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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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 ...
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32 views

Locally nilpotent vector field is complete

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). I am trying to show that if the $V$ is ...
3
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0answers
41 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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1answer
43 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 ...
5
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0answers
45 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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2answers
28 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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0answers
24 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
25 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 } ) = \{ ...
3
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1answer
60 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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14 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 ...
4
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1answer
60 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 ...
1
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1answer
24 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 ...
2
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1answer
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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2answers
15 views

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 $F(\infty)=0.$ ...
3
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0answers
28 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 + ...
1
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1answer
49 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 ...
2
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1answer
41 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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0answers
42 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 ...
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0answers
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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2answers
111 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 ...
2
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1answer
99 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
58 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 ...
2
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32 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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0answers
38 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 ...
3
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1answer
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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0answers
38 views

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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2answers
131 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 ...
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1answer
60 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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1answer
61 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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1answer
25 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 ...
1
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1answer
38 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 ...
4
votes
1answer
54 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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1answer
27 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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1answer
30 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
votes
1answer
55 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 ...
2
votes
1answer
43 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 $\left|f\right|$ ...
4
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2answers
111 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 ...
0
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0answers
35 views

Geometrical interpretation for curvatures

What is the geometric interpretation for Ricci and Holomorphic Bisectional curvatures in the two dimensional space,like an open ball in the real plane??Any intuitive idea or source will be helpful.
0
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0answers
58 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, ...
2
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1answer
29 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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1answer
52 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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1answer
43 views

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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1answer
60 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 ...
0
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0answers
51 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 ...
3
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0answers
35 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 ...
2
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1answer
57 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 ...
3
votes
1answer
48 views

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 ...
2
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1answer
78 views

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) ...