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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Bode phase calculus with frequency to $0$ and $\infty$ in a quadratic term

The tangent to calculate is: $$\phi=-\tan^{-1}\left[\frac{2\zeta\frac{\omega}{\omega_n}}{1-(\frac{\omega}{\omega_n})^2}\right]$$ Where $\omega_n$ is constant. If $\omega=0 \to\phi=0$ ...
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211 views

Is every complex (smooth) manifold a scheme?

The question in the title doesn't quite make sense. I was always wondering if the scheme is the generalization of manifold. The precise statement should be like following: If $X$ is a complex ...
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35 views

Smooth compact subvariety from a germ of an embedding?

I am interested in the extent to which a germ of an embedding determines a subvariety given certain global hypotheses. My intuition is that the answer should be yes under more general conditions than ...
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2answers
157 views

Continuous complex functions

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq ...
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339 views

Learning Complex Geometry - Textbook Recommendation Request

I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in ...
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25 views

Comparing the norm of a trace of a curvature tensor with the full norm

Let $V$ and $E$ be complex vector spaces of dimensions $n$ and $r$, equipped with hermitian inner products $\omega$ and $h$ respectively. Let $R$ be a curvature-type tensor, that is an element of ...
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51 views

Question on zero locus of holomorphic function.

I am trying to figure out a statement in Girffiths and Harris' book "Principles of Algebraic Geometry: Given $f:U\rightarrow V$ a holomorphic map of open sets in $\mathbb{C}^{n}$, and let ...
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639 views

Show that the function $f(z)=1/z$ transforms a circle centered at the origin in the $xy$ plane in a circle centered at the origin in the $uv$ plane

Show that the function $f (z) = 1 / z$ transforms a circle centered at the origin in the $xy$ plane in a circle centered at the origin in the $uv$ plane. Someone can give me a clue to start the ...
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37 views

How do we calculate the Euler numbers of this

Suppose we are given two cubics X(a) and Y(a) in $CP^2$; $X(a)={ (4-a^3) xyz-a^3(x^3+y^3+z^3) =0 }$ $Y(a)={ a(x^3+y^3+z^3)-(2+a^3)xyz =0 }$ where a is a parameter in C satisfying $a^3 \not=1$ and ...
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68 views

quotient of 2-torus by antiholomorphic involution is annulus?

I would like to study what the quotient $$T^2 / \Omega $$ of a closed compact Riemann surface with $g=1$ handles, once a complex structure is chosen, over an antiholomorphic involution $\Omega,$ can ...
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55 views

Do there exist double points on a surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In ...
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167 views

Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
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1answer
88 views

Singular points of an analytic variety

I am struggling with a question that was posed here before, about why a reducible analytic variety $V=V_1\cup V_2$ must be singular in $V_1\cap V_2$. I must say I didn't really figure out the ...
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39 views

Bode diagram: phase of factor $(j\omega T)^{-+ n}$

The phase of the factor $(j\omega)^{1}$ is equal to: $$ tan^{-1}(\frac{\omega}{0})=tan^{-1}(\infty)=90^{o}$$ for every value of $\omega$ and $T$. But, if we have the factor $(j\omega)^{n}$, the ...
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107 views

Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
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22 views

definite integral of Abel-Jacobi map on Riemann surface

Suppose you're given the Riemann surface $$ 0= e^{-u}+e^{-v}+e^{u-v-t}+1,$$ where $u,v$ are complex variables. Can anyone explain what is the Abel-Jacobi map on this surface, what is its relation to ...
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84 views

Quotient of $\mathbb{CP}^1$ by antiholomorphic involution

On $\mathbb{CP}^1\ni(X_1:X_2)$ let $z=X_1/X_2$ be one of the two charts, and define an involution map $$I^-:z \mapsto -\frac{1}{\overline{z}}.$$ Question: how to prove that the quotient ...
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2answers
70 views

Residue of a complex function at some pole.

How can one visualize residue of a complex valued function at some given pole? I know how to find it. but I want to know its significance and its geometric nature. Why do we study it? thank you.
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23 views

the dimension problem of complex projection

Is it true that $\operatorname{dim }H^{0}(P^{n},P(T^{*}P^{n}))>0$? That is, is there a global holomorphic section? Here $P^{n}$ is $n$-dimensional complex projection space and ...
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139 views

Mapping unit disc onto upper half plane

How can I map the unit disc onto the upper half plane? I tried mapping $(1,i,-1)\rightarrow(1,0,\infty)$ using cross-ratio: $z\rightarrow \frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, but didn't give ...
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1answer
57 views

Finding local normal form of a holomorphic function

So I'm trying to find local coordinates to compute the local normal form of a holomorphic function. I have $f : \mathbb{P}^1 \to \mathbb{P}^1$ given by $f(z) = \frac{z}{(z-1)^2}$. Now we have a nice ...
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49 views

Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
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73 views

$\bar{\partial}$-Poincare' lemma in one variable

my question regards the proof, as it is described in Principles of Algebraic Geometry by Griffiths and Harris, on pages 5-6: why does last equality in last equation hold true, namely $g_1(z)=g(z)$?
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83 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
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48 views

k-differentials and their residues.

