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Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

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2answers
70 views

Reality check on integrability of almost-complex surface.

Let $(M^2,J)$ be an almost-complex surface. I do not assume integrability of $J$ now. I understand that from the Newlander-Nirenberg theorem, one of the several equivalences for the integrability of $...
2
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0answers
33 views

The defining polynomials of complex abelian varieties

By Chow's theorem, the complex abelian varieties should be algebraic. So, what is the defining polynomial? (It is easy when $g=1$) I have searched it on arxiv and google scholar, but I get nothing. ...
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0answers
11 views

A question about the preserving order degeneracy maps in simplicial sets

I am reading the paper Cyclic Homology Theory written by Prof. Loday. On page 7, I don't know why the third diagram about the Degeneracy map is not allowed. Any help will be thanks.
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0answers
28 views

On the preimage of injective holomorphic map

I am hoping the following is true. Mention of related ideas/topics are appreciated. Suppose $F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_n$ ...
2
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1answer
55 views

Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
0
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0answers
24 views

Pull backs along rational maps

Let $M^m$ be a compact complex $m$-dimensional manifold and $f: M \dashrightarrow C\mathbb{P}^n$ a rational map (i.e. holomorphic map defined away from a subvariety, $V$, of codimension at least 2). ...
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1answer
33 views

Product representation of Weierstrass $\wp$-function

Let $\Lambda = w_1\mathbb Z+w_2\mathbb Z$ be a lattice and $$\wp(z)=\frac1{z^2}+\sum_{w\in\Lambda - 0}\frac1{(z-w)^2}-\frac1{w^2}$$ its associated Weierstrass $\wp$-function. Let $n>1$ be an ...
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0answers
33 views

Cocompact lattice

We have a complex 1-form $Ω$.suppose class of real part $ [Ω_{R}] $and imagine part $ [Ω_{I}] $ are linearly independent in $ \in H^1(M,\mathbb{R}) $ and There is a closed form $Ω^{\prime}\in Ω^1(Μ,\...
3
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1answer
78 views

Difference between several books on complex geometry

I would like to learn some complex geometry, especially the interaction between algebraic geometry and complex geometry. I found that there are several famous books: Huybrechts, Complex Geometry; ...
1
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1answer
129 views

Complex structure is determined in a compatible triple in a Kähler space

Let $V$ be a real (finite dimensional) vector space, $\Omega$ be a symplectic form on $V$, and $g$ be a pseudo-Euclidean scalar product. Using $g$ to obtain an isomorphism $\sharp\colon V^* \to V$, we ...
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1answer
19 views

Complex Differential Forms and Notation

If I have a complex form $\Omega$, what does the notation $\operatorname{Re}(\Omega)$ and $\operatorname{Im}(\Omega)$ mean? How does this relate to the decomposition of the space of $(p,q)$-forms and ...
3
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1answer
30 views

Inducing almost complex structure in tensor bundle

Let $E\to M$ be a (smooth) vector bundle and $J$ be a section of ${\rm End}(E)$ with $J^2= -{\rm Id}$. Can $J$ induce something in $$\mathscr{T}^{(r,s)}(E)=\bigsqcup_{x\in M} E^{\otimes r}\otimes (E^*...
4
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0answers
76 views

What is the fundamental group of this complex surface?

Let $X\subset A^3_{\mathbb{C}}$ be the affine complex surface defined by $x^r+y^s+z^t=0 $, and let $Y=X-(0,0,0)$. Here $r,s,t\geq2$ are positive integers. Then what is the fondamental group of $Y$? ...
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0answers
50 views

Is a finite cover of Kähler manifold again Kähler?

It is obvious that a branched finite cover of projective variety is projective. But I don’t see whether it is true that a smooth branched cover of Kähler manifold is Kähler.
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0answers
45 views

Chow ring isomorphic to homology ring?

I heard an algebraic geometry professor say that the Chow ring is usually isomorphic to the homology ring for cases we care about in application. However, I cannot find many results about this, beyond ...
2
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0answers
28 views

Visualising a particular line bundle on $S^2$

I saw this general construction of a line bundle in the book Integrable Systems: Twistors, Loop Groups and Riemann Surfaces by Hitchin, Segal and Ward (Chapter 2 Section 1 before Definition 1.5). ...
6
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0answers
91 views

references for algebraic topology of (complex) manifolds

Now I want to study some fundamental theorems of cohomology of (complex) manifolds. e.g., Poincare duality, Kunneth formula, weak and hard Lefschetz, cup product, cycle maps, "de Rham cohomology = ...
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2answers
53 views

Find Möbius transformation for half-plane to unit disk $|w|<1$?

