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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Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
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1answer
158 views

Holomorphic structures on associated bundles

Suppose $M$ is Kähler. Let $P \to M$ be the principal $U(n)$-frame bundle of $M$. Let $(\pi,V)$ be a finite dimensional unitary representation of $U(n)$ and let $E = P \times_\pi V$ be the ...
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3answers
418 views

Graphing the complex function

I'm looking for some software that my help me to graph some complex functions on unit circle. I.e. let say if I have $\ f(z)=1/(1-z)$ I want to see to give an input an image with unit circle and want ...
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1answer
179 views

Complex Space : Why unit disc?

This may sound a newbie question anyway I'm yet new to this area, In complex space for simplicity the properties of the functions on curves are sometimes considered on the unit circle on complex ...
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70 views

Complex structure of hyper-Kähler manifold

Let $X$ be a hyper-Kähler 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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233 views

Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in ...
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3answers
156 views

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
2k 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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0answers
59 views

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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327 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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40 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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200 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 ...
2
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1answer
86 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
427 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: ...
2
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1answer
154 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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298 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
124 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
84 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
234 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
191 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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0answers
146 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 to the homology class [Y]. I read that ...
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702 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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333 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
250 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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49 views

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
119 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 ...
6
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1answer
219 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
221 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
42 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
127 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
84 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 Kähler case, $h^{p,q}=h^{q,p}$, so the example cannot be ...
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99 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. ...
8
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1answer
214 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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2answers
253 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
198 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.
5
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1answer
232 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
230 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
132 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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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 ...
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402 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
341 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, ...
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1answer
188 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
492 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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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
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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, $\mathbb{C}$ ...
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1answer
541 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
304 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
288 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
158 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 ...