# Questions tagged [vector-bundles]

A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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### Vector bundles on blowups

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\ker{ker}$Let $X$ be a (projective) variety, which is singular at a point $p$ of $X$ (we can also replace $p$ with some closed subvariety). Suppose $Y$...

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### Equivalence of two sections of some vector bundle

Given a nice manifold $M$ and a vector bundle $V$ over $M$. Let $u,v$ be 2 smooth sections of $V$. We say $u$, $v$ are equivalent if there is an isomorphism $f$ of $V$, so that $u=f^*v$. Is there any ...

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### Complex vector bundles on compact complex manifolds

The complex vector bundles on complex projective space $\mathbb{CP}^n$ are explicitly classified for low dimensions. When $n\leq 3$, they are exactly the holomorphic vector bundles; when $n\geq 4$ we ...

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### Derivations on the continuous functions of a manifold

For a manifold $M$ a vector field is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I ...

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### Existence of non-trivial "line-symplectic" manifolds

One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...

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### The module of section of a vector bundle and the module of section of the pullbackbundle are isomorphic [closed]

Let $\pi: E \rightarrow M$ be a vector bundle and let $f: N \rightarrow M $ be a continuous map. Let $f^{*} E$ be pullback bundle
Let $\Gamma(E)$ be the module of section of the vector bundle $E$ and ...

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### Eigenvalues of the Casimir operator and the space $L^2(E)$

I have read something about harmonic analysis on a homogeneous vector bundle in a very interesting paper Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds by J. DeWitt in my M.Sc. ...

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### Fundamental result on the projective tensor product of sections of a vector bundle

Definition Let $E_{1} \xrightarrow{\pi_{1}} M_{1}$, $E_{2} \xrightarrow{\pi_{2}} M_{2}$ be two vector bundles over $M_{1}$ and $M_{2}$ with fibers $V_{1}$, $V_{2}$ respectively. The exterior tensor ...

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### Manifolds and Riemannian geometry with a bundle viewpoint

I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...

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### Projective bundle is stable under twisting by a line bundle [closed]

I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)≅P(E⊗L)$".
The statement can also be seen on the ...

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### Surjective map from a globally generated vector bundle to its determinant

Let $X$ be a non-singular quasi-projective variety over $\mathbb{C}$ and $E$ be a globally generated locally-free sheaf on $X$ of rank $r$. Denote by $L$ the determinant of $E$. Does there exists $r-1$...

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### Stiefel Whitney number of a fiber bundle

I was going through this paper, and the author rights the following
The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...

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### First Chern form of line subbundle

Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...

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### Direct image of a sheaf with nowhere vanishing sections

Suppose that $f: X \to Y$ is a morphism of schemes over the complex numbers and $E$ is a vector bundle on $X$ such that all the sections of $E$ are nowhere vanishing sections. Furthermore, assume that ...

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### What is known about the moduli of stable rank 3 bundles on the projective plane?

What is known about the moduli space of stable rank $3$ bundles on the projective plane $\mathbb{CP}^2$?
Ideally, there is a concrete complex manifold which is a fine moduli space for such bundles for ...

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### Coherent sheaves that are isomorphic on every fibre

Let $X$ be a smooth projective variety and $E$ be a line bundle on $X$. Let $F$ be a line bundle on $X\times \mathbb{P}^1$. It is known that, if $p_1^*E|_{X_t}\cong F|_{X_t}$ for every $t\in \mathbb{P}...

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### Projectivization of a coherent sheaf using resolution by vector bundles

Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles.
Is there a way to ...

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### Behaviour of a certain full-flag bundle under a finite group quotient

Let $X$ be a smooth complex projective curve of genus at least 2, and let $\mathcal{M}(r,\xi)$ denote the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $\xi$. Let $\Gamma$ ...

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### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...

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### For each $k$, is there a vector bundle $E$ such that $E\oplus\varepsilon^k$ is trivial but $E\oplus\varepsilon^{k-1}$ is not?

A vector bundle $E$ is stably trivial if $E\oplus\varepsilon^k$ is trivial for some $k \geq 0$; here $\varepsilon^k$ denotes the trivial bundle of rank $k$. For such a bundle, let $s(E)$ denote the ...

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### Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...

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### Quantitative results for stabilizing tangent bundles of homology spheres

I'll begin with a broad question: if $M$ is a smooth manifold and $E \to M$ is a stably trivial bundle, can one determine lower bounds on the rank $k$ of the trivial bundle needed such that $E \oplus \...

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### Induced homomorphism on tautological line bundles $\mathcal{O}_E(1),\mathcal{O}_F(1)$

Let $E,F$ be two holomorphic vector bundles on a compact Kahler manifold $X$. Denote by $\mathbb{P}(E), \mathbb{P}(F)$ the associated projective bundles and $L_E=\mathcal{O}_E(-1), L_F=\mathcal{O}_F(-...

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### For a given $E\to X$ find $\xi\to \mathbb{P}E^*$ such that $\pi_*\operatorname{ch}(\xi)=\operatorname{ch}(E)$

$\DeclareMathOperator\ch{ch}$This is a specification of the question For given bundle $E\to X$ find $\xi\to Y$ such that $\pi_*ch(\xi)=ch(E)$.
Let $E\to X$ be a holomorphic vector bundle over a ...

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### Examples and properties of spaces with only trivial vector bundles

Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties ...

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### For given bundle $E\to X$ find $\xi\to Y$ such that $\pi_*ch(\xi)=ch(E)$

Let $\pi:X\to Y$ a projective morphism and $F\to X$ a vector bundle. The Grothendieck-Riemann-Roch theorem states that
$$\pi_*(ch(F)td(\pi))=ch(\pi_!F)$$
where $td(\pi)$ denotes the relative Todd ...

