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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.

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Some problem of complex multi-function

Assume that $f:U\rightarrow \mathbb{C}$ is a holomorphic function on a connected open subset of $\mathbb{C}^n$. Then can we prove that for every point $x\in U$. There would be a local coordinates $(...
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Structure group simply connected implies first chern class vanishes

The situation is something like this: Let $M$ be a compact complex surface and $TM$ its tangent bundle. Assume that the structure group of $TM$ can be reduced to a simply connected, simple Lie group. ...
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Pulling-back functions that vanish of order one respectively two in $x$ yields a commutative diagram

Let $X$ be a complex manifold, $\sigma:\hat{X}\to X$ is the blow up of $X$ at $x$. Define $E:=\sigma^{-1}(x)$. $\mathcal{I}_{\{x\}}$ is the ideal sheaf at $x$. we compare the two exact sequences $0\...
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Finite etale coverings and torsion in Picard group

Let $X$ be a smooth projective complex manifold, suppose $L$ is a torsion element in Picard group of $X$, let $d>1$ be the order of $L$. In other words $L$ is a line bundle on $X$ and $L^{\otimes d}...
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How to linearize Quaternions?

Based on an answer to one of my questions and the comments exchanged here earlier I noticed that I cannot uniformly sample Quaternion vectors for rotation even though if I know the bounds of each ...
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A question about pullback bundle and sheaf

Let $X$ be a compact complex manifold, $\sigma:\hat{X}\to X$ is the blow up of a point $x\in X$. Let $E:=\sigma^{-1}(x)$ and $L\to X$ be a line bundle, then how to give a rigorous proof to show that ...
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Currents as differential forms and positivity

Currents can be regarded as "differential forms with distribution coefficients". My understanding is the following: Given a distribution $T$ we define a current $Tdx_I$ by $$ Tdx_I(\phi dx_J) = T(\phi)...
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Extension of holomorphic function to exceptional divisors

Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it. Let $\pi:E \to \...
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Flat G-bundles with canonical metrics: Computations

I am going through Corlette's paper "Flat G-bundles with canonical metrics", and I am having some difficulty with the computations of Proposition 2.1. Let $P$ be a principal $SL(n, \mathbb{C})$-...
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The fundamental form is real of type $(1,1)$

Given a real vector space $V$ with an almost complex structure $I$, and an inner product $\langle, \rangle$. Can define a fundamental form $(,):= \langle I(),() \rangle$. The claim is that is real of ...
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Simply connected surfaces and torsion in Grothendieck group

Let $X$ be a projective complex surface (complex manifold of dimension 2). In a paper I met the following claim: if $X$ is simply connected $\pi_1(X) \cong \{1\}$ then the torsion of the Grothendieck ...
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If $w_1$ and $w_2$ have unit modulus and arguments $0<\alpha_1<\alpha_2<\pi/2$, then $\arg(w_1-w_2)=\frac12(\alpha_1+\alpha_2-\pi)$

The complex numbers $w_1$ and $w_2$ have modulus 1, and arguments $\alpha_1$ and $\alpha_2$ respectively, where $0 < \alpha_1 < \alpha_2 < \frac{\pi}{2}$. Show that $\arg(w_1 - w_2) = \frac1{...
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The cokernel of $H^0(\hat{X},\sigma^*L^k)\to H^0(E,\mathcal{O}_E)\otimes L^k(x)$

Let $X$ be a compact manifold. $\sigma:\hat{X}\to X$ is the blow up of $X$ of $x\in X$. Denote $\sigma^{-1}(x)$ by $E$. $L^k\to X$ is a very ample line bundle. $L^k(x)$ is a fiber. And by $L^k\to X$ ...
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Different definitions of positive line bundle

Let $E$ be a holomorphic line bundle over a compact, complex manifold $X$. Then $E$ is said to be a positive line bundle if and only if there exists a hermitian metric $h_X$ on $X$ and a hermitian ...
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orthogonal complex structures on $\mathbb{R}^4$ and self-dual 2-forms

Let $V\cong \mathbb{R}^4$ be a Euclidean space, and let $\omega \in \bigwedge^2 V^*$ such that $||\omega||^2=2$ and assume further that $\omega$ s self-dual or anti-self-dual, i.e. $\ast \omega = \pm \...
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blowing-up preserves the first Betti number?

