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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563 views

A complex algebraic variety which is connected in the usual topology

Hartshorne wrote in his book's Appendix B that it can be easily proved that a complex algebraic variety is connected in the usual topology if and only if it is connected in Zariski topology. How can ...
13
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2answers
878 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
6
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1answer
91 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 ...
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1answer
88 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 ...
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6answers
1k views

Books for Hyperbolic Geometry.

I want to read hyperbolic geometry. Can any one suggest some good books on the topic.
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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 ...
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1answer
460 views

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

Let $X$ be a complex manifold and denote the space of all $(p, q)$-forms on $X$ by $\mathcal{E}^{p,q}(X)$. Forgetting about the complex structure, we can consider the real differential $k$-forms on ...
7
votes
2answers
1k views

Almost complex structures on spheres

It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex ...
5
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1answer
266 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?
3
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1answer
95 views

set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
7
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1answer
236 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on ...
3
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2answers
305 views

Are there many almost complex structures on a (complex) manifold?

I guess one can have many almost complex structures on a manifold, can someone give me an example? How about when the manifold is complex? is the almost complex structure induced by the complex ...
7
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1answer
539 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
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1answer
67 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
27
votes
2answers
2k views

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
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1answer
243 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
8
votes
1answer
174 views

Intuitive Aproach to Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
7
votes
1answer
789 views

Meaning of holomorphic Euler characteristics?

I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
6
votes
1answer
168 views

Blowup of $xy-z^2$

I try to compute the blowup of $f=xy-z^2 \subset \mathbb{A}^3$ at the origin, but got something I could not explain: Let $A=\mathbb{C}[x,y,z]/(f)$, and $m=(x,y,z)$ be the ideal corresponding to the ...
4
votes
1answer
49 views

Image of the Riemann-sphere

Let $S$ be the Riemann-sphere (the unit sphere in $\mathbb{R}^3$) and $\psi: S \rightarrow \mathbb{C}$ be defined by $$\psi(x_1, x_2, x_3)=\frac{x_1 + ix_2}{1-x_3}.$$ Let $\pi$ be a plane in ...
6
votes
2answers
644 views

When does a line bundle have a meromorphic section?

Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come ...
8
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1answer
183 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
6
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1answer
324 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
5
votes
1answer
114 views

Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line ...
5
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0answers
61 views

Which complex manifolds admit a collection of compatible 'charts' $\varphi_{\alpha} : U_{\alpha} \to \mathbb{C}^n$ which cover $X$?

A complex manifold of dimension $n$ is a $2n$-dimensional topological manifold $X$ together with a complex atlas which is a collection of compatible charts $\varphi_{\alpha} : U_{\alpha} \to B(0, 1) ...
5
votes
3answers
451 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
4
votes
1answer
142 views

An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
3
votes
1answer
1k views

What is a Holomorphic Vector Field?

On a smooth manifold $M$, a smooth vector field is an element of $\Gamma(M, TM)$ which is the space of all smooth sections of the bundle $TM \to M$. If $M$ is a complex manifold, then we have the ...
2
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1answer
197 views

Checking flat- and smoothness: enough to check on closed points?

I am currently studying varieties over $\mathbb{C}$, i know some scheme theory. Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition ...
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vote
2answers
351 views

Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
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vote
1answer
89 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
9
votes
1answer
190 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
8
votes
2answers
284 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
8
votes
2answers
432 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
8
votes
2answers
366 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
7
votes
1answer
680 views

Proof of Hartogs's theorem

I'd be very grateful if someone could help me understand the proof of Hartogs's theorem appearing in Huybrechts' "Complex Geometry." The statement is: Let $\mathbb{P}^n \subset \mathbb{C}^n$ be the ...
6
votes
2answers
95 views

(Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
6
votes
2answers
193 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
5
votes
2answers
522 views

Some references for potential theory and complex differential geometry

I am looking for references on two distinct (though related) topics. Potential theory : I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive ...
5
votes
1answer
1k views

Proof that the Nijenhuis tensor vanishes in a complex manifold

I'm in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any ...
4
votes
0answers
85 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
4
votes
1answer
455 views

When are two tori biholomorphic? [duplicate]

If $\Lambda \subset \mathbb{C}$ is a lattice, let $T_{\Lambda}$ be the torus $\mathbb{C}/\Lambda$. My question is: If $\Lambda_1, \Lambda_2 \subset \mathbb{C}$ are two lattices, when are ...
4
votes
1answer
115 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
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1answer
253 views

When is $CaCl(X) \to Pic(X)$ surjective?

I am curious about following similar statements in algebraic geometry and complex geometry: Algebraic Geometry Version: If $X$ is an integral scheme, the map from Cartier divisor group to ...
4
votes
1answer
132 views

Holomorphic Poincaré conjecture

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
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0answers
512 views

Why is an analytic variety irreducible provided that the set of its smooth points is connected?

Here is a proposition from Griffith and Harris, Principles of Algebraic Geometry. Let $V$ be an analytic variety on a complex manifold $M$, let $V^*$ be its points of smooth points. If $V^*$ is ...
3
votes
1answer
87 views

A linear system of a curve on a K3 surface.

Let $S$ be a K3 surface and $C \subset S$ be a smooth curve of genus $g$ (assume it represents a primitive homology class). Is it possible to compute the dimension of the linear system $|C|$ only from ...
2
votes
1answer
63 views

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex ...
2
votes
1answer
313 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
2
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1answer
591 views

(Continued:) finiteness of étale morphisms

I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem. First question: Checking flat- and smoothness: ...