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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Tangent fiber space of cone and vector fibered spaces

Suppose that $M$ is a complex analytic space. In the category of analytic spaces over $M$ one can consider cone and vector objects. Which I will call cones or linear fiber spaces over $M$. Every cone ...
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Question regarding $H^0$ of stable vector bundle

Let $M$ be a compact Riemann surface of genus $g$ and $E$ be a holomorphic vector bundle with the property that for any proper holomorphic subbundle $F$ we have $deg(F)/rank(F) < deg(E)/rank(E)$ ...
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Expression for the form $F^k\wedge \omega^{n-k}$

Let $E\to X$ be a holomorphic Hermitian vector bundle over a compact Kahler manifold. Let us write the curvature as follows $$F=\sum_{i,j}dz_i\wedge d\bar{z}_j\otimes f_{ij}$$ where $f_{ij}\in End(E)$ ...
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Examples of second fundamental forms for holomorphic vector bundles

I am looking for some hands on examples of short exact sequences of holomorphic vector bundles $$0\to S\xrightarrow{j} E \xrightarrow{p} Q\to 0$$ with explicit maps $j,p$ in order to compute ...
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Holomorphic sections of holomorphic vector bundles.

If $G:E\rightarrow F$ is a holomorphic bundle map over a smooth map $f:M\rightarrow N$ where $M$ and $N$ are two complex manifolds, there is a result that say that every local smooth section can be ...
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A nonzero holomorphic parallel section never vanishes

I'm reading a vanishing theorem from Kobayashis' differential geometry of complex bundles and I'm stuck with an argument. Let $E\rightarrow M$ a holomorphic vector bundle over a compact Kahler ...
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36 views

fiber of holomorphic bundle as pull back sheaf

Let $f:(X,\mathcal{R}_X)\to (Y,\mathcal{R}_Y)$ be the map between two topological space,with the sheaf of ring. Given a sheaf of $\mathcal{R}_Y-$module $\mathcal{N}$,we can define the inverse image ...
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Obtaining the Dolbeault operator on the pullbak of the holomorphic tangent bundle.

I have ran into a question while reading this paper by Witten. The question is mostly mathematical, so I thought it would be better posed here instead of on Physics SE. Let $X$ be a Kahler manifold, $\...
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Construction of a complex surface through line bundles

In §2 of the paper [1] below, there is something I don't quite understand fully. Here, we take $F(x_0,x_1,x_2)$ to be homogeneous of degree $2k$ with real coefficients, with the additional assumption ...
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Example of splitting vector bundle over projective line

Birkhoff-Grothendieck theorem states that any holomorphic vector bundle over $\mathbb P^1$ splits as a direct sum $E = \bigoplus \mathcal O_{\mathbb P^1}(a_i)$. I am trying to work out an explicit ...
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Proving that a quasi-isomorphism between complexes of trivial vector bundles is a cochain isomorphism.

Let $$ \cdots\rightarrow \mathcal{E}^{j-1}\xrightarrow{d_{j-1}}\mathcal{E}^{j}\xrightarrow{d_{j}}\mathcal{E}^{j+1}\rightarrow\cdots $$ be a complex of holomorphic vector bundles over a manifold $X$, ...
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Structure of the space of holomorphic structure on a vector bundle

It is well known that the space of connections on a vector bundle $E\rightarrow X$ is an affine space modeled on $\Omega^1(X,End(E))$. Let $Dol(E)$ denote the space of holomorphic structures $\bar{\...
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Determinant of the vertical tangent bundle of a projectivised bundle

Let $\pi : PE \to B$ be a projectivised rank $n$ vector bundle. In this MathOverflow answer, Michael Thaddeus says that for such a bundle, $T_{\pi} = \operatorname{Hom}(O(-1), E/O(-1)) \cong E(1)/O$, ...
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Confusion about Chern connection one form formula

Let $(E,h)$ be a holomorphic Hermitian vector bundle with local frame $\{e_i\}$. Denote by $H$ the matrix of $h$ in $\{e_i\}$. I found two different expression for the Chern connection 1-form $\omega$...
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Why is this bundle positive?

