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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Induced sheaf morphism from space of sections
Let $M$ be a complex manifold and let $E$ and $F$ be two holomorphic vector bundles on $E$. Let $\rho:E\to F$ be a bundle map.
This map induces a map $\tilde\rho:\Gamma(E)\to\Gamma(F)$ onto the ...
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47 views
Complex structure of HyperKaehler manifold
Let $X$ be a hyperKaehler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
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1answer
201 views
Is there a geometric projection for every complex function
I was wondering about the best way to visualize complex functions. As they're $$ R^2 \rightarrow R^2$$ i think best way are complex plane image/grid transforms like they used in the Dimensions movie ...
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3answers
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Definition of vector bundle
Everywhere i see definition of vector bundle as triple $(E, p, B)$, $B$ and $E$ are manifold and local trivialization condition holds. For example see the definition here. .
Local trivialization ...
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4answers
293 views
Number of points at which a tangent touches a curve
My teacher told me that we are mistaken coming out of school that tangent tocuhes a point at one point. According to him, a tangent is just a special type of secant where two points share the same ...
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Complex atlas and Zorn's lemma, maximal complex atlas [duplicate]
Possible Duplicate:
Is Zorn's lemma required to prove the existence of a maximal atlas on a manifold?
Could any one explain me in detail how to prove the following statements in ...
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139 views
Trivial canonical bundle
Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$.
We know that according to Enriques–Kodaira classification, $X$ is either a torus or ...
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30 views
Inequality of numerical invariants of complex algebraic surfaces?
Let $S, T$ be algebraic surfaces over $k=\mathbb{C}$, and $\phi: S \longrightarrow T$ a surjective morphism. Furthermore we have the numerical invariants:
\begin{align*}
q(S) &:= \dim H^1(S, ...
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3answers
132 views
Geometrically interpret the set $\{z \in \Bbb C :$ $|z+a|+|z-a| \leq 2b,$ $b\in\Bbb R^+,$ $|a|<b\}.$
Geometrically interpret and determine the following set of complex numbers:
$$\{z \in \Bbb C : |z+a|+|z-a| \leq 2b , b\in\Bbb R^+,|a|<b\}.$$
I understand it means the sum of distance between ...
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1answer
64 views
Exact sequence of four sheaves in Beauville: associated l.e.s.?
This question is about an exact sequence of four sheaves on a smooth projective surface $S$ over $k=\mathbb{C}$, to be found in Beauville: complex algebraic surfaces, theorem I.4, page 3 (second ...
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1answer
130 views
(Continued:) finiteness of étale morphisms
I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem.
First question: Checking flat- and smoothness: ...
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1answer
61 views
Checking flat- and smoothness: enough to check on closed points?
I am currently studying varieties over $\mathbb{C}$, i know some scheme theory.
Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition ...
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2answers
154 views
Fibres in algebraic geometry: multiplicity
Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory.
My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X ...
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1answer
75 views
global irreducible decomposition of an analytic set
Let $M$ be a complex manifold (or a complex analityc space) and $Z$ be an analytic subset of $M$. By Noetherianity of the rings of germs of analytic functions at a point we know that $Z$ has finitely ...
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1answer
73 views
Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions…
Let X be complex curve (complex manifold and $\dim X=1$).
For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions vanish at the points $x_1$ and ...
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1answer
166 views
When is $CaCl(X) \to Pic(X)$ surjective?
I am curious about following similar statements in algebraic geometry and complex geometry:
Algebraic Geometry Version:
If $X$ is an integral scheme, the map from Cartier divisor group to ...
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1answer
109 views
Drawing elliptic curve
Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional ...
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64 views
Thom class and the Poincaré dual.
Let $X$ be a complex manifold and $Y\subseteq X$ a submanifold. It is well known that the Thom class of the normal bundle of $Y$ over $X$ is the Poincaré dual [Y]. I read that this result is ...
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1answer
219 views
How to properly use GAGA correspondence
currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces.
Now i understood that by GAGA, ...
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174 views
understanding this differential operator on a tensor product
I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
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1answer
160 views
A consequence of Runge's theorem
I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it.
Fact. Let $U \subseteq V \subseteq ...
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why do i need convexity of the set of positive definite hermitian matrices for hermitian structure on vector bundle?
Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$.
I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it ...
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1answer
102 views
When does the vanishing wedge product of two forms require one form to be zero?
Let $\alpha$ and $\beta$ be two complex $(1,1)$ forms defined as:
$\alpha = \alpha_{ij} dx^i \wedge d\bar x^j$
$\beta= \beta_{ij} dx^i \wedge d\bar x^j$
Let's say, I know the following:
1) $\alpha ...
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1answer
153 views
gauss-manin connection for curves
Let $\pi: X \to Y$ be a finite morphism between smooth projective curves over the complex numbers. I would like to known:
(1) what the Gauss-Manin connection with respect to $\pi$ (that is, the ...
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2answers
130 views
riemann surface and discontinuous group action
If G is a group that acts properly discontinuously on a Riemann surface X , than we can give to the quotient X/G a structure of Riemann surface such that the projection p:X→X/G is holomorphic. How ...
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1answer
30 views
Rational embedded irreducible curves in a complex surface.
Given a complex surface $X$ and an embedded irreducible compact curve $C$ with its arithmetical genus $g(C) = 0$, how can one show that $C$ is non-singular ?
Thanks for your answers!
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1answer
99 views
Holomorphic Poincaré conjecture
Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
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1answer
60 views
Compact complex surfaces with $h^{1,0} < h^{0,1}$
I am looking for an example of a compact complex surface with $h^{1,0} < h^{0,1}$. The bound that $h^{1,0} \leq h^{0,1}$ is known. In the Kahler case, $h^{p,q}=h^{q,p}$, so the example cannot be ...
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2answers
85 views
are fibres of a flat bijective map reduced?
Let $f: X \to Y$ be a flat map of algebraic varieties or of complex analytic spaces which is bijective on closed points (or just bijective in the secnond case). Suppose both $X$ and $Y$ are reduced. ...
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1answer
131 views
How to prove (0,1) form is not $\overline\partial$-exact
On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
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163 views
connections on coherent sheaves
Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. ...
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1answer
108 views
Wedge product $d(u\, dz)= \bar{\partial}u \wedge dz$.
How to show that if $u \in C_0^\infty(\mathbb{C})$ then
$d(u\, dz)= \bar{\partial}u \wedge dz$.
Obrigado.
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1answer
96 views
Complex structure v.s. conformal structure in more than 1 complex dimension
I've recently been learning some complex geometry, mostly for my own edification. In the course of my studies I came across the following statement:
If $X$ is a Riemann surface then a choice of ...
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2answers
126 views
When do we have that Absolute hodge classes= Hodge classes for complex projective manifolds?
If X is a complex projective manifold we have in general that:
Algebraic classes $\subseteq$ Absolute Hodge classes $\subseteq$ Hodge classes
In the case of abelian varieties it was proved that ...
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1answer
72 views
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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1answer
707 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 ...
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137 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
175 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
71 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, ...
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1answer
129 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 ...
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1answer
255 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 ...
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2answers
293 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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Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.
I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in.
For example, ...
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1answer
216 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
171 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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136 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 ...
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1answer
92 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 ...
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1answer
172 views
Find locus of points in the plane [closed]
Find the locus of points $(x,y)$ in the plane such that $\sin^2 x+\sinh^2 y=1$
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1answer
154 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 ...
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1answer
608 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 ...
