compact Riemann surface. Is it possible to show that the map $$f:\text{Div}(\Sigma)^d_+\to \text{Div}(\Sigma)^{qd}_+$$ Given by $\sum_{i} n_ix_i\mapsto \sum_{i} qn_ix_i$, has ...
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83 views

Can we prove uniformization by solving the Yamabe problem directly?

One version of the uniformization theorem says that a simply connected complex manifold is biholomorphic to either the unit disc, $\Bbb C$, or $\Bbb{CP}^1$. The proof of this goes through potential ...
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15 views

Given two hermitian matrices of signature (2,1) there exists a Cayley transform between them?

Given a matrix $A\in M_{k\times l}(\mathbb{C})$ we define the hermitian transpose of $A$ as the matrix $A^*=\overline{A}^t\in M_{l\times k}(\mathbb{C})$. We say a matrix $H\in M_k(\mathbb{C})$ is ...
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8 views

Proper surjective holomorphic function is a topological bundle

If I have $u:M\rightarrow D\setminus 0$, $M\subseteq (\mathbb C^2,0)$ compact, $\mathbb C^2 = \mathbb C \times \mathbb C$ and $D$ the Poincare disk, $u$ is holomorphic, surjective, proper and every ...
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30 views

Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
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47 views

Volume of a hypersurface in flat families?

Suppose that $X_t$ is the vanishing of $x^2 + y^2 + t z^2 = 0$ in $\mathbb{CP}^2$. For $t \not = 0$, this is a submanifold, and it inherits a Riemannian metric from the Fubini-Studi metric on $\mathbb{...
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42 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
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46 views

Question about the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$ in Principles of Algebraic Geometry by Griffiths and Harris.

On page $115$ of Principles of Algebraic Geometry by Griffiths and Harris, in the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$, they state that $$\sqrt{-1}(\partial\bar{\...
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49 views

Intuition behind definition of Stable Bundles?

To my understanding, given a rank $r$ and degree $d$, we can fix a $C^{\infty}$ vector bundle $\mathcal{V}$ over some curve $\Sigma$ of genus $g$. We then want to study the moduli space $M_{g}(r,d)$ ...
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29 views

Mention the complex number with given condition on the place

How do i mention the set of complex numbers on the plane that satisfy the condition $$\arg(iz-1)= \frac \pi 3$$ I tried to assign $u=iz-1 $ then $\arg(u)= \pi/3$, but I don't know how to continue ...
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30 views

How does an almost complex structure on a manifold induce an orientation?

I have read that given a smooth even dimensional manifold $M$ with an almost complex structure $J$, then $M$ is orientable and there is a canonical choice of orientation. Why is this the case? How ...
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1answer
47 views

The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
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Tangent Space to Moduli Space of Vector Bundles on Curve

Let $X$ be a curve of genus $g \geq 2$. Using Geometric Invariant Theory, we can construct a moduli space $\mathcal{M}(r,d)$ of vector bundles on $X$ of rank $r$ and degree $d$. The details of this ...
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17 views

Degree of line bundles with a non-trivial morphism between them

We assume that $X$ is a K\"ahler manifold. $L_1$ is a holomorphic line bundle and $F$ is a subsheaf of rank $1$. If there is a non-trivial morphism $F\to L_1$, then $\operatorname{deg}F\leq \...
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23 views

Sketching image of paths in $\mathbb{C}$

In each of the following describe the image of $f \circ \gamma$ for all $t$ in the indicated range. Given $$\gamma (t) = it : -1 \le t \le 1 $$ and $$ f(z) = \frac{z + 1}{z - 1}$$ It's clear to me ...
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67 views

Serre duality explicitly on curves

Consider a Riemann surface $X$, with genus and marking so that a suitable moduli space exists. It's a well-known fact that the tangent space to that moduli space at $X$ (in other words, the space of ...
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25 views

What are the relations between complex numbers and visual representation on $\mathbb{R^2}$?

In a second order polynomial function, the discriminant say how many solution there are e what are their type. Set the discriminant $> 0$, there are two real solutions, which are the points where ...
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33 views

Geometric genus of a surface, Exercise 21.5 D in Ravi Vakil's book

I get stuck in Exercise 21.5 D in Ravi Vakil's book, Suppose $Z$ is a regular degree $d$ surface in $\mathbb{P}^3_{\bar{k}}$, compute the geometric genus of $Z$. The geometric genus is defined to be ...
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Is infinity allowed in the definition of a fractional linear map?

I need to create a fractional linear map of the form $$ F(z) = \frac {az+b}{cz+d}, $$ where a,b,c, and d are complex numbers such that $ad-bc \neq 0$ and $F(0) = 1$, $F(1)=\infty$, and $F(\infty)=0.$ ...
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36 views

Degree of the Divisor of a Theta Function

Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
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52 views

Surface in complex projective space

Let $\mathbb CP^3$ be projective space. Consider polynomial $f$ of degree $k$ satisfying $f(a \mathbb v)=a^k f(\mathbb v)$ for complex number $a$ and vector $v \in \mathbb C^4$. For example $f=xy+z^2-...
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How many complex structures are on $\mathbb{R}^{2n}$

By a complex structure on $\mathbb{R}^{2n} $ I mean $\mathbb{R}$-linear $J:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ such that $JJ=-I$. I asked myself how many nonisomorphic such structures are on $...
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50 views

A Complex Torus is generally simple!

Lets parametrize the set of lattices inside $\mathbb C^g$ with the open dense subset $U= GL_{2g}(\mathbb R)$ of $\mathbb R^{4g^2}$. Show that there exists a coutable family $(Z_n)_{n \in \mathbb ...
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How do I go from $\check{C}^1(\mathscr{U}, \mathcal{F})$ to $H^1(\mathbb{P}^1, \mathcal{F})$?

