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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Are holomorphic maps regular maps of varieties?

Is a holomorphic map of complex algebraic varieties always a regular map?
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53 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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58 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
117 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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133 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 ...
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75 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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40 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 ...
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94 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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126 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 ...
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1answer
60 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 $p$ ...
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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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72 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 ...
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1answer
147 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 ...
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1answer
108 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 ...
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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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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?
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127 views

Why the total space of $\mathcal{O}(K)$ has trivial canonical bundle?

Let $X$ be a smooth variety, and $K$ be the canonical divisor on $X$. Let $\mathcal{O}(K)$ be the corresponding canonical sheaf. Then why the total space $Spec\mathcal{O}(K)$ has trivial canonical ...
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67 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
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1answer
81 views

Resolution of direct image functor

Let $i: X \to Y$ be an embedding of compact complex manifolds (not necessarily projective) and $E\to X$ a holomorphic vector bundle. I've seen it stated that the direct image sheaf $i_* E$ has a ...
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108 views

A question in Forster

Here is a question which has been bothering me for a week. It comes from Otto Forster's lectures on Riemann surfaces. The question comes from Exercise 10.1(c): Let $X$ be a riemann surface and ...
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About an automorphism of an algebraic curve.

Let $C \subset \mathbb{P}^2$ the Riemann surface given by the equation $X^6+Y^6+2Z^6=0$. Let $\phi:C \to C$ be the automorphism defined by $\phi([X:Y:Z])=[Y:X:Z]$ and consider $C/\tilde{}$, where ...
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35 views

The trivial loci of a family of sheaves is closed subvariety

Let $V, T$ be varieties, $V$ is complete, and $L$ be an invertible sheaf on $V \times T$. Then why $\{t \in T | L_t \rm{is~ trivial} \}$ is closed in $T$? The reason given is "Because this set is the ...
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SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or ...
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197 views

Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous ...
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1answer
43 views

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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198 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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34 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
156 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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309 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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50 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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532 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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1answer
66 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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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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166 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
85 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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38 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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102 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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76 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
67 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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25 views

Ituitive meaning of holomorphic bisectional curvature.

I would like to know the intuition behind the holomorphic bisectional curvature of Hermitian manifolds. I already know that the classical sectional curvature of a Riemannian (not necessarily complex) ...
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1answer
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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1answer
127 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
54 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 $g_1(z)=g(z)$?