I am a theoretical physics student currently working on dualities in quantum field theories, which apparently are very well described by complex geometry. I have a question concerning this. In one of ...
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3answers
204 views

Compact Riemann surfaces are projective varieties.

We know that every compact Riemann surface is a complex compact manifold of dimention one. But why every compact Riemann surface is a projective variety?
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258 views

Complete linear systems of divisors

I looked up the web and did not find any good answer to this, so I thought about asking it here. Let $X$ be a smooth projective complex surface and $L$ a line bundle. The complete linear system $|L|$ ...
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1answer
123 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 ...
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98 views

The Wirtinger theorem proof

Let $M$ be a complex hermitian manifold with symplectic form $\Omega$ and $N \subseteq M$ be its smooth $2n$-dimensional (real dimensions, $2n \geq 2$) compact submanifold. I want to show the ...
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75 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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41 views

Linear Complex Structure and Kähler Angles

I am trying to read Donaldson's paper on symplectic submanifolds http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214459407 and am getting a bit ...
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126 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
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143 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?
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Is tautological bundle $\mathcal{O}(1)$ or $\mathcal{O}(-1)$?

I always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ...
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1answer
39 views

Meaning of CR-Automorphism

What is the meaning of the CR-Automorphism and CR-Manifold? I tried to find the definition from the web. Is it Continuous Real ....? Thanks.
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1answer
110 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 ...
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Miranda book Proposition 3.10 pag. 40

Let $F$ and $G$ be two holomorphic maps between Riemann surfaces $X$ and $Y$. If $F=G$ on a subset $S$ of $X$ with a limit point in $X$, then $F=G$. Choose two charts $(U_\alpha,\phi_\alpha)$ and ...
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43 views

geometry of Automorphism groups

It is proved that Aut group of a compact complex manifold is a lie transformation group. How do we show that a automorphism group of a tubular geometry is a lie group?
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89 views

Does the tangent bundle of an almost complex manifold split into the direct sum of $n$ two-dimensional $J$-invariant subbundles?

Is it true that for a manifold (of dimension $2n$) with an almost complex structure $J$, the tangent bundle splits as sum of $n$ $2$-dimensional invariant sub bundles? At least is it true that for a ...
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113 views

Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what ...
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1answer
124 views

Deformation of family of divisors

Given a family of complex compact surfaces $X \to B$ over a smooth curve $B$ and a one-dimensional family of effective Cartier divisors $\{D_t\}$ in the central fiber $X_0$. For simplicity, let's just ...
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113 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
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114 views

Isomorphism between Picard group and a Sheaf cohomology group

I would like to know how to prove that : $ \mathrm{Pic} ( X ) \simeq H^1 ( X , \mathcal{O}_{X}^* ) $. I specially want to know how to prove that $ \mathrm{Pic} ( X ) \to H^1 ( X , \mathcal{O}_{X}^* ) ...
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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 ...
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180 views

Questions on Chern characters.

Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf. How can one compute the Chern character $ch(\mathcal{O}_p)$? For vecoto bundles, we have $ch(V)\cup ch(W)=ch(V\otimes W)$, but ...
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113 views

When do local isomorphisms extend to global isomorphisms

Let $U$ be a smooth quasi-projective variety over $\mathbf C$, and let $V\subset U$ be a dense open subvariety. Let $X\to U$ and $Y\to U$ be smooth projective morphisms such that their restrictions ...
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1answer
86 views

A question on Aut$(N)$ and Aut$(N/G)$

Let $N$ be a complex manifold and $G$ is a finite group freely acting on $N$. Define another complex manifold $M$ as $M=N/G$. I would like to study Aut$(M)$, the (holomorphic) automorphism group of ...
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68 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...
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Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
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126 views

Finding two curves with intersection included in a specified neighborhood

Let $V$ be a complex projective surface and $U$ an analytic neighborhood of $x\in V$. How can I prove that there are two smooth curves $C_1$ and $C_2$ s.t. $C_1\cap C_2\subset U$ ?