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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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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 frame ...
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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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Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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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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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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Picard Group coincides with Chern Classes via $H^1(X, \mathcal{C}_X ^*) = Pic(X)$

Let $X$ be a topological space, $\mathcal{C}_X$ the sheaf of $\mathbb{C}$-valued continuous functions. Let $Pic(X)$ the Picard groups, so the group of invertible line bundles $L \to X$ (or ...
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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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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_{\...
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Isomorphism of Holomorphic Vector Bundles

Given two holomorphic vector bundles $E,F \rightarrow X$ over a complex manifold $X$, an isomorphism is defined to be a holomorphic map $f: E \to F$ such that $f_x: E_x \to F_x$ is a linear ...
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Subbundles of Direct Sum of Holomorphic Line Bundles

Given a holomorphic line bundle $L$ on a Riemann surface $M\,,$ with $L^2$ nontrivial, when is it true that the only holomorphic subbundles of $$L\oplus L^{-1}$$ are $L\oplus\{0\}$ and $\{0\}\oplus L^{...
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meaning of positive line bundle $\mathcal{O}_X(Y)= (\mathcal{I}_Y)$*

I am reading Voisin Hodge theory part 2. On page 37 a condition to a theorem (theorem 1.29) is written as: positive line bundle $\mathcal{O}_X(Y)= (\mathcal{I}_Y)$*. Y a smooth hyper surface of X= $\...
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Which line bundles on a product of projective spaces are ample?

We know that $\text{Pic}(\mathbb{P}^r \times \mathbb{P}^s) \cong \mathbb{Z}^2$ via the pullbacks of the hyperplane bundles from the factors, namely $\mathcal{O}_{\mathbb{P}^r \times \mathbb{P}^s}(1,0)$...
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Exact sequences of vector bundles/spaces and isomorphisms between exterior powers

$$\DeclareMathOperator{\rk}{rk}$$ Remark. Given a short exact sequence of holomorphic vector bundles: $$0\to E\to F\to G\to0,$$ we cannot, in general, write $F\cong E\oplus G$, but it ...
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Roots of a canonical line bundle on a compact Riemann surface

Suppose we have a compact Riemann surface $X$ of genus $g$. Let $K$ denote the canonical line bundle on $X$, it's well known that $deg\ K=2g-2$. A square root of $K$ by definition is a holomorphic ...
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Bundle cohomology of a tensor product of line bundles

Consider a complex manifold $X$, and let $V$ be a holomorphic vector bundle on $X$ given by a Whitney sum of $n$ holomorphic line bundles $L_i$, i.e. let $V=\oplus_{i=1}^n L_i$. Further, if it is ...
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Sections of holomorphic vector bundles

Let $f:E\rightarrow F$ be a surjective morphism of holomorphic vector bundles on a complex manifold $X$. How I can prove that for each holomorphic section $s: U\rightarrow F$ there exists a section $t:...
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Available techniques for computing normal bundles of rational curves

If you have a complex manifold $Z$, and if you have a $\mathbb{C}P^1$ embedded in $Z$, what are the available techniques/methods for computing the normal bundle of such a $\mathbb{C}P^1$? I am aware ...
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Neron-Severi group= $H^{(1,1)}(X,\mathbb{Z})$

Where does my confusion arise from? I am getting $H^{(1,1)}(X,\mathbb{Z})=NS(X)$, the Neron-Severi group, however in any reference I don't find this remarkable thing. Let $X$ be a Kahler manifold. ...
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Natural line bundle over $\mathbb{P}^n$

What follows is from the book "Mirror Symmetry" by Hori et. al. From the definition of $\mathbb{P}^n$ we see there is a natural line bundle over $\mathbb{P}^n$ whose fiber over a point $l$ in $\...
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Vanishing of $H^0(X,\mathcal{L})$ and $H^0(X,\mathcal{L}^{-1})$

We have this fact: If $X$ is a smooth projective variety, and $\mathcal{L}$ is a line bundle, and $\mathcal{L}^{-1}$ its dual, then $H^0(X,\mathcal{L})$ and $H^0(X,\mathcal{L}^{-1})$ cannot both ...
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Why is every holomorphic line bundle over $\mathbb{C}$ trivial?

Why is every holomorphic line bundle over $\mathbb{C}$ necessarily trivial? I am having a hard time finding a proof to this seemingly innocuous fact. I have tried showing that such a line bundle ...
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1answer
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Determinants of exact sequence of bundles

Let $X$ be a complex manifold. Let $$0\rightarrow E\rightarrow F\rightarrow G\rightarrow 0$$ be an exact sequence of holomorphic bundles over $X$. Show that $\det F\cong\det E\otimes\det G$. I ...
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When are two line bundles isomorphic?

Probably not a very deep question follows. I struggle to understand the idea of the Picard group. I want to understand when two line bundles are isomorphic. Is there a nice illustrative example? For ...