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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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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124 views

Top current research topics in Complex Algebraic and Differential Geometry

I am currently a first year PhD candidate and I need some help figuring out the top current research topics in Complex Algebraic and Differential Geometry (can be in both, or just one of the two). I ...
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37 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
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1answer
93 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
93 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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37 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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38 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
95 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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0answers
17 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 ...
3
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2answers
60 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 ...
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1answer
95 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. ...
2
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3answers
133 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
80 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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0answers
27 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
89 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
38 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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0answers
18 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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0answers
35 views

When is a map $\mathbb{CP}^1 \to \mathbb{CP}^2$ a holomorphic embedding?

Consider the map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x= u^2, \quad y=v^2, \quad z=uv.$$ Is it a holomorphic embedding? What is to be checked, perhaps via some ...
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1answer
120 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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0answers
15 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
35 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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0answers
66 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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0answers
32 views

Graphing complex plane curves

This may be just as much a question about computers as a question about math. Suppose we have a complex curve $C\subset\mathbb{CP}^2,$ given by some $f(r,s)=0.$ Picking an affine chart, we can view ...
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23 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
25 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
81 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 ...
2
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0answers
27 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 ...
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0answers
44 views

holomorphic Lie group automorphisms of the complex Heisenberg group preserving the lattice

I don't understand Lie group,but I need an elementary result about it,so I ask for help for a simple question. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & z_1 & ...
2
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1answer
44 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
74 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 ...
2
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0answers
79 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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33 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
47 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
52 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
57 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
104 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 ...
4
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1answer
126 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 ...
3
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1answer
73 views

Dolbeault cohomology and analytic regularity

Let $M$ be a complex analytic $n$-manifold. The Dolbeault cohomology complex is defined using a quotient space of smooth differential forms. My question is : would it make a big difference if we were ...
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0answers
39 views

“Center of mass” of a complex hypersurface

Background: Suppose first that $f \in \mathbb{C}[z]$ and consider the "analytic hypersurfaces" $V(f)$ and $V(df)$, which are of course just sets of points with multiplicities. (That is, we think of ...
6
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1answer
91 views

Holomorphic sections of tensor product

I'm making a stupid mistake but I can't figure out what. Let $E,F$ be holomorphic vector bundles over a complex manifold $X$. Let $\mathcal O(E), \mathcal O(F)$ be the respective sheafs of ...
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122 views

Exercise $3.1.7$ from Huybrechts' Complex Geometry: An Introduction

I am trying to do the following question: Show that $L$, $d$, and $d^*$ acting on $\mathcal{A}^*(X)$ of a Kähler manifold $X$ determine the complex structure of $X$. Here $L$ is the Lefschetz ...
2
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1answer
56 views

Riemann Surface, existence of meromorphic function.

There is a question which has been perplexing me. Given $S$ a compact Riemann surface, $p$ and $q$ two distinct points. Is it always possible to find a meromorphic function on $S$ which is zero on $p$ ...
2
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1answer
64 views

calculate the group of all biholomorphic group automorphisms of complex tori

My backgrand is complex geometry,but when I confront complex tori,I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group. Let ...
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36 views

Monodromy action induced by higher direct images.

Let $X$ be a n-dimensional complex variety and let $f:X\longrightarrow \Delta$ be a Lefschetz degeneration (f is proper, with non zero differential on $\Delta^*$ and with one ODP in the fibre ...
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87 views

Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite resolution by locally free modules. By GAGA, I believe this should ...
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1answer
59 views

What is a rational elliptic surface?

I know what is an elliptic surface, and I also know what is a Rational surface. However I can't find the definition of Rational elliptic surface. Can any one help me? I read this on the Kulikov ...
2
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1answer
142 views

What are imaginary triangles?

The condition is that $x,y,z \in \mathbb{R}$ are the sides of a triangle inscribed in a circle of unit diameter determines a (symmetric) compact 2-dimensional manifold in $\mathbb{R}^3$ whose equation ...
2
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1answer
106 views

Normal bundle of a hyperplane section

Let $Y\subset \mathbb{P}^n$ be a smooth projective variety and let $H$ be a smooth hypersurface in $\mathbb{P}^n$ such that $Z=Y\cap H$ is smooth. How are the normal bundles of the various embeddings ...
4
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0answers
74 views

Why $H^{1,1}(X,\mathbb{C}) = Pic(X) \otimes \mathbb{C}$ for Calabi-Yau 3-folds?

Let $X$ be a Calabi-Yau 3-fold, that is $\omega_X = 0$ and $h^{1,0}=h^{2,0}=0$. Let $\operatorname{Pic}(X)$ be the group of line bundles on $X$. Then why the following isomorphism is true ...
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1answer
55 views

Non-hyperelliptic Riemann surface

Let $S$ be the Riemann surface of the plane algebraic curve $XYZ^3+X^5+Y^5 = 0$. How can I prove that $S$ isn't a hyperelliptic Riemann surface?