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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Question on Intersection Theory of Effective Divisors

I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...
4
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592 views

Can one give me some concrete examples explaining Picard's Great Theorem

Picard's Great Theorem Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one ...
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1answer
257 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 ...
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95 views

Application of Kodaira Embedding Theorem

I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem. I also wish to give some applications of the theorem. One of the application ...
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48 views

Space of almost complex structures on a compact manifold

According to the book by Huybrechts, Complex Geometry: An Introduction, this is a nice space and may be regarded, after some form of completion, as an infinite-dimensional manifold. How is this done, ...
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77 views

Automorphisms of Complex Projective Space

Each automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ (where $\mathbb{C}P^n$ is regarded as a complex manifold) is induced by a linear map $\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$. I know ...
2
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1answer
92 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
5
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2answers
223 views

Complex manifold with no divisors

I read in Griffith Harris P132 that a complex manifold of dimension greater than one can have no divisors on it at all. I want to find examples. Is there an example? Does the Hopf manifolds $S^1\times ...
3
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1answer
486 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
4
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1answer
172 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
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1answer
70 views

Branch locus of a projection of algebraic set

Let $X$ a algebraic cone in $\mathbb{C}^n$ with $\dim_0 X=p$. def.: if $f:A\to B$ is smooth map between smooth manifolds, then $br(f)$ is the points $x$ that $df_x$ is not surjective. def.: if $\pi: ...
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129 views

Deformation polarized abelian variety

In "$C$ is not algebraically equivalent to $C^{-}$ in its Jacobian" by Ceresa and then in "On the periods of certain rational integrals, II" by Griffiths (which is quoted as a reference in the first ...
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64 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
3
votes
1answer
90 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 ...
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1answer
179 views

Dolbeault cohomology on torus

Let $T=\mathbb{C}/\Gamma$ where $\Gamma$ is a lattice of $\mathbb C$. Given that $H_{dR}^1(T)=\mathbb{C}^2$. Prove that $H^{1,0}_\bar{\partial}(T)=\mathbb{C}$. I have no idea what to do. Can someone ...
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0answers
37 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
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75 views

2-forms represented by a first Chern class?

Let $M$ be a complex manifold and $\omega$ be a 2-form on $M$. Is there a good way to see whether $\omega$ is represented by the first Chern class of a line bundle on $M$? In other words, when is it ...
6
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1answer
69 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
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58 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
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1answer
123 views

Cohomology to compute number of holes?

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong ...
6
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1answer
209 views

Level sets of holomorphic functions

It is a somewhat well known fact that any closed set (say in the plane) can be realized as the level set of a smooth ($C^\infty$) function, so level sets of smooth functions are as general as they can ...
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42 views

Properness on an open subvariety

I was wondering if the following property holds in general For any morphism $f: X \to Y$ between two varieties over $\mathbb{C}$, there exists a nonempty open subvariety $U \subseteq Y$ such ...
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68 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
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1answer
162 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
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20 views

Maximal immerged disk on a hermitian Riemann surface

Let $S$ be a Riemann surface equipped with a hyperbolic metric, and $z_0 \in S$. Is there a way to estimate the maximal radius $r>0$ such that there is a holomorphic embedding $\psi : \Delta_r ...
4
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2answers
91 views

Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
4
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1answer
142 views

some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
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3answers
303 views

Are the smooth points dense in a projection of a complex variety?

Let $X=V(I)\subset \mathbb{C}^{n}$ be the vanishing set of an ideal of complex polynomials, let $\pi \colon \mathbb{C}^{n} \to \mathbb{C}^{n-1}$ be the projection onto the first $n-1$ coordinates and ...
2
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1answer
100 views

Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$ h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10 $$ Do you know why this is the case? In ...
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66 views

Why is every one dimensional Complex Manifold paracompact?

I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue ...
5
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1answer
156 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
3
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1answer
57 views

Boundedness of the operator $[\Lambda, \Theta]$.