Consider the half-plane depicted in the following figure How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found? What are the steps and things to think ...
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0answers
46 views

Tangent bundle of complex torus is trivial

I've seen this post and I've been wondering if it's also true in the complex case. So is the holomorphic tangent bundle of the complex torus also trivial? And can one adapt the proof of the real case? ...
3
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0answers
33 views

Multiplication in Deligne cohomology: explicit formula for p = q= 1

In the very beginning of [1] the geometric meaning of Deligne cohomology $H^q(X, \mathbb{Z}(p))_D$ and multiplicative structure on it is being discussed. In particular, it is not hard to see that $H^q(...
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0answers
40 views

Traditional way to study deformation theory of complex manifold

I want to study the deformation theory established by Kodaira - Spencer. In Kodaira book or his corrected paper is the only one what I сome up with. Someone can tell me recent topics of deformation ...
7
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1answer
67 views

Example of no non-constant meromorphic functions

This is a basic question but I have found it annoyingly hard to find an answer (which means its incredibly trivial or non-trivial!) Are there examples of compact complex manifolds (of dimension at ...
2
votes
1answer
36 views

Injective holomorphic endomorphism whose image is the complement of a proper analytic subset is surjective.

Let $f: M \rightarrow M$ be an endomorphism of a connected complex manifold $M$. Assume that $f(M)$ is a dense open set in $M$ and that $M \setminus f(M)$ is an analytic subset of $M$ Question: Can I ...
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0answers
18 views

the modular curve of level 7 is klein quartic

Does anyone know a simple proof that the modular curve of level 7 is the Klein quartic? I'm looking for a book that has a proof of this fact. Thank you!
3
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0answers
46 views

Condition for a submanifold of complex Euclidean space to be analytic

I have seen, but don't fully understand, the following statement and sketch proof: Statement: A codimension $2$ submanifold $C \subset \mathbb{C}^2$ such that $C$ has positive intersection index with ...
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0answers
45 views

Complex bundles and (quasi)-complex structures of manifolds

Could you help me with some hint or reference for the following questions? I'm reviewing the Milnor-Stasheff for references. Is there some $(2k)$-manifold with stably quasi-complex structure, such ...
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0answers
33 views

Representation of fundamental form by positive function

I want to solve the following exercise: Let $\omega = \frac{i}{2\pi} \sum dz_i \wedge d\overline{z_i}$ be the standard fundamental form on $\mathbb{C}^n$. Show that one can write $\omega = \frac{i}{...
3
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0answers
31 views

Existence of J-Holomorphic Curves in Almost-Complex (or Symplectic) Manifolds

Suppose I have a (let's say compact) symplectic manifold $(M, \omega)$ and I choose an $\omega$-tame almost-complex structure $J$ on $M$. (Edit: Actually, I think I only really care about the almost-...
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0answers
25 views

Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$. The complex line generated by $v$ is $$\{r[(\...
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0answers
28 views

Conditions for a Connection to be a Metric(or Chern) Connection

Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$, $$[De=\mathcal{A}e,]$$ where $e$ is a holomorphic ...
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0answers
25 views

Sequence of differentation is exact

I want to proof the following statement: Let $B \subset \mathbb{C}^n, p,q \geq 1$. The following sequence is exact: $\mathcal{A}^{p-1,q-1}(B) \xrightarrow{\partial \overline{\partial}} \mathcal{A}...
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3answers
55 views

inverse point with respect to circle

I'm reading Silverman's Complex variable with application. at page 78, the author says "We say that $s$ and $s^*$ are the inverse points with respect to circle in $\mathbb{C}$ if every line or circle ...
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0answers
21 views

Extension of a rational map in codimension one - relative version

Suppose $X$ and $Y$ are smooth projective $T$-varieties, where $T$ is a smooth affine curve. Let $\phi:X\dashrightarrow Y/T$ be a rational map over $T$. My question is: is there a closed subset $Z\...
2
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1answer
34 views

Almost complex structure question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are ...
2
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0answers
40 views

Complex structure of branched cover over Riemann surface [closed]

Suppose $X$ is a Riemann surface, $Y$ is a Hausdorff topological space and $p: Y\to X$ is a local homeomorphism. Then there is a unique complex structure on $Y$ such that $p$ is holomorphic. Now if $\...
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0answers
33 views

