# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

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### IMO complex number geometry problems

I've been trying to master complex number geometry for some time and now I'm having a hard time finding problems suitable for complex numbers. Can anyone suggest some IMO or other olympiad problem ...
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### Geometry of the complex Gauge group

This is a pretty naive question: Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$. ...
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### When is a line bundle the pullback of another line bundle?

Let $X$ be a compact Riemann surface, $Y$ a smooth complex variety and $\pi : X \times Y \rightarrow Y$ the projection. Given a line bundle $L$ on $X \times Y$ which restricts to the trivial bundle ...
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### What is the coordinate definition of holomorphic vector fields?

We can write a holomorphic vector field in local co-ordinates as $X=X^i\dfrac{\partial}{\partial z_i}$, where $\dfrac{\partial}{\partial z_i}$ forms a local frame for $T^{(1,0)}M$. My questions are ...
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### Find the image of D on Möbius transformation.

I am not sure how to solve the following exercise on Möbius transformations: Let $D=\{z:|z-1|\le\sqrt{2}\wedge|z+1|\le\sqrt{2}\}$ and $f(z)=\frac{-2}{z+i}$. Find the image of the set D through the ...
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### Is a compact kahler manifold with non-zero Neron-Severi groups projective?

In Huybrechts' book 《complex geometry》 p251, there is a statement: A compact Kahler manifold $X$ is projective if and only if $K_X\cap H^2(X,\Bbb Z)\neq 0$, where $K_X$ means the Kahler cone of $X$. ...
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### Chern connection local structure

Let $E$ be a holomorphic vector bundle over a hermitian complex manifold $(X, J,h)$ where $J$ is the complex structure. It is well kown that for every hermitian holomorphic bundle $E \to M$ over a ...
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### Question about holomorphic maps between

Given a holomorphic map between compact Riemannian surfaces $f:S\to S'$, it's said that, for any $p\in S$, we can find local coordinate $z$ around $p$ in $S$ and $w$ in $S'$ such that the map $f$ is ...
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### Example of non-algebraic compact Kahler surface?

We call a Kahler manifold of dimension 2 a Kahler surface. Kodaira has proved a famous theorem:every compact Kahler surface is a deformation of an algebraic surface. We know every algebraic surface ...
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### How to see $S^6$ has a non-integrable almost complex structure?

Let $G_2$ denote the exceptional simple Lie group of 14 dimension and $SU(3)\subset G_2$. Consider $S^6\cong G_2/SU(3)$. $\textbf{Q1:}$ How do I see above isomorphism? $\textbf{Q2:}$ "From ...
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### Construction of a resolution for a coherent sheaf

Let $\mathcal{S}$ be a coherent sheaf over a complex manifold $M$. How do I construct a resolution of $\mathcal{S}$ by holomorphic vector bundles? Is this construction "unique"? Are the answers the ...
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### The proof of that the Riemman sphere is not a complex submanifold of $\mathbb{C}^n$

As is well known , according to the maximal principle we can easily conclude that every compact connected complex manifold of $\mathbb{C}^n$ degenerate to a point . From the above point of view, we ...
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### Visualizing a set of points in $\mathbb{C}^2$

As the title says, I want to find a way to visualize the set $S=\{p=(p_1,p_2)\in\mathbb{C}^2 : |p_1|=|p_2|\}$. I thought of maybe tori? is it a Riemann surface?. How could I find some other ...
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### Prove that the points A,B,C are collinear with the point $A$ lying between the points $B$ and $C$

Let $A(z_{1}),B(z_{2}),C(z_{3})$ be three points in a plane such that $$z_{1}|z_{2}-z_{3}|-z_{2}|z_{3}-z_{1}|-z_{3}|z_{1}-z_{2}|=0$$ Then prove that if the points A,B,C are collinear then the point $A$...
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### Show that every curve of genus 2 can be expressed as a fourth degree plane curve possessing a double point.

Show that every curve of genus 2 can be expressed as a fourth degree plane curve possessing a double point. This curve is of course a hyperelliptic curve. In order to find a map into $\mathbb{P}^3$, ...
Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves $$\... 0answers 24 views ### Convergence of F_2(z) I want to show that i) F_2(z)=\cfrac{1}{4}\pi i-\cfrac{1}{2}ln2+\left(\cfrac{z-i}{1-i}\right)+\cfrac{1}{2}\left(\cfrac{z-i}{1-i}\right)^2+\cfrac{1}{3}\left(\cfrac{z-i}{1-i}\right)^3+\ldots converges ... 0answers 14 views ### Fin the region of convergence of \int_0^\infty e^{-(z+1)^2t}dt. And compute the value of the analytic prolongation F_1(z) of z=2-4i. I tried to get the radio of convergence of the funcion F(z)=\int_0^\infty e^{-(z+1)^2t}dt with the root test, but I failed. I don't know how to find the "region" of convergence. Any help is welcome! 0answers 84 views ### Fibration and a morphism which is homotopic to a fiber bundle Let f:X\to Y be a surjective holomorphic map between two compact complex manifolds. Suppose that R^if_*\mathbb{Q}_X are locally constant sheaves over Y for all i, and f is homotopic to a C^{... 1answer 62 views ### Compute the value F(5). I am in trouble with the following problem from the Schaum's outline of Complex variables: "A function F(z) is represented in |z-1|<2 by the series \sum_0^\infty\cfrac{(-1)^n(z-1)^{2n}}{2^{2n+... 1answer 46 views ### zero extension of positive currents are always positive In Demailly's Complex Analytic and Differential Geometry page 139: He said the trivial (zero) extension of the positive current T (on X\setminus E), which denoted by \tilde T is always positive ... 1answer 71 views ### Does every path in SO(2k) from 1 to -1 pass through the space of complex structures? Recall the space of (normalised) complex structures \mathcal{J}_{2k} : = \{J \in SO(2k) \mid J^2 = -1\} on \mathbb{R}^{2k}. I am curious to know if every path from 1 to -1 in SO(2k) must ... 0answers 21 views ### Confusion regarding analytic extension of series. I am worinkg with the complex function \sum_{n=0}^\infty z^{3^n} . How can I prove that the function cannot be analytically continued past the unit circle. Any help is welcome! 1answer 48 views ### The differential of Kähler form The Kähler form is defined as$$k=-\frac{i}{2}h_{ij}dz^i\wedge d\overline {z^j} We differential the Kähler form to get the condition of Kähler manifold \begin{align} dk&=-\frac{i}{2}(\frac{\...
I have a problem where I need to find a locus of all points in the complex plane that satisfy $|z-ia|=\lambda|z+ia|$, where $z=x+iy$, and $\lambda>0$. I know I need to get a circle with the radius ...