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Questions tagged [holomorphic-bundles]

A complex vector bundle $\pi \colon E \to M$ is a vector bundle whose fiber bundles $\pi^{-1}(m)$ are a copy of $\mathbb{C}^k$. $\pi$ is a holomorphic vector bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic. The simplest example is a holomorphic line bundle, where the fiber is simply a copy of $\mathbb{C}$.

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Classifying space of finite group as a complex manifold

Suppose $G$ be a finite group. Then, is it always possible to construct a classifying space which is a finite dimensional(if not, possibly infinite dimensional) complex manifold? More precisely, are ...
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Complex manifold structure on moduli space of holomorphic bundles on Riemann surface

Consider the space $\mathcal N$ of rank $r$, degree $d$ holomorphic bundles on a compact Riemann surface $X$ of genus $g\geq 2$. Neitzke constructs this by considering $L^2_k$-completions and taking ...
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Finding a holomorphic representing section

Let $E\rightarrow M$ a holomorphic vector bundle over a complex manifold M, and $S\subset E$ a holomorphic subbundle. Let $\tilde{\eta}$ a local holomorphic section on the quotient bundle $Q=E/S$ (...
5 votes
1 answer
237 views

Algebraic vector bundles which are analytically but not algebraically isomorphic

I am looking for an example of two algebraic vector bundles on an algebraic complex manifold / smooth complex algebraic variety which are analytically isomorphic, but not algebraically isomorphic. By ...
• 2,532
1 vote
1 answer
96 views

Does a compact Kahler manifolds with negative first Chern class admits any nontrivial holomorphic vector field?

Let $M$ be a compact Kahler manifold with $c_{1}(M)\lt 0$. It seems that I can prove the following claim: there is no nontrivial holomorphic vector field on $M$. Here is my proof: let $T^{1,0}M$ be ...
1 vote
0 answers
47 views

Equivalent definition to holomorphic vector bundle

Let $\pi:E \rightarrow M$ be a complex vector bundle over a complex manifold $M$. Prove that if there exists a complex structure on $E$ as a manifold, such that the projection $\pi$ is holomorphic ...
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2 votes
1 answer
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What happens when we compactify every fibre of a holomorphic $\Bbb C$ bundle over a compact Riemann surface?

I am a beginner in learning about holomorphic line bundles over Riemann surfaces. In particular, I am a complete naïf as regards sheaf cohomology (so I hope the answer(s) won't involve that too deeply)...
• 1,156
4 votes
2 answers
267 views

Complex vector bundle that does not admit holomorphic structure

Is there any example of a complex vector bundle $E \to M$ whose base space $M$ is also a complex manifold, but where the total space $E$ does not admit a holomorphic structure compatible with the ...
• 1,755
5 votes
1 answer
315 views

Is flat connection holomorphic?

Let $M$ be a complex manifold and $E\to M$ be a holomorphic vector bundle. Let $\nabla:E\to E\otimes A_M^1$ be a flat connection, where $A_M^1$ stands for the sheaf/complex vector bundle of smooth 1-...
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3 votes
1 answer
151 views

Zero Chern class and trivial line bundle

Let $X$ be a compact complex Kähler manifold and $L$ an holomorphic line bundle over $X$. Let suppose $c_{1}(L) = 0$ in $H^{1, 1}(X, \mathbb{C})$. I would like to show that if $L$ is not the trivial ...
2 votes
1 answer
104 views

Are there holomorphic vector bundles that do not holomorphically embed in any free one?

Are there any examples of holomorphic vector bundles $E \to M$ of rank $k$ that do not admit a holomorphic classifying map $M \to \mathrm{Gr}_k(\mathbb C^n)$ for any $n \in \mathbb N$? Or, ...
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1 vote
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• 1,755
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1 answer
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Are holomorphic structures on complex vector bundles unique up to isomorphism?

I'm looking for an example of the following situation: $M$ is a complex manifold, $E_1, E_2$ are non-isomorphic holomorphic vector bundles on $M$, but $E_1$ is isomorphic to $E_2$ as a complex smooth ...
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2 votes
1 answer
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Clarifications on working with holomorphic vector bundles

I am at a loss as to several points made in A. Moroianu's "Lectures on Kähler Geometry", Chapter 9, section 9.2, from where I am trying to understand how to work with holomorphic vector ...
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3 votes
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Complex manifolds $X$ with $TX \to X$ smoothly but not holomorphically trivial

I am aware that there exist holomorphic vector bundles $E \to X$ that are smoothly trivial but not holomorphically so. For example, I think the following example works: let $X$ be a compact Riemann ...
• 1,755
2 votes
0 answers
79 views

Harmonic sections of $\Omega_X^p\otimes E$ over a Kähler manifold

I am confused by an exercise in Huybrechts Complex geometry: an introduction regarding sections of a certain holomorphic vector bundle over a Kähler manifold, related to Hodge theory. The exercise is ...
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2 votes
1 answer
70 views

Holomorphic forms on product spaces

Let $C$ be a compact Riemann surface without boundary. In particular, $C$ is a surface of a genus $g$. Then the vector space of holomorphic 1-forms $\mathrm{H}^0(C,\mathrm{K}_C)$, where $\mathrm{K}_C$ ...
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bundle valued form over $\mathbb{C}P^1$

Suppose $L$ is a line bundle over $\mathbb{C}P^1$, I wonder how to determine whether $H^{0,1}(\mathbb{C}P^1,L)$ is $0$ or not. I want to use kodaira vanishing theorem at first, but in my case the ...
• 1,157
1 vote
0 answers
45 views

Topological equivalent vs Holomorphic equivalent of line bundle.

I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows ...
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1 vote
1 answer
58 views

holomorphic line bundle over germ of complex space

I wonder if holomorphic line bundle over zero of holomorphic function(over $\mathbb{C}^n$) is trivial?(We can assume it to be a manifold if necessary) I Know there is a principle that, for Stein ...
• 1,157
2 votes
1 answer
193 views

When determinant line bundle is holomorphically trivial

I'm learning the deformation theory of holomorphic structure over given smooth vector bundle by the book Smooth four - manifolds and complex surfaces. However, when talk about holomorphic vector ...
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1 vote
1 answer
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