Complex geometry, complex numbers

The locus of any point $p(z)$ on Argand plane is $\;\arg((z-5i)/(z+5i))=\pi/4$, then the total number of integral points inside the region bounded by the locus of $p(z)$ and the imaginary axis is? Any ...
3
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2answers
65 views

Existence of non-constant holomorphic map between two given compact Riemann surfaces

Given two compact Riemann surfaces $X,Y$, can we always find a non-constant holomorphic map from $X$ to $Y$? In particular, when $Y$ is a elliptic curve, does that map exist? Michael Albanese has ...
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0answers
24 views

Nontrivial holomorphic vector bundle on $\mathbb{C}^n$

Does there exist any nontrivial holomorphic vector bundle on $\mathbb{C}^n$? I know $(1)$ Every line bundle on $\mathbb{C}^n$ is trivial, $(2)$ Every holomorphic vector bundle on $\mathbb{C}$ is ...
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0answers
24 views

Global Kodaira correspondence for analytic families of compact complex manifolds

First of a necessary definition. An analytic family of compact complex submanifolds of a complex manifold $Z$ with parameter space $M$, which is a complex manifold, is a complex submanifold $F\subset ...
3
votes
1answer
58 views

Why does a pointed surface minus a countable set of points contain a curve?

Let $S$ be a surface over $\mathbb{C}$ and let $s_1,\ldots, s_n$ be closed points of $S$. We consider this data as fixed. It is not hard to see that there is a curve passing through $s_1,\ldots,s_n$....
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0answers
55 views

Singular maps in into projective spaces

This may be a very basic question, so my apologies if that is the case. But I was interested in having some examples of meromorphic (singular) maps into complex projective space from complex surfaces ...
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0answers
42 views

Indeterminacy locus of an Iitaka fibration

This might be a trivial question, in this case I apologize. Let $X$ be a smooth projective complex algebraic variety of dimension $n$ and Kodaira dimension $n-1$. Let $\phi:X\dashrightarrow Z$ be the ...
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1answer
20 views

Plotting Complex Parametric Curves

I have to plot out by hand the following curve $z(t) = 3+ie^{it}$ for $0\leq t \leq \pi$ I know that circles in $\mathbb{C}$ can be parameterized as $z(t) = c + re^{it}$ where the circle has radius $...
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1answer
27 views

Does every point of a complex manifold have a neighbourhood basis of stein manifolds?

In Grauert & Remmert - "Coherent Analytic Sheaves" it is stated on page 34 that "every point of an arbitrary complex space has a neighborhood basis of STEIN spaces". Since I am not familiar with ...
3
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1answer
72 views

Chern class of tautological line bundle over the projectivization of a vector bundle

Let $\mathbb{C}^k\hookrightarrow E\to B$ be a complex vector bundle. Let $\mathbb{CP}^{k-1}\hookrightarrow\mathbb{P}(E)\to B$ be its projectivization. We can consider the tautological line bundle $L$...
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1answer
16 views

Question about equivalent definitions of (holomorphic) line bundles

Definition: A complex line bundle is a smooth manifold $L$ with a projection map $\pi:L\to M$, so that each fiber $\pi^{-1}(p)$ has the structure of a complex vector space of dimension 1. And for each ...
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2answers
68 views

Definition of divisor associated to a meromorphic function

Let $X$ be a complex manifold and let $f$ be a meromorphic function defined on it. Let $Div(X)$ be the group of locally finite sum of analytic irreducible hypersurfaxces of $X$. One would like to ...
2
votes
1answer
40 views

Is every complex manifold that's homeomorphic to $\mathbb C\mathbb P^n$ also isomorphic to $\mathbb C\mathbb P^n$?

Put another way, is the complex structure on $\mathbb C\mathbb P^n$ unique? I know that this is the case for $n\in\{1,2\}$, so I'm curious as to whether it's the case in general. If it's not known, is ...
1
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1answer
50 views

Finding the singular locus of the given complex space

This problem is from Greuel et al., Introduction to Singularities and Deformations. Determine the singular locus of the complex spaces defined by the following $\mathcal{O}_{\mathbb{C}^n}$-ideals: (...
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1answer
70 views

How should I interpret $H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}\oplus i\mathbb{Z})$?

I am reading A. Zorich's article "Square Tiled Surfaces and Teichmuller Volumes of the Moduli Spaces of Abelian Differentials" and there a main object is the space $$H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}...