# 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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### Hermitian holomorphic Connection and Dolbeault operator

Let $E \to M$ a Hermitian holomorphic vector bundle of rank $m$ over an almost-$\mathbb{C}$ hermitian Kähler manifold $(M,h)$ of dimesnion $n$. By a well known theorem $E \to M$ has a unique ...
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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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### Kummer suface ; the resolution has a holomorhic (2,0)-form .

At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $\mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori . Now ...
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### Definition of holomorphic vector bundle using sheafs [duplicate]

Can somebody give me a general idea how to define a vector bundle of rank $N$ if we have a sheaf, which is locally given by $N$ holomorphic functions?
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### Direct image of sheafs

I am trying to understand the definition of direct images of sheafs in Hitchin lecture notes. Here $f:\tilde{M} \rightarrow M$ is a holomorphic map between Riemann surfaces. My problem appears in the ...
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### Higgs bundle and spectral curve

I have trouble understanding the following part in E. Wittens paper "More On Gauge Theory And Geometric Langlands": The context is the following: $(E, \varphi)$ is a Higgs bundle with structure group ...
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### Definition of Higgs bundle

I currently try to deal with the definition of a Higgs bundle: The definition is: $(E, \varphi)$ is called a Higgs bundle, if $E$ is a holomorphic vector bundle and $\varphi$ is a holomorphic 1-...
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### Holomorphic functions on the singular locus unique liftable

The example I would like to discuss in this question is introduced by Moret-Bailly in a MO question serving an example of a non-algebraic singularity: Let $U\subset \mathbb{C}$ be open. Choose two ...
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### Conditions for a Connection to be a Metric(or Chern) Connection

Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$, $$[De=\mathcal{A}e,]$$ where $e$ is a holomorphic ...
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### Question about equivalent definitions of (holomorphic) line bundles

Definition: A complex line bundle is a smooth manifold $L$ with a projection map $\pi:L\to M$, so that each fiber $\pi^{-1}(p)$ has the structure of a complex vector space of dimension 1. And for each ...
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### Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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### Canonical section of a Hirzebruch surface

What is the definition of the canonical section of the Hirzebruch surface $\mathbb{F}_2=\mathbb{P}(\mathcal{O}(-2)\oplus \mathcal{O})$?
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Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $... 0answers 41 views ### Can we ignore the holomorphic trivialisation'' in the definition of a holomorphic vector bundle? I have learnt two definitions about holomorphic vector bundles over a complex manifold$M$.$E\to M$is a smooth complex vector bundle with a trivialisation such that the transition functions are ... 0answers 63 views ### A homotopy where all intermediate maps have holomorphic antiderivatives I will denote by$\mathbb{C}^*$the punctured complex plane,$\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane,$w: \mathbb{C}^* \to \mathbb{C}$, such that the ... 0answers 54 views ### Every holomorphic vector bundle on a stein manifold is Nakano positive? I encounter this statement on page 53 of Takeo Ohsawa's L2 Approaches in Several Complex Variables but I don't know how to prove it. Could anyone explain it to me or give me a reference for it? 1answer 199 views ### Zeros of a global section on a holomorphic line bundle I'm studying an introduction to line bundles and I'm struggling with a particular proof. I'm following some notes that present this theorem: Let$ L \rightarrow X $be a holomorphic line bundle on a ... 0answers 57 views ### (1,0)-forms/bundle problem I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 69 is this exercise: Can some one give me a hint? I'm kinda new to the subject. 1answer 33 views ### Holomorphic bundle - holomorphic structure problem I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 72 is this theorem: And the proof gose like this: And so on. My question is at the second to last proposition. ... 0answers 66 views ### Deformation theory of holomorphic vector bundles in Donaldson-Kronheimer There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows: Proposition (6.4.3) (i) There is a holomorphic map$\psi$from a neighborhood of$0$in$H^1(\operatorname{End} \mathscr{E})...
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I see a statement of the theorem in Jonathan Evans' lecture note 13 An indecomposable Hermitian holomorphic vector bundle $\mathcal E$ on a Riemann surface $M$ is stable if and only if there is a ...
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### (Real) holomorphic vector fields on Compact Kähler manifolds

I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (pag. 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
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### Is $\bigwedge^{p,0}M$ a holomorphic vector bundle?

I have difficulties in understanding how to show that a vector bundle is holomorphic. For instance, how can I prove that $\bigwedge^{p,0}M$ is a holomorphic vector bundle, where $M$ is a complex ...
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### Let $M$ be a complex manifold. Why are the bundles $\bigwedge^{p, q}M$ not holomorphic vector bundles over $M$ for $q \neq 0$ ?

Let $M$ be a complex manifold. Can anyone help me understand why $\bigwedge^{p, q}M$ are not holomorphic vector bundles over $M$ for $q \neq 0$ ? I think it suffices to show that the transition maps ...
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### Pullback of local sections over the total space

Let $\pi\colon P\to X$ be some $G$-principal bundle; $\{U_\alpha\}$ a cover of $X$; and $\{s_\alpha\colon U_\alpha\to\pi^{-1}(U_\alpha)\}$ a collection of local sections. Claim: The pullback $\pi^*P$ ...
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### Rank and smooth section of a tautological vector bundle on $\mathbb{C}P^{1}$

The tautological vector bundle on $\mathbb{C}P^{1}=S^2$ is given by $L=\{(l,v) \ \vert \ l \subset \mathbb{C}^{2},\ \dim_{\mathbb{C}}l=1,\ v \in l\}$. How do I find it's rank as a real vector bundle; ...
### The definition of the line bundle $E^k\to P_1(\mathbb{C})$
Here is an example from the book of differential analysis on compact manifold. But I feel confused with using the transition function $g_{0,1}^k$ to define the line bundle $E^k\to P_1(\mathbb{C})$. ...