Questions tagged [vector-bundles]
For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.
1,080
questions with no upvoted or accepted answers
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Show that $\mathbb{CP}^{2n}$ is not the boundary of a $4n+1$ dimensional Manifold using Chern classes
I'm currently studying with the Book "From Calculus to Cohomology" by Madsen & Tornehave(free PDF here).
Unfortunately I am really struggling to understand the Example 18.14 where Chern classes ...
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92
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Curved vector bundles and central extensions of $\pi_1$
Every flat vector bundle on a manifold comes from a representation of $\pi_1$, by the equivalence between flat vector bundles and local systems.
However, I've seen it mentioned that non-flat vector ...
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659
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Direct image of vector bundle under projection map
Let $\pi: Y \to X$ be a smooth projection map between manifolds, and assume the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~ \forall x \in X$. Given a smooth map ...
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275
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A compact complex manifold admits an ample line bundle if and only if it is projective
Given a holomorphic line bundle $L$ on a complex manifold $X$, a point $x\in X$ is called a base point of $L$ if $s(x)=0$ for all $s\in H^0(X,L)$ (the space of global holomorphic sections of $L$). The ...
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Line bundle on a curve is positive iff it has positive degree
Let $C$ a complex curve and $L$ a holomorphic line bundle on it.
I want to show that $L$ is positive iff it has positive degree.
Here the degree is defined as $\int_C c_1(L)$ and positive means that ...
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281
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Section of pullback bundle isomorphic to the sheaf pullback of sections
We have $p: E \to X$ an holomorphic vector bundle,where $X$ is a complex manifold and $f: Y \to X$ where $Y$ is anothet complex manifold. We can build up $f^* E$ pullback bundle on $Y$. Now,is it true ...
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Normal bundle of the diagonal
The normal bundle of a smooth submanifold $X \subset Y \subset R^N$ is defined as $N(X,Y)=\{(x,v) : x \in X, v \in T_xY, v \perp T_xX\}$. I want to show that $TX \cong N(\Delta,X\times X)$, where $\...
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Representation of fundamental group and flat bundle
My question is inspired by this discussion where the two notions of flatness for vector bundle are discussed.
I would like to understand the one-to-one correspondence between flat bundles over a ...
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165
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Invariant differential operators on equivariant vector bundles over Lie groups
This is a quick and dirty formulation of my question, with the hope that experts will quickly figure out what I am looking for and provide a reference.
Let $G$ be a Lie group, and let $\pi:E\to G$ be ...
4
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2
answers
793
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Local Form of Covariant Derivative Induced from a Connection one-form
Let $P\rightarrow M$ be a Principal G-Bundle with $E=P\times_\rho V$ the associated vector bundle with $\rho$ a representation of $G$ on $GL(V)$. Also let $\omega$ be a connection one-form on $P$ i.e. ...
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answer
401
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Sections of a Vector Bundle and Equivariant Maps on the Frame Bundle
Throughout we work in the smooth setting.
Let $\pi:E\to M$ be a rank $k$ real vector bundle and $F(E)\to M$ denote the corresponding frame bundle.
I am trying to understand the following statement, ...
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Is $df$ exact for a map between manifolds?
Let $M,N$ be $d$-dimensional Riemannian manifolds, $f:M \to N$ a smooth map.
Then $df \in \Omega^1\big({M,f^*TN}\big)=\Gamma(T^*M \otimes f^*TN)$.
Let $\nabla$ be the pullback connection on $f^*TN$ ...
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Do equivariant vector bundles always descend to the quotient?
Let $G$ be a topological group and $p:E\rightarrow X$ a $G$-equivariant topological vector bundle over the topological space $X$, where equivariant means that for any $v\in E$ and $g\in G$,
$$p(g\...
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Why is a Dirac operator involutive only if the curvature and torsion are in the image of the kernel of the symbol under exterior multiplication?
I am trying to understand the classic paper by Atiyah, Hitchin, and Singer
https://www.jstor.org/stable/79638
and I'm getting stuck on part of the proof of proposition 3.1. The proposition is ...
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810
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Riemann-Roch Theorem over Algebraic Surface
Let $X$ be an algebraic surface, $E$ be a rank $r$ vector bundle over $X$, then the Riemann-Roch formula
$$\chi(E)=r\chi(\mathscr O_X)+\frac12\left(c^2_1(E)-c_1(E)c_1(K_X)\right)-c_2(E),$$
holds.
Q:
...
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414
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Proving euler class of trivial bundle is zero
I am trying to prove that if $p:E \to B$ is a rank $n$ trivial bundle then the euler class is zero. The definition I am using for euler class is that it is a pull back of the Thom class in $H^n(E,E_0)$...
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856
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Easy way to see that the Chern class is the Poincaré dual of the vanishing locus of a section?
Let $\mathscr{L}$ be a holomorphic line bundle (perhaps smooth suffices, but I'm concerned with the holomorphic case anyway) on a smooth complex manifold $X$. We can define its Chern class via the ...
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Finite dimensionality of space of meromorphic sections with prescrbes poles
Disclaimer: I hope but do not guarantee I am using correct terminology!
Let $M$ be a compact complex manifold, and let $D$ be an effective divisor (by this I mean a finite formal sum over positive ...
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If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$
Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
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396
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When parallel transport determines a connection
Consider a vector bundle $E \to M$. Given a connection $\nabla$, it induces a parallel transport, which (in particular) is a choice of isomorphism $T_{\gamma} : E_{\gamma(0)} \to E_{\gamma(1)}$ for ...
