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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4
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2answers
96 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 ...
1
vote
0answers
41 views

Conformal Mapping: Is this correct?

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: $$x^2+(y−1)^2=1\\x^2+(y−2)^2=4 $$ What I now want to do ...
0
votes
0answers
24 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
votes
0answers
46 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
votes
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 ...
0
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0answers
29 views

Is this complex vector bundle 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 $L \oplus L^{-1} \rightarrow \Sigma$ ...
0
votes
0answers
25 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 ...
1
vote
1answer
48 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
votes
0answers
30 views

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

I think 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 space ...
2
votes
0answers
24 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
votes
1answer
41 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
45 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
votes
1answer
58 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) ...
6
votes
0answers
64 views
+100

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 ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
16 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 ...
0
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0answers
12 views

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 ...
3
votes
0answers
24 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 ...
1
vote
0answers
24 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 ...
0
votes
1answer
26 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 ...
3
votes
1answer
35 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 ...
0
votes
1answer
50 views

Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the ...
2
votes
1answer
50 views

Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
-1
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1answer
22 views

Why there is no biholomorphism between complex plane and unit disk? [closed]

Why there is no biholomorphism between complex plane and the unit disk?
0
votes
1answer
40 views

Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
2
votes
1answer
35 views

Polyakov action in complex coordinates

Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define $$ S(f,g) = ...
0
votes
1answer
21 views

Natural map from cokernel of a monad

If I have a monad $$ U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W $$ then there should be a natural map $$ \text{cokernel}(\alpha) \rightarrow W $$ but I can't think of ...
0
votes
2answers
57 views

Reference request on complex projective algebraic geometry

I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic ...
0
votes
0answers
14 views

Cohomology with adding an infinite point

Let $F$ be a closed subset of $\mathbb{C}$. (We assume $F \neq \emptyset, F \neq \mathbb{C},$ and $0 \notin F$.) Of course $F$ is not a closed subset of $\overline{\mathbb{C}}$ in general but it ...
1
vote
1answer
59 views

Kähler differential and higher derivations (geometric interpretation of diagonal here)

I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The ...
0
votes
1answer
33 views

On the proof of Riemann extension theorem in Huybrechts

In the proof of the generalized Riemann extension theorem in Daniel Huybrechts's Complex Geometry, which is: Proposition 1.1.7 (Riemann extension theorem) Let $f$ be a holomorphic function on an ...
6
votes
1answer
192 views

Find type of a differential form on an almost complex manifold

If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms $$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad ...
0
votes
0answers
26 views

Show that $|1-h \lambda| <1 $ is a disc

From the stability region of Euler, show that $|1-h \lambda| <1$ is a disc, where $\lambda$ is imaginary. I am wondering why it is a disc with center at $h \lambda = -1$
4
votes
1answer
99 views

How much algebraic geometry is there in complex geometry (for example, Demailly)?

I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld ...
4
votes
1answer
78 views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
6
votes
0answers
83 views

Is there an irreducible projective hypersurface such that its complement has zero Euler characteristic?

We know that, if $f=X_0X_1...X_n \in \mathbb{C}[X_0,...,X_n]$ and $Z(f)\subset \mathbb{CP}^n$, then the Euler characteristic of its complement is zero, i.e. $$ \chi(\mathbb{CP}^n\setminus Z(f))=0. $$ ...
3
votes
1answer
58 views

Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
3
votes
1answer
58 views

Analytic proof of Serre vanishing theorem

Consider the following equivalent statement of Serre vanishing theorem (replacing ampleness condition on the line bundle with postivity condition). Let $X$ be a compact complex manifold. Let $L$ be ...
6
votes
1answer
92 views

Imaginary lines and tangents

My geometry text seems to say that the tangent to a circle from an interior point in the real plane is imaginary. Further...it seems that when a double cone is intersected by a plane with an angle ...
3
votes
0answers
41 views

valid definition of complex geodesic

Let $X$ be a complex manifold, and let $Y$ be a complex submanifold of $X$. If $X$ has an hermitian structure(on its tangent bundle), we can consider the Chern connection $\nabla$ on the holomorphic ...
2
votes
0answers
53 views

Functions that maps unit circle into unit circle

This and This problems discuses the characterization of Analytic functions which maps unit circle on to itself. I would like to know the characterization of functions which map unit circle into ...
0
votes
1answer
67 views

Horrocks-Mumford bundle

Let $E$ be the Horrocks-Mumford bundle, which is a rank-2 vector bundle on $\mathbb P^4$ with $c_1(E)=5$ and $c_2(E)=10$, defined by some combinatorical construction (see Okonek, Schneider, Schindler, ...
0
votes
1answer
40 views

How do we get from $\Delta f = \rho$ to $\partial\bar{\partial}f = \text{Const.} \rho\,dz\wedge d\bar{z}$?

I asked this question about the Kähler potential on MathOverflow. Donu Arapura left a comment saying Classically, a potential satisfies $\Delta f = \rho$. In the plane, this can be rewritten as ...
2
votes
1answer
40 views

Neron-Severi group as the image of first Chern class

Let $X$ be a smooth projective variety over $\mathbb{C}$, then the Neron-Severi group $NS(X)$ of $X$ is defined to be the Picard group of $X$ modulo algebraically equivalent relations. On the other ...
1
vote
0answers
19 views

The Levi-Civita and the Covariantly Constant Tensors in Kahler Manifold?

Please scroll down to the bold section if you are too bored to read the whole details. Aiming to explain the mathematical structure of Kahler manifolds, Freedman and Van Proeyen, in their book ...
3
votes
0answers
21 views

almost complex structure on a surface

Let $M$ be an oriented smooth surface, $GL(M) \to M$ the bundle of oriented frames of $M$. Why is the space $S(M)$ of almost complex structures on $S$ equal to smooth sections of $GL(M) ...
1
vote
0answers
24 views

One of Hermitian metric's properties?

We now define a Hermitian manifold is a complex manifold in which unmixed components of metric tensor vanish $g_{ij}=g_{\bar{i}\bar{j}}=0$. Is this a propert of a Hermitian manifold? Or is it an extra ...
3
votes
1answer
65 views

first chern class of holomorphic tangent bundle $T\mathbb{C}P^n$

Let $L$ be tautopological bundle of $\mathbb{C}P^n$ and $L^{-1}$ be its duality. Because $L$ is a subbundle of $\underline{\mathbb{C}}^{n+1}$, $\underline{\mathbb{C}}=L\otimes L^{-1}$ is a subbundle ...
4
votes
1answer
51 views

Algebraic surface with infinitely many exceptional curves

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point. If I understood correctly every surface $X$ ...
1
vote
1answer
72 views

the $\partial\bar{\partial}$-lemma dilemma

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge ...