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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Connected Reinhardt Domain which is not complete

Can i have an example of connected Reinhardt domain in $C^n$ which contains zero. But it is not complete. Complete means: For $w= (w_1,..w_n)\in D$, if $z$ is such that $|z_j|\leq |w_j$ for all $j$ ...
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2answers
2k views

Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
4
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0answers
385 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 ...
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1answer
325 views

complex coordinate system

what is the actual definition of "complex coordinate system"? I am not referring for 'complex number' nor 'polar coordinate system'. These terms are overlapping in my mind and i am unable to get ...
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1answer
95 views

A complex function

I just need some help with the penultimate question of my coursework: Let $w=f(z)=\coth(z/2)$. Show that $w=f(z)=h(C) = (C+1)/(C-1)$ where $C=g(z)=e^z$. Find the image of the given points, ...
6
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1answer
185 views

extending Zariski closed sets

Let $U \hookrightarrow X$ be an embedding of algebraic varieties such that $U$ is dense in $X$. Then any Zariski closed subset of $U$ is a trace of a Zariski closed subset of $X$. It escapes me why ...
15
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1answer
478 views

Is there an algebraic reason why a torus can't contain a projective space?

Let $X$ be an abelian variety. As abelian varieties are projective then $X$ contains lots and lots of subvarieties. Why can't one of them be a projective space? If $X$ is defined over the complex ...
5
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2answers
309 views

what does following matrix says geometrically

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} ...
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0answers
117 views

Length of $\frac{\partial }{\partial z}$ in Kähler geometry.

I am taking a Kähler geometry course this semester. The book we use is Tian's Canonical Metrics in Kähler Geometry. I got a little confused about the calculation there in. For example, $\mathbb{C}$ ...
4
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1answer
515 views

A question of Hirzebruch surface $ \mathbb{P}(\mathcal{O}(1) \bigoplus \mathcal{O})$

My question comes from my professor. I try my best to understand what the question means, but it doesn’t work! I even cannot understand the question meaning! I think I need some hints to answer the ...
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2answers
299 views

Non-degenerate quadratic form

I came across one sentence below, I am not able to see it... Any comment suggestion, reference is welcome. Thanks in advance. "Let $V$ be a finite dimensional inner product space. The ...
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1answer
283 views

volume form on a Riemann Surface

Suppose I have a Riemann Surface (i.e. an oriented manifold) and I have an integral that uses the volume density $|dx|$ instead of the volume form $dx = dx_1dx_2$ (here the $(x_1,x_2)$ are local real ...
5
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1answer
155 views

product of harmonic forms in a kähler manifold

In general, the product of two harmonic differential forms is not harmonic. However, for Kähler manifolds, the product of two harmonic forms is harmonic. What is a counterexample for the first ...
9
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1answer
240 views

Hodge theory for toric varieties

Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various ...
18
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1answer
2k views

Why is the hard Lefschetz theorem “hard”?

Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear ...
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215 views

Interesting applications of Kähler Identities

The Kähler identities give commutation relations between the Lefschetz operator ($\alpha \mapsto \omega \wedge \alpha$), the differential operators $\partial, \bar{\partial}$ and its adjoints ...
0
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1answer
188 views

Vector bundle morphism defined by cocycle

Holomorphic tangent bundle can be defined by cocycle of holomorphic Jacobians of transition maps. But this method will give different bundles which only agree up to isomorphism. I see in some text ...
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1answer
86 views

The $0$-section of sheaf

I ran into a problem with a defintion in Complex Analysis as follows: A sheaf $\mathscr S$ over a paracompact Hausdorff space $X$ with a map $f: \mathscr S \to X$ such that (1) $f$ is surjective and ...
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1answer
70 views

Does the restriction to hyperplanes determines a line bundle?

Supose $L$ and $L'$ are holomorphic line bundles over $\mathbb{CP}^n$ such that $L|_{H} \simeq L'|_{H}$ for every hyperplane $H \subset \mathbb{CP}^n$. Does it follow that $L \simeq L'$? ...
3
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2answers
447 views

Adjunction formula (Griffiths & Harris proof)

I'm having trouble understanding the proof of the adjunction formula on Griffiths & Harris book (p. 146). The formula states that if $V \subset M$ is a smooth analytic hypersurface then we have ...
2
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1answer
107 views

Example of a variation of Hodge structure such that the gradation does not vary holomorphically

The motivation for considering the Hodge filtration as opposed to just the grading is supposed to be that the filtration varies holomorphically in a family, whereas the grading does not always. Can ...
4
votes
2answers
413 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 ...
4
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1answer
997 views

Doing Complex Analysis on the Riemann Sphere?