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80
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Classifying vector bundles with a reduction of its structure group
Let $Bun(X)$ denote the set of equivalence classes of complex rank 2 vector bundles with a reduction of its structure group to $\mathbb{H}^*$. How can I proof that there is a bijection between $Bun(X)$...
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170
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Pullback of a semistable sheaf to a product is semistable?
Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $F$ be a $\mu_L$ semistable rank 2 vector bundle on $X$ (semistability in the sense of Mumford-...
4
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344
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Continuous Functor on Vector Spaces induces Functor on Vector Bundles
Let $T$ be a continuous functor between two different categories of finite dimensional vector spaces, $\mathcal{C}$ and $\mathcal{D}$.
Then we would like to define a functor between categories of ...
4
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1
answer
973
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Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)
Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$
Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
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190
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Extending a vertical vector to a vertical vector field
Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, ...
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254
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What is the co-kernel of the morphism of vector bundles?
Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$.
Suppose there is a surjection: $E\longrightarrow i_*A\...
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639
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Smooth sections of smooth vector bundle
Suppose that $E \to B$ is a (real for example) smooth vector bundle ($B$ is assumed to be a smooth manifold). There is a important notion of the smooth section $s:B \to E$: is has to satisfy $s(x) \in ...
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Homogeneous polynomials and line bundles
In Huybrechts' Complex Geometry text, he makes the following claim:
(Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map $(\mathbb{C}^{n+1})^{\...
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2k
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On the Chern connection
It is well kown that
If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection $...
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Pushing forward vector bundles on a plane curve via projection from a point
Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point
$$
\pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1,
$$
and restrict this ...
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How many charts?
My question is rather vaque but I hope that You will feel what I would like to know. A manifold (say: real, $n$-dimensional) is something which locally looks like an open set in $\mathbb{R}^n$. A rank ...
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253
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measurable function induces a measurable bundle
Greetings I am preparing a work on bundles and I found this statement
Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space).
If $G_k(...
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156
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Total space of a geometric vector bundle
Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X \...
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A question of extension of vector bundles.
Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong \...
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Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?
Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?
Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that $(...
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Flat Maurer-Cartan connection iff flat Berry connection
I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...
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If $M$ is a smooth manifold and $E \to M$ a vector bundle is it possible to recover $M$ from the sheaf $\Gamma$ of smooth sections of $E$?
If $M$ is a smooth manifold and $E \to M$ a vector bundle is it possible to recover $M$ from the sheaf $\Gamma$ of smooth sections of $E$?
This might need some additional conditions on $M$ such as ...
3
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Cohomology of twisted tangent bundle of fake projective plane
Let $X$ be a fake projective plane, id est $X$ is a smooth projective surface with the same Betti numbers of $\mathbb{P}^2_{\mathbb{C}}$ but not isomorphic to $\mathbb{P}^2_{\mathbb{C}}$. These ...
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When does a distribution admit a closed top-dimensional differential form?
Let $M^n$ be a smooth manifold. Let $\mathcal{D}$ be a $k$-dimensional integrable distribution on $M$. Denote by $\mathcal{D}^\bot\to M$ the vector bundle which is spanned by the differentials of the ...
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57
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Parametrizing unstable extensions of line bundles
Suppose $L$ and $M$ are holomorphic line bundles over the same compact Riemann surface $X$. I want to study which extensions of $M$ by $L$, i.e., vector bundles $E$ in a short exact sequence of the ...
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55
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Structure on moduli space of topological/smooth vector bundles
If $M$ is paracompact (let us assume smooth manifold of dimension $d$), then one has that the set of isomorphism classes of vector bundles of rank $n$ over $M$ is isomorphic to the set of homotopy ...
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Realization of the $2$-form as the curvature of some bundle.
Suppose $M$ be a manifold, and $\omega$ be any smooth $2$-form. My question is about the existence of the Lie group $G$ which satisfies that there is always a principal $G$-bundles $P$(or its ...
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Extending a section of vector bundle to union of open sets
I'm having some trouble with one part of Bott–Tu Theorem 6.8, which proves that homotopic maps induce isomorphic vector bundles under pullback.
The setup is that $\pi:Y\times I\to Y$ is the projection,...
3
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78
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Computing Theta Characteristics of Hyperelliptic Curves
Take a hyperelliptic curve $C$ (nonsingular, algebraically closed field) of genus $g$. The theta characteristic(s) of the curve are defined to be the line bundles $\{\theta\in Pic^{g-1}(C)|\theta^{\...
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97
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Line bundle of complex tori
Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus.
According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the ...
3
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47
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Determinant of certain rank-2 bundle on product of curves
Let $X_1,X_2\subset\mathbb{P}^n$ be two disjoint smooth projective and irreducible curves. Then we have a $\mathbb{P}^1$-bundle $B$ on the product $X_1\times X_2$ defined by $$B=\{(p,q,r)\in X_1\times ...
3
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0
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42
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Connections on indefinite inner product space bundles
Let $K$ denote a finite-dimensional vector space over $\mathbb{C}$. If $K$ is paired with a map $(\cdot,\cdot):K\times K\to\mathbb{C}$ such that, for all $\varphi,\psi,\chi\in K$ and $z,w\in\mathbb{C}$...
3
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40
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Why are these two definitions of conjugate vector bundle the same?
Let $E$ be a complex vector bundle over some space $X$.
Definition 1:
Let $E'$ be the complex vector bundle with the same total space as $E$, but with conjugate complex multiplication.
That is, if $\...
3
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0
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156
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Definition of uniqueness of tubular neighbourhoods
The tubular neighbourhood theorem states that if $M \subset N$ is an embedding of smooth manifolds without boundary and $\nu: E \to M$ is the normal bundle of $M$ in $N$, then there is a smooth ...