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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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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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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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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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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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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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 ...
What is the definition of the canonical section of the Hirzebruch surface $\mathbb{F}_2=\mathbb{P}(\mathcal{O}(-2)\oplus \mathcal{O})$?