Let's say I have $\mathcal{F}= \mathcal{O}(a)\oplus \mathcal{O}(b)$ for concreteness. Then what does a closed element of $\check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))$ look like as an ...
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117 views

$S^1$ cannot be the zero locus of polynomials: is this argument correct?

Suppose I want to rigorously prove that the unit circle $S^1 \subseteq \mathbb C$ can't be the zero locus of any polynomials. Can one just observe that polynomials with domain $\mathbb C$ are ...
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111 views

Variation of argument of a complex function

Variation of Argument : Definition( Collect from my book ) : Let $f$ be analytic inside and on a sinple closed contour $C$ except possibly for poles inside $C$ and $f(z)\not=0$ on $C$. As $z$ ...
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1answer
78 views

Picture of elliptic curve double covers Rieman sphere

For simplicity, let us assume we live in the field $\mathbb{C}$. Then all elliptic curves are hyperelliptic, so they admit a double cover of $\mathbb{P}^1$ branched over four points, whose preimage ...
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Complete Intersection does not lie in a hyperplane?

Suppose there are $k$, $k \leq n-2$, irreducible hypersurfaces of in $\mathbb{P}^n,n \geq 3$ cut out by $F_1,\cdots, F_{k}$ with degree $d_i \geq 2$ and they form a complete intersection which is ...
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45 views

Picard Group of Universal Plane Conic

This is from 18.4.5 of Vakil's book Foundations of Algebraic Geometry. We work over a fixed field $k$, if $\mathbb{P}^2$ has projective coordinates $x_0,~x_1,~x_2$ and $\mathbb{P}^5$ has coordinates $...
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Differential in Huybrechts

the differential $df|x:T_x\mathbb{R}^2\to T_{f(x)}\mathbb{R}^2$ of a differentiable map $f:\mathbb{R}^2\to \mathbb{R}^2$ is given by the usual jacobian matrix. However, if we complexify the tangent ...
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Definition of Numerical Trivial Invertible Sheaf

I am reading Ravi Vakil's book, Foundations of Algebraic Geometry and in section 18.4.9, the definition of numberically trivial invertible sheaf is, Suppose $X$ is a proper $k-$ variety, and $\...
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How is the smoothness of the space of deformations related to unobstructedness?

As a beginning differential geometer, I've been trying to learn about deformation theory. Other than Kodaira's book, I've found virtually no references from the point of view of differential geometry....
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62 views

Difference between $\mathbb{C}P^2$ and $\overline{\mathbb{C}P^2}$

Sorry for the trouble, I got confused between $\mathbb{C}P^2$ and $\overline{\mathbb{C}P^2}$, do they have the same Euler characteristic? the same signature? does $\overline{\mathbb{C}P^2}$ admit a ...
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63 views

Divisors and their meaning

I have something that has been bothering me for a while: The concept of divisor. Let X be an affine variety, let's say smooth. Let $E_i $ be a prime divisor (i.e of codim 1). Let's $ D= \sum a_i E_i $....
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38 views

Intuition of section of a hermitian line bundle

Can someone explain to me intuitively and without much technical stuff the following: A hermitian line bundle is a complex line bundle with a hermitian metric. I think of this as a bundle over my ...
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1answer
60 views

On Dolbeault cohomology and Dolbeault operator

I'm trying to construct ladder operators on cohomology space, I searched for a similar procedure but I can't find anything. To be clearer, I consider the cohomology space of a compact Kähler manifold ...
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1answer
55 views

When is a $(p,q)$-form real?

Let $M$ be a complex manifold, and let $\omega$ be a $(p,q)$-form. Then, in holomorphic coordinates $(z^1,\dots , z^n)$, $\omega$ can be expressed as $\omega = f_{U,V} dz^U \wedge d\bar{z}^V$, where $...
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32 views

Prove that the sum of angles is equal to 90° using complex numbers

On the picture, we see three squares: $ABGH$, $BCFG$ and $CDEF$. Prove that the sum of angles: $\angle DAE$, $\angle CAF$ and $\angle BAG$ is equal to $90°$. The real problem is that we have to ...
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31 views

How to visualize complex domains

I was hoping if someone can help me visualize complex domains. I know how simplex ones like $|z|<1$ or $\text{Re}z < 1$ look like but for the more complicated ones such as $$\text{Im } z < 2|...
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1answer
65 views

Hodge numbers of a cartesian product of copies of $\mathbb{C}P^1$

I wonder if some works have been done in the context of cohomology space of projective complex manifolds. Specifically I want to study the Hodge diagrams of $\mathbb{C}P^1\times\mathbb{C}P^1$ and $\...
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1answer
50 views

Let $f$ holomorphic funcion in $U$ such that $\left|f\right|$ constant on the border of $K$. Show that $f$ is constant or $f$ have a zero in $K^{0}$.

Let $U\subseteq\mathbb{C}$ be an open and connected set and $K\subset U$ a compact subset with nonempty interior $K^{o}$. Let $f:U\rightarrow \mathbb{C}$ holomorphic funcion such that $\left|f\right|$ ...
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119 views

What is the derivative of $z^{-1}$ with respect to $\bar{z}$?

I asked a question here a few days ago but it wasn't answered and, as often happens with me, in trying to answer it myself I just confused myself out of understanding what I thought I knew. What is ...
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64 views

Mapping circles in the complex plane

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1 \\ x^2+(y−2)^2=4 $$ That is, I believe, ...