Everyone: This is my first post. Sorry if I break some protocol. I do know some complex analysis and how to tell when a function from $\mathbb C \rightarrow\mathbb C$ . But I am confused when I hear ...
9
votes
2answers
506 views

Riemann surfaces are algebraic

Via a very ample divisor one can embed a Riemann surface holomorphically into some $\mathbb{P}^n$. Now, we can then project the Riemann surface to $\mathbb{P}^3$, and we can even go until ...
6
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1answer
162 views

Deformation of the Kähler structure on $CP^n$

Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as $$ ...
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2answers
226 views

Section of a holomorphic vector bundle

I am stuck on this problem. Let $E$ be a holomorphic vector bundle on a complex manifold $M$, and $N$ a complex submanifold of codimension at least two. Prove that every section of $E$ on $M\backslash ...
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3answers
391 views

Definition of the order of a meromorphic function

Let $X$ be a complex manifold and $Y \subset X$ a hypersurface. Let $x \in Y$ and $f$ a meromorphic function on $X$ near $x$. In Huybrecht's Complex Geometry the order of $f$ along $Y$ at $x$ is ...
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0answers
175 views

Definition of a complex analytic space

A complex analytic space is a topological space (say, Hausdorff and second countable) such that each point has an open neighborhood homeomorphic to some zero set $V(f_1,\ldots,f_k)$ of finitely many ...
2
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1answer
110 views

A question about complex manifolds

Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost ...
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2answers
310 views

Complex manifolds without compact submanifolds

It is well known that $\mathbb{C}^n$ does not admit any compact complex submanifold, I was wondering if this can happen for compact manifolds, i.e., does there exist an example of compact complex ...
2
votes
1answer
307 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
3
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1answer
443 views

Constructing the jacobian of a curve

I was just wondering, is there a general way to construct explicitly the jacobian of a curve, giving explicitly the vector space $\mathbb{C}^g$ and the lattice? (over the complex numbers is enough for ...
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1answer
209 views

Torsion elements in H^1 of a complex manifold

If $X$ is a compact complex manifold, the exponential sequence gives an injective map $H^1(X,\mathbb{Z}) \to H^1(X,\mathcal{O}_X)$. I think that this shows that $H^1(X,\mathbb{Z})$ is torsion free. ...
3
votes
2answers
376 views

Calabi-Yau Manifolds

In short, I'm hoping for some reading recommendations. I'm starting to do some work with Calabi-Yau manifolds, though my prerequisites are fairly minimal in differential geometry. I've taken a ...
2
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0answers
124 views

Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology. Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. ...
9
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3answers
752 views

Where can I read about elliptic operators on manifolds?

In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators: Let $P ...
0
votes
1answer
68 views

Differentiable connections

Let $E$ a vector bundle on a differentiable manifold and $D: E\rightarrow E\otimes \Omega^1$ an homomorphism, with $\Omega^1$ differential 1-forms. If I take the map $D\wedge D$ which is the target ...
2
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0answers
184 views

Construction of a symplectic basis for a lattice

Let $(T,E)$ be a polarized abelian variety ($T=V/L$, $\dim_\mathbb{C} V=g$, $E:V\times V\to\mathbb{R}$ a nondegenerate real alternating bilinear form, with $E(L\times L)\subseteq\mathbb{Z}$ and ...
3
votes
2answers
160 views

Complex sets of the form $|f(z)| = c$

I'm looking for information on sets of the form $|f(z)| = c$, possibly with additional restrictions (e.g. $|z| < d$). I can probably also assume that $f$ is as nice as you'd like. I'm interested ...
4
votes
2answers
163 views

Why is the Hodge-Deligne polynomial a polynomial?

Let $X$ be a compact complex manifold. Its Hodge-Deligne polynomial is then defined to be $\sum_{p, q \geq 0} (-1)^{p+q} h^{p, q}(X)$ where $h^{p, q}(X):= \mbox{dim}_{\mathbb{C}}H^{p, q}(X)$. The ...
2
votes
1answer
161 views

is the differential of the distance function holomorphic?

i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ ...
2
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1answer
328 views

Geometric Interpretation of Complexified Tangent Vectors on a Real Manifold

What is a good geometric way of thinking of complex tangent vectors on a manifold? I can convince myself that I understand tangent vectors by thinking of them as paths on the manifold. Is there a nice ...
4
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2answers
306 views

Equivalent definitions of Kähler metric

Two different ways to define a Kähler metric on a complex manifold are: 1) The fundamental form $\omega = g(J\cdot,\cdot)$ is closed, ie, $d\omega=0$; 2)The complex structure $J$ is parallel with ...
4
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0answers
259 views

Why the moduli space of complex structure in a compact complex manifold is of finite dimension

I admit the statement in the title might be much too unclear. I just heard from my teacher that we can form a finite dimension moduli space of all the complex structure in a compact complex manifold. ...
10
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4answers
168 views

holomorphic exoticness

A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...
5
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1answer
906 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 ...
2
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0answers
148 views

Describe Geometrically the action of a discrete subgroup on the complex Heisenberg Group

I am new to the study of lie groups, nilmanifolds etc. but the following question can be described very basically I think. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & ...
4
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1answer
194 views

Proof of Moisezon Theorem

We call a compact complex manifold Moisezon manifold, if its dimension coincides with the algebraic dimension, i.e. it has as many algebraically independent meromorphical functions as its complex ...
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1answer
163 views

Is there a “zero section” of any fibration of tori?

I'm sure this is a silly question, but suppose we have a proper holomorphic fibration $f : X \to B$ of complex manifolds where the fibers of $f$ are complex tori. Each torus $X_b = f^{-1}(b)$ has a ...
7
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1answer
231 views

Holomorphic mappings, Plücker's formula (Mistake in Rick Miranda's book?)

In Rick Miranda's book "Algebraic Curves and Riemann Surfaces", in order to prove Plücker's formula for smooth projective plane curves, he first defines a projective plane curve by the formula ...