Let $E$ be a holomorphic bundle (with rank $r$) over a compact complex manifold $X$. $E^{*}$ will denote its dual. $\mathbb{P}(E^{*})$ will denote the projective bundle associated to the dual of $E$ ...
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The identification of $iX$ with $JX$

I am reading the book, Lectures on Kahler Geometry, by Andrei Moroianu. Here a link, https://books.google.com.hk/books?id=oqmroUc9E8YC&printsec=frontcover&hl=zh-CN&source=gbs_ge_summary_r&...
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Decision problem for a bundle in a line bundle extension of $\dim Ext^1=1$.

Let $E$ be a vector bundle over a nonsingular irreducible surface $X$ given by an extension $$0 \rightarrow A \rightarrow E \rightarrow B \rightarrow 0,$$ where both $A$ and $B$ are a line bundle over ...
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Does the natural exact sequence of the holomorphic jet bundle spilt?

Let $X$ be a complex manifold. Let us consider the jet bundle of the trivial line bundle on $X$. We denote it as $J_{1}(\mathbb{C})$. We have the short exact sequence: $$0\rightarrow\Omega_{X}\...
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35 views

Why $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\partial x}=0$?

I passed through this answer here and I don't understand the following. We are at the moment: If $u^2+v^2 \neq 0$, then $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\...
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Non-degenerate pairing in the context of holomorphic vector bundles

I'm trying to do exercise 2.2.3 on Huybrechts, Complex Geometry: Show that for any holomorphic vector bundle $E$ of rank $r$ there exists a non-degenerate pairing $$\Lambda^k E\times \Lambda^{r-k} E\...
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Triviality of holomorphic vector bundles over $\mathbb{C}$

Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle. I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
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Holomorphic subbundle and second fundamental form

Setup: Let $E\rightarrow X$ be a holomorphic Hermitian vector bundle and $S\hookrightarrow E$ a holomorphic subbundle. In Kobayashi's DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES, in proposition 1....
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130 views

Hermitian metric that induces a holomorphic splitting

Let $E\rightarrow X$ be a holomorphic vector bundle and $S\hookrightarrow E$ a holomorphic subbundle. Can we always pick a Hermitian metric $h$ on $E$ such that $S^{\perp}$ is holomorphic and the ...
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Lefschetz hyperplane theorem through Morse Theory in G-H p158

I am reading the Morse theoretic proof of the Lefschetz Hyperplane theorem in Griffiths-Harris and I am missing a transition. They claim that since the matrix $$\dfrac{1}{4}\left(\left(\dfrac{\partial^...
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Slope-stability: subsheaves vs. subbundles

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as $\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
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Checking that a line bundle generated by global section is positive

Let $L\rightarrow X$ be a holomorphic line bundle generated by global sections $s_1,...,s_k$. One can define a Hermitian metric on $L$ by $$|s|^2 = \dfrac{|\psi(s)|^2}{\sum_i |\psi(s_i)|^2}$$ in a ...
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Hermitian metric induced by short exact sequence of sheaves

Let $L\rightarrow X$ be a holomorphic vector bundle generated by global sections $s_1,...,s_k$. It is well known that there exists a Hermitian metric on $L$ defined in a local trivialization $\psi$ by ...
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536 views

Fubini Study metric on the complex projective space $\mathbb{PC}^m$

I study Andrei Moroianu's Lectures on Kähler Geometry and have some troubles to understand the proof of Lemma 7.1. The claim is that the tensor $h$ that gives some called Fubini–Study metric on the ...
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105 views

Why is $\int_M i \partial\bar \partial u \wedge \Phi^{n-1}$ zero?

I was studying vanishing theorems on holomorphic sections of holomorphic Hermitian vector bundles on Kähler manifolds. Here $(E,h)$ is a Hermitian holomorphic bundle on a compact Kähler manifold $M$. ...
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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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132 views

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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234 views

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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152 views

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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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 ...