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### Global choice of eigenvectors on an open surface

Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...

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### About the construction of the associated complex vector bundle of an orbifold one

My question has to do with the general construction that associates to each complex orbifold vector bundle $\mathscr E\rightarrow\mathscr X$ over an orbifold Riemann surface, a complex vector bundle $...

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### A map relevant to polystable extension of holomorphic line bundles

Let $E$ be a rank two holomorphic vector bundle on a compact Riemann surface $X$. Consider the following extension of two holomorphic line bundles
$$
\mathbb{E}:\ 0\rightarrow L\stackrel{i}{\...

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### Are pure sheaves actually vector bundles on projective curves?

A coherent sheaf is pure if every non-trivial coherent subsheaf has the same dimension, where the dimension of a sheaf is the dimension of its support.
As in the title, I wonder if the notion of pure ...

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### Proving "by hand" Grothendieck-Riemann-Roch for $\mathcal{O}_E(1)\to \mathbb{P}E\to X$

Let $E\to X$ be a holomorphic vector bundle of rank $r+1$ over a compact Kahler manifold. Let $\mathbb{P}E\to X$ be the associate projective bundle and $\mathcal{O}_E(1)\to \mathbb{P}E$ the ...

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### Sheafification of presheaf of trivial vector bundles is the stack of vector bundles

This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...

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### Do we know anything about Harder-Narasimhan filtrations of tensor products of vector bundles?

I am interested in vector bundles over a nonsingular complete algebraic curve $C$ over $\mathbb C$. For a vector bundle $E$, its Harder-Narasimhan filtration is a filtration of subbundles
$$0=E_0\...

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### For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section?

Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is
$$p(t)=rt+r(1-g)+d$$
where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and ...

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### Characterize Hermitian-Einstein metric on $E$ using the tautological bundle $\mathcal{O}_E(1)$

Let $E\to X$ be a holomorphic vector bundle. Denote by $\mathbb{P}(E)\to X$ its projectivisation and $\mathcal{O}_E(1)\to \mathbb{P}(E)$ the associated tautological line bundle.
I would like to know ...

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### Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...

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### Surfaces of general type such that $\operatorname{Sym}^n \Omega_X$ is globally generated (but $\Omega_X$ is not)

Let $X$ be a minimal surface of general type. Recall that a vector bundle $\mathscr{E}$ on $X$ is called globally generated if the evaluation map of global sections $$e \colon H^0(X, \, \mathscr{E}) \...

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### Semi-stability of $S^n\Omega_S$ with respect to $K_S$

Let $S$ be a minimal compact complex surface of general type with ample canonical class $K_S$. In [1, Theorem 3] the following result is stated:
Theorem. Every symmetric power $S^n \Omega_S$ of the ...

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### Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic

Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...

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### Surfaces of general type with globally generated cotangent bundle

There is a lot of work about compact complex surfaces of general type $X$ having ample cotangent bundle $\Omega_X$: for instance, one can read the recent works of Damian Brotbeck and collaborators in ...

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### Why is the space of smooth sections complete?

This page is about the space of sections:
Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...

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### How to apply theories of line bundles to arbitrary bundles

Many concepts in geometry apply a priori specifically to line bundles. For certain theories like ampleness, nef, positivity I know that in order to generalize to arbitrary ranked vector bundles $E\to ...

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### Relative Todd class of projective bundle fibration

Let $E\to X$ be a holomorphic vector bundle of rank $r+1$ over a complex manifold $X$. Denote by
$$\pi: \mathbb{P}(E)\to X$$
the projectivization of $E$.
I would like to compute the relative Todd ...

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### What is the "right" tautological line bundle over a projective bundle

Let $E\to X$ be a holomorphic vector bundle over a complex manifold $X$ and denote by $\mathbb{P}(E)$ its projectivization. There is then a notion of a tautological line bundle $\mathcal{O}_E(1)\to \...

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### Number of small eigenvalues for flat unitary bundles

It is known that the number of small eigenvalues (eigenvalues less than $1/4$) of the Laplacian on hyperbolic surfaces may be topologically bounded from above. If $X$ is a finite area hyperbolic ...

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### Curvature of the tautological line bundle of a projectivized vector bundle

Let $E\to X$ be a holomorphic Hermitian vector bundle and $\mathcal{O}_E(1)\to \mathbb{P}(E^*)$ the tautological line bundle over the projectivization of $E^*$. In Demailly's Complex Analytic and
...

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### Identification of tangent spaces by parallel transport along geodesics [closed]

Given a geodesically complete manifold M, can we define a global identification of tangent spaces by starting from a base point, and parallel transporting along smooth geodesics? For this to be ...

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### Extension of holomorphic line bundles

Let $E$ be a rank two holomorphic vector bundle, consider the following extension of two holomorphic line bundles
$$
\mathbb{E}:\ 0\rightarrow S\stackrel{j}{\rightarrow}E\stackrel{g}{\rightarrow} Q\...

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### Is there a stable reduction for a family of vector bundles?

I. General Question
Consider a one-parameter family of vector bundles $E_t$ on a smooth projective variety $X$ with fixed Chern character $v$. Suppose $E_t$ is Gieseker stable when $t\neq 0$ and $E_0$ ...

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### A tale of two maps into a Grassmannian

I suspect that the answer to this question is well-known to the experts. However, I was not able to find it in the literature, so let me ask here.
Setup. In the sequel, $X$ is a compact complex ...