# 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 ...
391 views

### 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 ...
129 views

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 $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\... -3 votes 0 answers 36 views ### The module of section of a vector bundle and the module of section of the pullbackbundle are isomorphic [closed] Let$\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$and ... -1 votes 0 answers 105 views ### Eigenvalues of the Casimir operator and the space$L^2(E)$I have read something about harmonic analysis on a homogeneous vector bundle in a very interesting paper Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds by J. DeWitt in my M.Sc. ... 3 votes 1 answer 127 views ### Fundamental result on the projective tensor product of sections of a vector bundle Definition Let$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}$respectively. The exterior tensor ... 2 votes 0 answers 99 views ### Manifolds and Riemannian geometry with a bundle viewpoint I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ... 1 vote 1 answer 90 views ### Projective bundle is stable under twisting by a line bundle [closed] I want to prove that "Given a bundle$E$, for any line bundle$L$the projectivizations of$E$and$E$tensor$L$are isomorphic i.e$P(E)≅P(E⊗L)$". The statement can also be seen on the ... 2 votes 0 answers 98 views ### Surjective map from a globally generated vector bundle to its determinant Let$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$... 2 votes 0 answers 139 views ### Stiefel Whitney number of a fiber bundle I was going through this paper, and the author rights the following The Stiefel-Whitney class of$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}\... 2 votes 0 answers 75 views ### 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} ... 0 votes 1 answer 194 views ### 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 ... 2 votes 0 answers 59 views ### 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 ... 2 votes 1 answer 151 views ### 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}... 2 votes 0 answers 123 views ### 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 ... 0 votes 0 answers 34 views ### 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 ... 4 votes 1 answer 115 views ### 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 ... 8 votes 1 answer 328 views ### 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 ... 2 votes 0 answers 49 views ### 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 ... 2 votes 1 answer 114 views ### 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 \... 0 votes 1 answer 117 views ### 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(-... 1 vote 0 answers 93 views ### 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 ... 7 votes 3 answers 476 views ### 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 ... 0 votes 1 answer 148 views ### 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)$$where td(\pi) denotes the relative Todd ... 3 votes 1 answer 190 views ### Global choice of eigenvectors on an open surface Let (M^2,g) be a noncompact orientable Riemannian surface without boundary. Let A \in \Gamma(\operatorname{Sym}(TM)) be a section of the bundle of symmetric endomorphisms of TM, that is, for ... 4 votes 0 answers 83 views ### About the construction of the associated complex vector bundle of an orbifold one My question has to do with the general construction that associates to each complex orbifold vector bundle \mathscr E\rightarrow\mathscr X over an orbifold Riemann surface, a complex vector bundle ... 2 votes 0 answers 41 views ### A map relevant to polystable extension of holomorphic line bundles Let E be a rank two holomorphic vector bundle on a compact Riemann surface X. Consider the following extension of two holomorphic line bundles$$ \mathbb{E}:\ 0\rightarrow L\stackrel{i}{\... 0 votes 0 answers 137 views ### 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 ... 2 votes 0 answers 188 views ### 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 ... 4 votes 1 answer 197 views ### 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$S$... 2 votes 1 answer 189 views ### 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 $$0=E_0\... 1 vote 1 answer 123 views ### For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section? Assume we are over \mathbb C. Let C be a complete algebraic curve, and E an algebraic vector bundle. Its Hilbert polynomial is$$p(t)=rt+r(1-g)+d$$where r=\mathrm{rank}(E) and d=\deg(E) and ... 0 votes 1 answer 103 views ### Characterize Hermitian-Einstein metric on E using the tautological bundle \mathcal{O}_E(1) Let E\to X be a holomorphic vector bundle. Denote by \mathbb{P}(E)\to X its projectivisation and \mathcal{O}_E(1)\to \mathbb{P}(E) the associated tautological line bundle. I would like to know ... 5 votes 0 answers 182 views ### Non-trivial line bundle on \mathbb{C}^{\ast} \times \mathbb{C}^{\ast} A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold. The complex manifold X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast} admits a non-trivial line bundle for the ... 3 votes 1 answer 271 views ### Surfaces of general type such that \operatorname{Sym}^n \Omega_X is globally generated (but \Omega_X is not) Let X be a minimal surface of general type. Recall that a vector bundle \mathscr{E} on X is called globally generated if the evaluation map of global sections$$e \colon H^0(X, \, \mathscr{E}) \... 4 votes 1 answer 230 views ### 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 ... 2 votes 1 answer 83 views ### 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$by ... 5 votes 0 answers 204 views ### Surfaces of general type with globally generated cotangent bundle There is a lot of work about compact complex surfaces of general type$X$having ample cotangent bundle$\Omega_X$: for instance, one can read the recent works of Damian Brotbeck and collaborators in ... 3 votes 0 answers 152 views ### Why is the space of smooth sections complete? This page is about the space of sections: Let$E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$be a smooth vector bundle. On its real vector space$\Gamma_{\Sigma}(E)$of smooth sections consider the ... 4 votes 0 answers 266 views ### How to apply theories of line bundles to arbitrary bundles Many concepts in geometry apply a priori specifically to line bundles. For certain theories like ampleness, nef, positivity I know that in order to generalize to arbitrary ranked vector bundles$E\to ...
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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 ...