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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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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77 views

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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15 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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64 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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35 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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51 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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42 views

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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47 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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39 views

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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113 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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107 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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68 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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33 views

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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37 views

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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142 views

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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61 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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62 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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33 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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49 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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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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58 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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48 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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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, ...
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33 views

Torsion in cotangent sheaf of the cusp

Let $X$ be the cuspidal curve $y^2 = x^3$. This is a singular curve with a unique singularity in the origin. One can easily see this using the Jacobi criterion. How does one show explicitly that the ...
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57 views

Let $L$ be a complex line bundle on a Riemann surface. Is $L\oplus L^{-1}$ trivial?

Let $\Sigma$ be any Riemann surface, and let $L \rightarrow \Sigma$ be a complex line bundle (which is classified according to its degree). Then the vector bundle $L \oplus L^{-1} \rightarrow \Sigma$ ...
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29 views

Curvatres specialized on disc

Consider an open disk of unit radius in the real (two dimensional) plane.If we want to define the Ricci curvature and bisectional curvature on that disc,what will be their equivalent forms and the ...
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How do I find a smooth map from complex Gr(k, n) to real Gr(2k, 2n)?

I am trying to find a smooth bijective map from complex Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ to the Grassmannian $2k$-dimensional subspaces of $\mathbb{R}^{2n}$, but I do not ...
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80 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
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55 views

Why the canonical bundle of a complex manifold is a line bundle?

I think that I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1, that is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector ...
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38 views

Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
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60 views

Complex Hopf Fibration

The Hopf construction gives a circle bundle $p$ : $S^{3}$ → $\mathbb{CP}^1$. The equation of a 3-sphere in $\mathbb{R}^4$ is $X^2+Y^2+V^2+W^2=R^2$, where $R$ is the radius of the 3-sphere. We may ...
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Why $V_{\mathbb{C}} = V_{1,0}\oplus V_{0,1}$?

I am having a little problem with elementary linear algebra. Let $V$ being a real vector space. Lets call $V_{\mathbb{C}} := V\otimes \mathbb{C}$. Consider $J: V \to V$ an automorphism such that $J^2 =...
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Holomorphic Frobenius Theorem

I'm trying to understand a proof of the Holomorphic Frobenius Theorem using the smooth version as seen in Voisin's Complex Geometry book: (pg 51) http://www.amazon.com/Hodge-Theory-Complex-Algebraic-...
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Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
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Find the image of the sector $|z|\lt 1, 0\lt \arg z\lt \frac{\pi}{n}$, for the function $w=\frac{z^n+1}{z^n-1}$.

Find the image of the sector $|z|\lt 1, 0\lt \arg z\lt \frac{\pi}{n}$, for the function $w=\frac{z^n+1}{z^n-1}$. $w$ is a composite of two functions $\phi_1=\frac{z+1}{z-1}$ and $\phi_2z^n$, and ...
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21 views

Find the image of $f(S)$ and draw what's happening

$$w=f(z)=z^2$$ $$S=\{z\mid Re(z)=a\}$$ I don't get what this is asking and what that second part means. What is the concept here? If I let $z=x+iy$, then I can square it. The real term is then $x^2-y^...
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What is the name of these elliptic surfaces E(n)?

I am referring to the elliptic surfaces $E(n)$, with fibration over $\mathrm{C}\mathbb{P}^1$. They are common in 4-manifold theory and complex geometry. See for example Chapter 7 in Akbulut`s "4-...
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39 views

Good book on quantum probability

Who can suggest me a book on quantum probability? I'm mostly interested in its geometric aspects (complex projective space, Fubini-Study...). I have a background in quantum mechanics and differential ...
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30 views

Draw Regions On the Complex Plane that Satisfy this Relation

I'm looking to draw a region that satisfies the following: $$ Im\left(\frac{z-z_1}{z-z_2}\right)=0 $$ What I know so far is this: the expression as it's given is not of the form $ a + bi $, as there'...
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38 views

Order of Contact for a general tangent line of a cubic threefold

I am trying to solve Exercise 18.21 of Harris "Algebraic Geometry". In the proof of the unirationality of a smooth cubic threefold X he claims that a general tangent line to X at a general point p $\...
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1answer
52 views

Vector bundle morphisms and determinants

Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$. Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is ...