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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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves

The following passage is from a thesis I'm reading: Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
0 votes
0 answers
77 views

I'm looking for the NLab page on particle species

This is just a reference request. I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently. If someone can point ...
2 votes
1 answer
157 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
2 votes
0 answers
125 views

Cohomology of equivariant toric vector bundles using Klyachko's filtration

I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties. Whereas detailed literature ...
5 votes
1 answer
230 views

Bochner Laplacian in coordinates

Sorry if this is a too basic question, but I didn't find an answer anywhere: The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
1 vote
0 answers
71 views

Are coherent modules with integrable log-connections locally free?

Let $X$ be a smooth Noetherian scheme over a field $K$. It is known that every coherent module with integrable connection on $X$ is locally free. Is the same true for coherent modules with log-...
21 votes
6 answers
3k views

A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
2 votes
0 answers
374 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
1 vote
1 answer
423 views

Vector bundles on $\mathbb{P}^1$

I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
1 vote
0 answers
89 views

Stable equivalence and stability theorem of vector bundles

I am going through this paper by Tanaka. In the proof of Proposition 3.2(1) given below The author says that by the stability theorem as $\dim (B)\le m$ we have $\alpha\oplus1\cong m\oplus1$. But I ...
3 votes
1 answer
118 views

Derived flat bundles

I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
4 votes
0 answers
93 views

category of vector bundles with connections and its K-theory

For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
8 votes
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296 views

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 $H$-...
3 votes
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73 views

Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
6 votes
0 answers
142 views

Extending topological vector bundles and obstruction theory

This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
3 votes
1 answer
165 views

Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism

Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle $$ \mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
0 votes
1 answer
636 views

Slopes of the Harder-Narasimhan filtration of a subbundle

Suppose $(p_1, ..., p_n)$ and $(q_1, ..., q_n)$ are two (non-strictly) decreasing sequences of positive real numbers. We say that $(q_1, ..., q_n)$ dominates $(p_1, ..., p_n)$ if $q_n \geq p_n$, $q_n+...
2 votes
0 answers
57 views

Rank and determinant of the image of a vector bundle after applying a Schur functor?

Let $\mathcal{E}$ be a vector bundle of rank $r$ and degree $d$ over some smooth projective variety $X$. Furthermore, let $\lambda$ be a partition of $n$. We apply the $\lambda$-th Schur functor to $\...
2 votes
0 answers
104 views

Reference for numerically non-negative polynomials for nef vector bundles

Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
4 votes
2 answers
310 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
0 answers
64 views

Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$

I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well. Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:...
0 votes
1 answer
260 views

Self-intersection of zero section of line bundle over elliptic base curve

Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
3 votes
2 answers
390 views

Obstructions to the existence of a flat connection on a vector bundle

Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
0 votes
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220 views

A question on the proof of pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes

Here is a question about proving the pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes Bundles, Homotopy, and Manifolds. The proof is in page 73 of the notes. For me, ...
0 votes
0 answers
223 views

Pushforwards in vector bundles over a topological spaces

I have been reading the discussion from Pushforward and pullback.. I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
0 votes
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129 views

A question about vector bundle isomorphisms

Let $E\to X\times[0, 1]$ be a vector bundle. Define a map $i_t:X\to X\times[0, 1]$ by $i_t(x)=(x, t)$. Then we know the vector bundles $i_0^*E$ and $i_1^*E$ are isomorphic, and probably there are many ...
2 votes
0 answers
139 views

The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds

$\def\sO{\mathcal{O}} \def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
2 votes
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93 views

Differential graded modules and the Serre-Swan theorem

I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules. If $E \rightarrow B$ is a vector bundle, or even just any ...
1 vote
0 answers
42 views

Homology groups of moduli of parabolic bundles with fixed determinant

I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant. In particular, what is the second homology group of such ...
7 votes
1 answer
265 views

Non-homogeneous line bundles over a homogeneous space

Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form $$ G \times_{\...
1 vote
0 answers
62 views

The curvature of the induced connection on the antidual bundle

Let $E\to M$ be a complex vector bundle over a (real, smooth) manifold and $\nabla$ a connection on $E\to M$ whose curvature is $R$. From Section 1.5 of "Differential Geometry of Complex Vector ...
4 votes
2 answers
370 views

Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?

Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$? Motivation: I had intention to consider this question ...
3 votes
0 answers
248 views

Equality of topologies in the spaces of section of a vector bundle

In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...
2 votes
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239 views

The definition of the determinant of a coherent sheaf

Let $ X $ be a smooth (projective) variety and $ \mathcal{F} $ a torsion-free coherent sheaf of rank $ r $ on $ X $. The determinant $ \det \mathcal{F} $ can be defined by (1) $ \det \mathcal F := ( \...
2 votes
0 answers
414 views

Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?

Let $M$ be a smooth manifold and let $A=C^\infty(M).$ We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
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122 views

Rank of a tangent map related to holomorphic line bundles

Let $L,\,\,J$ be two holomorphic line bundles over a compact Riemann surface $X$ of genus $g_X>0$ such that (1) $d_1:=\dim H^0\big(\operatorname{Hom}(L,J)\big)>0$ and $d_2:=\dim H^0\big(\...
2 votes
1 answer
98 views

Normal bundle of veronese as iteration extension of symmetric powers

In this post, an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But ...
2 votes
0 answers
106 views

Unicity of modifications of vector bundle on a regular base

I think that I have overheard the following statement, and would be grateful for either a reference or an explanation about why hearing must be slightly faulty and a clarification about what must ...
1 vote
0 answers
80 views

Algebraizable image of a morphism of Galois cohomology stacks

Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
5 votes
2 answers
345 views

Exterior differentiation of foliations

Let $M$ be a differentiable manifold. Let $T^*M$ be the cotangent bundle of $M$. Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
0 votes
1 answer
162 views

Equivariant sheaves on $\mathbb P^1$

Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
2 votes
1 answer
347 views

Pullback of a vector bundle extension class

Let $ X $ be a variety with an automorphism $ \phi : X \rightarrow X $. Suppose there is a short exact sequence of vector bundles $ 0 \rightarrow F \rightarrow E \rightarrow G \rightarrow 0 $ on $ X $ ...
2 votes
0 answers
99 views

Stability of cotangent bundle of hypersurface

Let $ X $ be a smooth hypersurface of degree $ d > 1 $ in $ \mathbb{P}^{n+1} $. What can be said about the stability (Slope/Gieseker) of the cotangent bundle of $ X $? The closest reference I could ...
5 votes
2 answers
217 views

A smooth family of lattices on the tangent bundle?

I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...
3 votes
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147 views

$N$th-order approximation of point stabilizing diffeomorphisms by $N$th-order jet group?

NOTE: migrated from math SE. I was wondering if ever higher jet groups of frames on a (possibly pseudo) Riemannian manifold $M$ approximate the point stabilizing subgroup of diffeomorphisms on $M$ as ...
1 vote
0 answers
108 views

Pushforward of locally free sheaf by open immersion

Say $X$ is a smooth variety (even just $\mathbb{A}^n$) and $j\colon U\hookrightarrow X$ is an open immersion with $X - U$ of codimension 2 such that $E$ is a locally free sheaf on $U$. Since $X$ is ...
15 votes
3 answers
2k views

Tensor product of vector bundles

The Whitney sum (where fibre dimensions add) of two real, or two complex, vector bundles $\pi : E \to X$ and $\pi' : E' \to X$ over a topological space $X$ is not hard to get an intuitive grasp of. ...
3 votes
0 answers
282 views

The definition of a homogeneous vector bundle

For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
1 vote
1 answer
115 views

Singularities of fibrations in conics

Consider a rank two vector bundle $E = \mathcal{O}(a)\oplus \mathcal{O}(b)\oplus \mathcal{O}(c)$ over $\mathbb{P}^1$. Fix coordinates $u_0,u_1$ on the base $\mathbb{P}^1$ and $v_0,v_1,v_2$ on the ...
0 votes
1 answer
105 views

Integral mean value property

Let $V$ be the space of all continuous functions $f$ on the real line with $f(x)=\frac12\big(f(x-1)+f(x+1)\big)$. It contains the space of periodic functions. The latter equals the space of ...

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