I am reading Griffiths-Harris book on algebraic geometry and in "Theorem B", where he proves (the analogue of Serre's vanishing theorem) that Let $M$ be a compact, complex manifold, $L\rightarrow ...
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22 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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2answers
400 views

Complex manifolds and Hermitian metrics

I've been trying to learn some complex geometry, and was getting confused in thinking about Hermitian metrics. In this post, I've written up my current understanding, in hopes that someone can look ...
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29 views

lagrangian locus in the quintic as fixed point set of involution

Consider the Fermat quintic $$X = \left\{ \sum_{i=1}^5 x_i^5 = 0 \right\} \subset \mathbb{CP}^4$$ and a map $\sigma: X \to X$ given by $(x_1:x_2:x_3:x_4:x_5)\mapsto ...
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0answers
69 views

Visualizing a complex curve

This may be as much a question about computers as about math. Let $C=\{f(r,s,t)=0\}$ be a curve in $\mathbb{CP}^2.$ By forgetting about the points at infinity we can view $C$ as a surface in ...
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78 views

equivariant map $\mathbb{CP}^1 \to \mathbb{CP}^2:$ where does it send the boundary

Start from a map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x = au^2, \quad y = av^2, \quad z = uv,$$ where $a \in \mathbb{R}$ is a parameter. The image of the map is the ...
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51 views

Analytic (sub)variety and connected components

I am trying to figure out the proof of Bertini's theorem in Griffiths-Harris book on Principles of Algebraic Geometry, page 138: But by the calculation above, the ratio $f/g$ is constant on every ...
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1answer
33 views

holomorphic disk and crosscap as quotients of $\mathbb{CP}^1$ by antiholomorphic involutions

Consider $\mathbb{CP}^1 \ni (u:v)$ and the maps $$ \sigma_{\pm}: \quad (u:v) \mapsto (\overline{v}:\pm\overline{u})$$ How do we show that the quotient $\mathbb{CP}^1/\sigma_+$ gives a disk, and ...
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1answer
146 views

regarding finite integral ring extension

My question regards understanding (and possibly a source for proof) of the following, cited in the book Complex Geometry by Huybrechts (Theorem 1.1.30.) (Also, it is there stated that this is a ...
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0answers
105 views

relation between structure group of holomorphic ,antiholomorphic, complex tangent bundle

I Know that a complex tangent bundle $T_{\mathbb{C}}M$ can be written as direct sum of holomorphic and anti- holomorphic tangent bundles $i. e.$ $T_{\mathbb{C}}M = T_{\mathbb{C}}'M \oplus ...
2
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1answer
84 views

Explicit diffeomorphism between complex tori

Let $Im(\tau) > 0$ and $X_{\tau}$ be the complex torus given by $\mathbb{C}/\mathbb{Z}\oplus \tau\mathbb{Z}$. How do I go about constructing an explicit diffeomorphism (as real manifolds) between ...
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2answers
76 views

A subset of $\mathbb C\times\mathbb C$

I'm trying to think if the space $\{(z,\,i\overline{z})\,:\,z\in\mathbb{C}\}$, where $\overline{z}$ is the complex conjugate of $z$ and $i$ is the imaginary number, is topologically equivalent to ...
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147 views

All almost complex structures on a manifold

I read the statement of the Newlander-Nirenberg theorem, which says that "any integrable almost complex structure is induced by a complex structure". To make sense of the statement, I was wondering ...
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37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
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1answer
67 views

Are holomorphic maps regular maps of varieties?

Is a holomorphic map of complex algebraic varieties always a regular map?
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1answer
121 views

The Grassmannian of 2-planes in complex n-space is hyperkahler

I have seen it mentioned that the Grassmann manifold of complex 2-planes in complex n-space is a hyperkahler manifold, but I can't find a reference for a proof. Does anyone know the proof of this or ...
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1answer
101 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 ...
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1answer
281 views

Sheaf cohomology of projective spaces

I came across Bott's formula in "Vector bundles on complex projective spaces" by Okonek, Schneider & Spindler. The formula is a formula for $h^q(\mathbb P^n,\Omega^p(k))$, where $\Omega^p(k)$ is ...
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282 views

Cotangent bundle of complex manifold is Calabi-Yau manifold

We say that a complex manifold $M$ is Calabi-Yau if the canonical bunlde is trivial $K_M=0$. How can we prove that the total space of the cotangent bundle of a compact complex manifold $N$ is ...