Let $G$ be a finite group and $S$ be a K3 surface. $G$ acts effectively and symplectically(fix the nowhere vanishing 2-form of $S$) on $S$. Since the action is symplectic,quotient surface $S/G$ has ...
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The symmetry group / isometry group of the complex projective space

question: For the complex projective space of $n$-complex dimensions, $$\mathbb{P}^n,$$ what is the symmetry group / isometry group of this complex projective space $\mathbb{P}^n$? Attempt: ...
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Application of Cartan's theorem

The question is as follows: Let $X$ be a Stein space. Show that any epimorphism $\mathcal{S}\rightarrow \mathcal{T}$ of coherent analytic sheaves induces an epimorphism $\mathcal{S}(X) \rightarrow \...
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Vanishing of totally holomorphic components of curvature tensor of complex Hermitian manifold

I'm trying to show that on a complex manifold $M$ with Hermitian connection the fully holomorphic components of the curvature tensor vanish, i.e. $R^\lambda_{\ \kappa\mu\nu}=0$. So far I have found $$...
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1answer
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Existence of isomorphism from the complex vector bundle to the dual

I know for a complex vector bundle $E\to X$, we have Chern class $c_i(E)=(-1)^ic_i(E^*)$. Therefore, in many cases, $E\to X$ and $E^*\to X$ are not isomorphic. I wonder if there exists some non-...
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Parametric/ homotopy version of Mergelyan theorem

Mergelyan's theorem says the following: Let $K$ be a compact subset of $\mathbb{C}$ with connected complement. Then any continuous complex-valued function on $K$ which is holomorphic in the interior ...
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Very ample line bundle isomorphic to a restriction of $\mathcal{O}(-1)$ under an embedding?

I know this claim this wrong since a very ample line bundle isomorphic to a restriction of $\mathcal{O}(1)$ under an embedding. But I can still construct an isomorphism below: Let $L\to X$ be a very ...
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a torsion-free connection that preserves a complex structure

Let $(M,I)$ be a complex manifold with a complex structure $I$, i.e. an endomorphism $I$ of the tangent bundle such that $I^2 = -Id$ and such that the subbundle $T^{1,0}$ of eigenvectors of $I \otimes ...
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1answer
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Is this exact: $0\to\mathcal{O}^{hol}\to\mathcal{O}(p)\to \Bbb{C}_p,$

Let $X$ be a Riemann surface and $p\in M$ some point. Let $\mathcal{O}(p)=\mathcal{O}((-p))(U)=\{f\in \mathcal{O}^{hol}(U)\mid f\text{ has a zero of order atleast 1 at p}\}$ I.e. we have the divisor ...
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Prove point of triangle lies on the base of other two similar triangles with complex system

I am trying to solve the following with complex numbers. In the diagram A,B,C are arbitrary points on the line $l$ and $\alpha$ is a given fixed angle. Isosceles triangles $APB$ and $BQC$ are ...
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1answer
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Projective algebraic manifold admits a positive line bundle

By the theorem: Let $X$ be a compact Hodge manifold. Then $X$ is a projective algebraic manifold, it follows that any compact complex manifold $X$ is projective algebraic iff it admits a positive line ...
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1answer
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are top holomorphic forms exterior products of holomorphic forms on almost complex manifolds?

Let $M$ be a manifold with a non-integrable almost complex structure, and let the form $\omega \in \Lambda^{n,0} TM$ be a holomorphic form, i.e. $\bar\partial \omega = 0$. Is it true then that there ...
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Singular cohomology of a limit of topological spaces

Let $(X_\lambda)$ be a filtered projective system of topological spaces, $X = \varprojlim X_\lambda$ and let $R$ be a finite ring, for example $R = \mathbb{Z}/l$. Assuming that $(X_\lambda)$ is ...
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What will be the graph of the straight line Arg (z-a)= $\pi$ + $\theta$.

I am not able of gaining proper geometry of the above equation, where a=m+ $\iota$ n, where m & n both are positive.
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Smooth automorphism preserves the Kähler class?

Let $S\subset \mathbb {CP^3}$ be a Kähler surface, $[\omega]$ be the Kähler class. Let $f:S\to S$ be a smooth orientation preserving automorphism (in the category of smooth manifolds, i.e. not has to ...
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A proving question based on DeMoivre's theorem.

Prove that$$\frac{1+\sin(1/8)π+i \cos(1/8)π}{1+\sin(1/8)π–i \cos(1/8)π} =\; –1$$ I tried to solve this by converting it into $e^{ik\alpha}$ but could not rationalize it please help me out.
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an exercise about integrability of almost complex structures

i have spent some time over the following problem from a problem sheet of a course on complex geometry. Let $M=G/H$ be a homogeneous space (where $G$ is a Lie group, $H$ a closed subgroup) and let $I$...
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Can we throw away points to make a Holomorphic injection into a homeomorphism?

Let $U\subset\mathbb{C}$ and let $\varphi:U\to\mathbb{C}^n$ be a holomorphic injection. Is it true that there is a discrete (or better yet finite) set $Z\subset U$ such that $\varphi|_{U\backslash Z}$ ...
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Why is $\bar\partial_E=\bar\partial+A^{0,1}$?

