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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23 views

Orthogonality and Chern classes in holomorphic bundles over the Riemann sphere

Let $E\to\mathbb{CP}^1$ be a smooth real vector bundle, $\langle\cdot,\cdot\rangle$ a Riemannian metric on $E$, $\nabla$ a metric connection on $E$. In the complexfication $E\otimes\mathbb{C}$, $\...
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Comparison of two line bundles

Let $L_1$ and $L_2$ be two line bundles on a complex smooth variety $X$. Suppose $L_1$ and $L_2$ have the same fibers on $X$ except on a prime divisor $D$ of $X$. How to prove that $$L_1 \cong L_2 \...
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map from $\mathbb{P}^{r_1} \times \mathbb{P}^{r_2} \rightarrow \mathbb{P}^{r_3}$ with finite fibers

For $n \in \mathbb{N}$, denote by $\mathbb{P}^n$ complex projective space, just as a set. Suppose we are given a map $\mathbb{P}^{r_1} \times \mathbb{P}^{r_2} \rightarrow \mathbb{P}^{r_3}$ that has ...
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Can positivity of currents implies positivity of forms?

Let $\alpha$ and $\beta$ be 2 continuous (or smooth) forms of $(1,1)$-type on a complex manifold $X$. Of course they can be considered as currents. Assume $\alpha\geq \beta$ in the sense of currents. ...
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Line bundles on projective space $K\mathbb{P}^n$

Let $K = \mathbb{R}, \mathbb{C}$. The tautological bundle $O(-1) \to K\mathbb{P}^n$ can be explicitly described as the set $$O(-1) = \{ ([x], v) \in K\mathbb{P}^n \times K^{n+1} \ \vert \ v \in K \...
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First Chern Class $c_1$ of $O(1)$ Bundle over $\mathbb{CP}^n$

Let $\mathbb{CP}^n$ the complex projective space and $U_0:= \{[1: z_1:...:z_n] \in \mathbb{CP}^n \ \vert \ z_i \in \mathbb{C} \} \subset \mathbb{CP}^n$ a standard open subset wrt first cordinate. ...
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How to show that two arcs are parallel with respect to poincare metric of the unit disc?

Show that two circular arcs in the unit disc with common end points on that unit circle are noneuclidean parallels in the sense that the points on one arc are at constant distance from the other. For ...
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1answer
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complex non algebraic manifold local ring of holomorphic functions is noetherian?

Consider $X$ a complex manifold. Denote $x\in X$ a point and $O_x$ as the local holomorphic function ring at $x$. Assume $X$ is not algebraic. $\textbf{Q1:}$ Is $O_x$ Noetherian? If it is ...
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Question about codimension of a special variety.

Let $G(k,n)$ denote Grassmannians. For a hypersurface $W\subset \mathbb{P}^n$ of degree $2$, we let $\tau(W)\subset G(2,n+1)$ denotes the set of lines in $\mathbb{P}^n$ lying on $W$. $\tau(W)$ is an ...
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estimates for tensors using local coordinates

Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
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Alternate proofs for Classifying complex structures on $\mathbb{R}^2$

We know that $\mathbb{R}^2$ admits 2 integrable almost complex structures (upto biholomorphism), one coming from a diffeomorphism with the disk and other being the standard complex structure on $\...
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Well-definedness of pull-back of divisor

I read this paragraph from Complex Geometry by Huybrechts: ...The pull-back of a divisor $D$ under a morphism $f:X\rightarrow Y$ is not always well-defined, one has to assume that the image of $f$ ...
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Question about restriction of holomorphic vector bundle.

There is a statement as follows: For $E\to M$ a holomorphic vector bundle and $V\subset M$ a subvariety, the kernel of the restriction map $\mathcal{O}_M(E)\to \mathcal{O}_V(E)$ is the sheaf of ...
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Lifting of meromorphic function along a finite morphism

I am currently reading the book "Geometry of algebraic curves II", by Arbarello, Cornalba and Griffiths, and I am having some difficulties understanding a passage p.105. The setting is the following:...
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Question about “G.A.G.A” theorem.

In the book Principles of Algebraic Geometry, there is a theorem: Every meromorphic function on an algebraic variety $V\subset \mathbb{P}^n$ is rational. And the proof of this assertion is in two ...
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Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity: $$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
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Wedge product of complexifications is complexification of wedge product

$V$ is a real finite dimensional vector space. Denote by $V_{\mathbb{C}}$ its complexification $V\otimes_{\mathbb{R}}\mathbb{C}$. I have already proved that $V_{\mathbb{C}}\otimes_{\mathbb{C}}V_{\...
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1answer
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Hodge star operator and exterior calculation

I am learning complex geometry by D. Huybrechts. Here is a formula that I can't understand $$\omega \wedge \beta\wedge \star \alpha=\beta\wedge(\omega \wedge \star \alpha)\tag 1$$ Here $\omega$ is ...
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Question Regarding Proof of Hodge Index Theorem

I am reading Voisin's proof of the Hodge Index Theorem on pp. 153-154 of her Hodge Theory and Complex Algebraic Geometry I. The proof is mostly clear except for one technical point. Let $n$ denote an ...
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1answer
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Fubini-Study metric on $\mathbb{CP}^n$

On $\mathbb{CP}^n$, we have $\phi_{\alpha}([z^1,...z^{n+1}])=(\omega_{\alpha}^1,...,\omega_{\alpha}^n)$ where $$\omega_{\alpha}^i=\begin{cases} \frac{z^i}{z^{\alpha}}, & \text{if $1\leqslant i \...
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Is Atiyah's periodicity Theorem related to splitting principle?

I assumed all vector bundles are over complex number. Let $V$ be a vector bundle over $X$. Then $P(V)$ denotes the projectivization of fibers of $V-0$ as a projective space bundle over $X$. Denote $K(...
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1answer
39 views

A complex vector bundle $E$ is a holomorphic vector bundle iff $(\overline{\partial^E})^2=0$ help with proof?

Ok so I am following a set of notes on complex differential geometry and there is a theorem that says the following: If $E$ is a complex vector bundle over a complex manifold and $\overline{\partial^...
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1answer
109 views

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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1answer
63 views

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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1answer
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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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$ \mathrm{ch}(F(A)) = \mathrm{ch}(F(B)) \implies \mathrm{ch}(A) = \mathrm{ch}(B) $ for autoequivalences?

Let $\mathcal{C} \subset D^b(X)$ be a subcategory of the derived category of coherent sheaves on a smooth projective variety $X$. Let $F : \mathcal{C} \to \mathcal{C}$ be an autoequivalence. Let $\...
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Direct proof for $\mathcal{O}_{\mathbb{CP}_{1}}(-2) \cong T^{*}\mathbb{CP}_{1}$

Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the ...
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Extension of a holomorphic function on a complex submanifold

Let $ M $ be a second-countable and Hausdorff complex manifold, and $ N $ be its complex submanifold. That is, we assume that for all $ x \in N $ there exist an open neighbourhood $ U $ of $ x $ in $ ...
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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 $\mathbb{P}^3$, ...
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1answer
76 views

Simple example of non-integrable holomorphic connection

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 $$\...
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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 ...
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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!
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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^{...
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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+...
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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 ...
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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 ...
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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!
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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{\...
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1answer
26 views

Locus of points in complex plane

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 ...

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