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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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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Cyclic vector of local system

Let $E\rightarrow X$ be a holomorphic vector bundle over a compact Riemann surface with a holomorphic connection $\nabla:E\rightarrow E\otimes K$, where $K$ is the canonical bundle of $X$ ( you may ...
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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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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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Show that exists a linear connection $D$ such that $D''=d''$

I need an help, I want to understand this theorem. Statement: Let $E$ be a a holomorphic vector bundle over a complex manifold $M$. There exists a connection $D$ such that $D''=d''$ Proof: Let $U=...
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Generators of a Graded Algebra

I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425): We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$. This ...
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Why are holomorphic vector bundles over $\mathbb{C}^n$ trivial?

Let $E$ be a holomorphic vector bundle over $\mathbb{C}^n$. How do I show that $E$ is trivial? I know this to be true for vector bundles over affine varieties but I don't know how to extend the ...
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Integrating the top Chern class, where am I going wrong?

Consider the vector bundle $\mathcal{E}\rightarrow\mathcal{D}$ over the open unit complex disc, where the fiber over $z\in\mathcal{D}$ is the span of the vector $(z, 1)$. Consider the Hermitian ...
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Finding a conformal mapping to the unit disc, from the unit disc missing the x axis

I am revising conformal transformations, and am confused in general as to how to find one between two sets. I know that circlines map to circlines, but not sure how I can use that to help me with this ...
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Multiplication in Deligne cohomology: explicit formula for p = q= 1

In the very beginning of [1] the geometric meaning of Deligne cohomology $H^q(X, \mathbb{Z}(p))_D$ and multiplicative structure on it is being discussed. In particular, it is not hard to see that $H^q(...
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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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109 views

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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Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

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 $...
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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 ...
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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 ...
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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?
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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 ...
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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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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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A question about Narasimhan-Seshadri Donaldson theorem.

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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Three 'equivalent' definitions of a holomorphic vector bundle

I'm aware that similar questions have been asked here and here but neither of these seems to have settled my issue. I have the following three definitions of a holomorphic vector bundle (in all cases $...
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Complex Vector Bundle vs Holomorphic Vector Bundle vs Almost Complex Structures

I am writing because I am extremely confused with the structure of complex vector bundles. Ok first of all I understand that a complex vector bundle is a just a vector bundle $\pi:E\to X$ such that ...
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Holomorphic functions on Complex Space

Let $E$ be a holomorphic $m$-dimensional vector bundle over complex (analytic) space $X$ with projection $p:E \to X$. We now want to endow $E$ with sheaf structure $\mathcal{O}_E$ as follows: We set ...
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Extension of a vector bundle

I've got two isomorphic holomorphic vector bundle $E,E' \to X$, where $X=X'-\{x_0\}$ where $X'$ is a complex variety. I know that $E'$ is the restriction of an holomorphic bundle over $X'$. Can I ...
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conjugate bundle

The conjugate bundle $\bar{E}$ of a complex vector bundle E is defined as a complex vector bundle which is the same smooth manifold as E and same projection map but on each fiber has the complex ...
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Chern classes and the Jacobian

I am under the impression that a holomorphic line bundle is determined by its first Chern class. I mean this in the sense that $c_1 : H^1(X,\mathcal{O}_X^\ast) \to H^2(X,\mathbb{Z})$ is injective. I ...
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Simple question about Kahler polarization

Differential geometry: Let $S$ be a Kahler surface. That is a 4 real dimensional surface or a complex 2 dimensional one. Since by definition $S$ is complex it comes equipped with a Kahler 2-form $J$. ...
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The global sections of the holomorphic line bundles $E^k \to P_1(\mathbb{C})$

Here $U_{r,n}$ is the disjoint union of the $r$-planes ($r$-dimensional $\mathbb{C}$-linear subspaces) in $\mathbb{C}^n$. I am confused with how he used $f_i\circ\phi_i^{-1}$ to get $\mathcal{O}(P_1(\...
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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; ...
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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 $E^k\to P_1(\mathbb{C})$. ...
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Applying Riemann-Roch on the tangent bundle

Recently, I encountered the version of the Riemann-Roch theorem for line bundles $\mathscr L$ on a compact Riemann surface $X$: $$\dim H^0(X,\mathscr L) - \dim H^1(X,\mathscr L) = 1 - g + c_1(\mathscr ...
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Connection of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$

In this question- Connection of $\mathcal{O}(n)$ on a toric manifold, it is explained that the covariant derivative on $\mathcal{O}(n)$ is given by $$ \nabla=d+nA, $$ where $A$ is the connection on a ...
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How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$. For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{...
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250 views

holomorphic line bundles

Consider the unit disk $\mathbb D \subset \mathbb R^2$, and a holomorphic map $f :\mathbb D \to \mathbb D$. $df$ can be thought of as $df: T^{0,1}D \to f^*(T^{0,1}D)$(the pull back), that is, an ...
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Narasimhan-Seshadri and Weil's theorem

I am trying to make sense of how the following two statements fit together. Here, $X$ is a compact Riemann surface and $E$ is a holomorphic vector bundle of rank $r$ over $X$. Theorem 1 (Weil): Let $...
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the natural isomorphism $H^0(\mathbb P^n, T_{\mathbb P^n}(-1)) \cong H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(1))^\vee $

I have trouble in proving the natural isomorphism $$H^0(\mathbb P^n, T_{\mathbb P^n}(-1)) \cong H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(1))^\vee $$ where '$\vee$' stands for the dual space, and $T_{\...