For a connection$\nabla$, we have $\nabla=d+A$, $\nabla=\nabla^{1,0}+\nabla^{0,1}$. In particular, for a Chern connection, we have $\nabla=\nabla^{1,0}+\bar\partial_E$, which means $\bar\partial_E=\...
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Isn't $∇^{0,1}=\bar\partial_E+A^{0,1}$?

A connection ∇ on a holo bundle $E$ is called compatible with holo structure if $∇^{0,1}=\bar\partial_E$. And such a connection is called a Chern connection. (reference) p.17 And we know $\nabla=d+A$....
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1answer
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Proving if $z$ is an n'th root, $\bar z$ is also an n'th root

Let $n>0$ be an even number, and let $z$ be an $n$'th root of a real number. Is $\bar z$ also an $n$'th root of this number? My answer is yes. The way I solved this was to consider a complex ...
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1answer
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Which point should I pick such that the shape is a parallelogram?

So I am stuck on the following problem from "Edexcel AS and A Level Modular Mathematics FP$1$": $z=\frac {1+7i}{4+3i}$ a Find the modulus and argument of $z$. b Write down the modulus and ...
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1answer
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Why is there no $\partial_E$?

On Hybrechts's book, there exists a natural linear operator $\overline{\partial}_E$: But why is there no ${\partial}_E$? Why doesn't ${\partial}_E:=\partial\otimes id_E$ make sense?
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Computation of cohomology with ideal sheaf involved.

Let $X$ be a complex projective surface an $Z\subset X$ be a finite set of points (reduced closed subscheme of dimension zero). Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$. Let $E$ be a vector ...
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Chern connections theorem problem

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 79 is this theorem: I don't understand the last proposition. namely the fact that $\nabla_Z(H(\sigma))=H(\...
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When are the vector bundles $E$ and $E^\vee \otimes \det(E)$ isomorphic?

Let $C$ be a smooth complex projective curve and let $E$ be a rank two vector bundle over $C$. If $E$ is decomposable, ie $E=L\oplus M$ for some line bundles $L$ and $M$ we have that $$ E^\vee \...
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1answer
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(Almost) complex manifold problem.

Can some one give me an exemple of an almost compplex manifold that is not a complex manifold and why? I know that an almost complex manifold is of even real dim and is orientable. I also heard that ...
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The definition of integral forms

A $d$-closed differential form $\phi$ on $X$ is said to be integral if its cohomology class in the de Rham group, $[\phi]\in H^*(X,\mathbb{C})$, is in the image of the natural mapping: $H^*(X,\mathbb{...
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1answer
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A differential form with integration $1$ has to be integral?

Let $X$ be a compact connected Riemann surface. If $\Omega\in H^2(X,\mathbb{C})$, s.t. $\int_X\Omega=1$, then why is $\Omega$ integral? A $d$-closed differential form $\phi$ on $X$ is said to be ...
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how do vanishing cycles change under blowups of the central fibre?

Let $X_t$ be a family of complex manifolds of dimension $n$ over a punctured disc $D^\circ=\{x \in \mathbb{C} \mid 0 < |x| < \epsilon\}$ and assume that we have chosen a model $\mathcal{X}_t$ ...
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(1,0)-forms/bundle problem

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 69 is this exercise: Can some one give me a hint? I'm kinda new to the subject.
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Holomorphic bundle - holomorphic structure problem

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 72 is this theorem: And the proof gose like this: And so on. My question is at the second to last proposition. ...
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Showing two tori are non-isomorphic as complex manifolds

I want to show that there exist two elliptic curves over $\Bbb C$ that are non-isomorphic. Let us write $\Gamma_1=\Bbb{Z}\oplus \Bbb{Z}\tau_1$ and $\Gamma_2=\Bbb{Z}\oplus\Bbb{Z}\tau_2$ where $\tau_1,\...
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Does the conjugate of $dz = d\bar{z}$ and complex differential forms

I've been calculating $|dz_1|^2$ of a function, $z_1 = \frac{e^{i\psi}z}{\sqrt{1+\left|z \right|^2}}$. I get that $dz_1 = \frac{ie^{i\psi}z}{\sqrt{1+\left|z \right|^2}}d\psi + \frac{e^{i\psi}}{\...
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1answer
47 views

Computation of cohomology of $End(TX)$ for a ruled surface $X$.

Let $C$ be a complex projective curve curve, let $E \longrightarrow C$ be a rank two vector bundle and let $X = \mathbb{P}(E)$ be the associated ruled surface. Then define the locally free sheaf $End(...