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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. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

3688 questions
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Question about Hodge-Riemann bilinear relations

In the book, Principles of Algebraic Geometry, written by Griffiths and Harris, they prove the Hodge-Riemann bilinear relations for manifold $M$ with complex dimension 2, which can be found on page 125. In the proof, they claim that for the…
taiat
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Relation between topological and holomorphic Euler characteristics

Let $X$ be a compact Kahler manifold. We denote it's topological Euler characteristic by $\chi$ and it's holomorphic Euler characteristic by \begin{align*} \chi(\mathcal{O})=\sum\limits_i (-1)^i\text{ dim}H^i(X;\mathcal{O}) \end{align*} Here…
Partha
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Multivariable Cauchy integral formula from Griffith and Harris

I don't understand how we get the integral in line 11 of the proof. Line 11 is as follows: $$ \frac{1}{2\pi \sqrt{-1}} \int_{\partial \Delta_{\epsilon}} f(w)\frac{dw} {(w - z)} = \int_{\partial \Delta} f(w)\frac{dw} {(w - z)} + \int_{\Delta -…
joe blacksmith
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Understanding Hermitian metrics on complex vector bundles and almost complex manifolds

A Hermitian metric $H$ on a complex vector bundle $E$ is a smooth family of Hermitian products on each fiber $E_x$ satisfying $H(u,v)$ is $\mathbb{C}$-linear in $u$; $H(u,v) = \overline{H(v,u)}$ for all $u,v\in E_x$; $H(u,u)>0$ for all $u\neq…
string
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Nontrivial holomorphic vector bundle on $\mathbb{C}^n$

Does there exist any nontrivial holomorphic vector bundle on $\mathbb{C}^n$? I know $(1)$ Every line bundle on $\mathbb{C}^n$ is trivial, $(2)$ Every holomorphic vector bundle on $\mathbb{C}$ is trivial, (3) There exist a nontrivial line bundle on…
Daebeom Choi
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Conformal homeomorphism

I was reading this paper https://arxiv.org/pdf/math/0106036.pdf and it talks about conformal homeomorphism, can someone give me the definition please?
Claudio Delfino
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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 $\bar\partial_E=\bar\partial+A^{0,1}$. But by $\nabla\xi(f)=\sum…
Danny
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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?
Danny
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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 $\nabla_Z(H(\sigma))=H(\nabla_{\over{Z}}\sigma)$
Hurjui Ionut
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Embeddings Complex Projective Space

Is there a quick way to see that $\mathbb{R}^8$ is the smallest space where $\mathbb{C}P^2$ can be embedded. I know that $\mathbb{C}P^2$ is an adjoint orbit of $SU(3)$, and therefore $SU(3)$ acts adjointly on an $\mathbb{R}^8$ vector, is this enough…
seckin
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Biholomorphically equivalence of germs of hypersurfaces with isolated singularity

Let $X$ and $X'$ be two hypersurfaces in $\mathbb{C}^n$ that contain the origin $0$. Consider the following two statements: The germs of $X$ and $X'$ at $0$ are biholomorphically equivalent, that is they have biholomorphic representatives or…
G.-S. Zhou
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Extension of hermitian metric on tensor bundles

I'm reaing the Griffiths & Harris "Principles of algebraic geometry", and I'm blocked at page 80. Previously it has defined the hermitian metric $ds^2$ on a complex manifold $M$ as $$(\ ,\ )_z:T'_z(M)\otimes\bar{T'_z(M)}\longrightarrow\mathbb{C}$$…
user402793
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Example about hyperbolicity.

$\def\abs#1{\left|#1\right|}$I would like to understand this example: Why is the following set a hyperbolic manifold? $X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w < \abs{\exp(1/z)}\}$ It's an examples given in the book Hyperbolic…
Andrea
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Question about the proof of Lefschetz hyperplane Theorem

I am reading page $159$ of Principles of Algebraic Geometry and a bit confused about Lefschetz hyperplane Theorem. They write: Let $M$ be an $n$-dimensional compact, complex manifold and $V \in M$ a smooth hypersurface with $L=[V]$…
unicornki
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Definition of Hodge star operator

When I read Griffths and Harris, the Hodge star operator is an operator form $A^{p,q}(M)$ to $A^{n-p,n-q}(M)$. But in Huybrechts' book it is from $A^{p,q}(M)$ to $A^{n-q,n-p}(M)$, because it takes conjugate before wedge product. So which definition…
user330928
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