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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-4
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25 views

Vector application [on hold]

A rope is hung at both ends from a horizontal beam, and a weight m is suspended from it. The left part of the rope exerts a force G at P, while the right part of the rope exerts a force H. Find the ...
4
votes
0answers
137 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
5
votes
1answer
233 views
+50

When is a general sheaf (on the projective plane) globally generated?

Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...
5
votes
0answers
88 views

Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that ...
1
vote
1answer
104 views

Transferring connection information to associated bundles and back

This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try. At the risk of repeating well known stuff I tried ...
3
votes
0answers
113 views

Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...
1
vote
1answer
91 views

the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
3
votes
0answers
154 views

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
0
votes
0answers
64 views

Horizontal vector fields and the push forward of the differential of the projection map

I am not very familiar with differential geometry but need to understand some aspects of it for my research. This includes in particular the notion of horizontal vector fields and I would like to ...
0
votes
1answer
153 views

Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...
1
vote
1answer
148 views

If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable?

Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$. Let $E$ be a vector ...
3
votes
2answers
463 views

Principal bundles that can't be detected by spheres

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are ...
6
votes
2answers
273 views

Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$

Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}_{\mathbb P^n}^{r} \longrightarrow ...
3
votes
1answer
84 views

self-Whitney sum of the canonical vector bundle on Grassmannians

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow ...
0
votes
0answers
85 views

Sheaf of Sections of Cone

Fulton's intersection theory book at Chapter 4 makes the following claim: If $\mathcal{E}$ is a locally free sheaf (on $X$), and $E:=Spec(Sym(\mathcal{E}))$ a total space of some cone/bundle on $X$, ...
8
votes
2answers
263 views

Maximal trivialising subspace for a vector bundle

Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...
8
votes
2answers
464 views

Vector bundles, finitely generated projective module?

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
2
votes
0answers
53 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
4
votes
0answers
164 views

Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type? [closed]

Using Stiefel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
5
votes
2answers
230 views

Relative Characteristic classes

A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...
16
votes
0answers
188 views

“High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
4
votes
0answers
46 views

Isometry theorem, exists homeomorphism that carries each fiber isomorphically onto itself

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically onto ...
3
votes
1answer
282 views

Tangent bundle of $S^2 \times S^1$ trivial or not [closed]

Is the tangent bundle of $S^2 \times S^1$ trivial or not?
2
votes
0answers
58 views

Are “vector spaces” over a smooth scheme with constant fiber dimension locally free?

I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism $p:V\to S$ of schemes of finite type over some base field. Assume that $p$ has all the ...
4
votes
0answers
87 views

Higher tangent bundles of manifolds with non integer dimension

One way to define the tangent space of a manifold at a point $p\in M$ is the following: We define an equivalent relation on the space of curves passing $p$ as follows: Two curves $\alpha, \beta$ are ...
2
votes
0answers
130 views

Doubt on elementary transformations in the paper - On a family of algebraic vector bundles by Maruyama

Let $S$ be a surface. $C$ be a curve on $S$ and $i:C\hookrightarrow S$ is the inclusion. Let $E$ be a rank 2 vector bundle on $S$ and $F$ a line bundle on $C$. Suppose we have a surjection ...
6
votes
4answers
506 views

What does “higher monodromy” tell us about a principal bundle

Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to ...
1
vote
1answer
51 views

Spaces of Killing spinors for different orientation

Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors. To be more precise: Let $M$ be a spin manifold (i.e. the first and ...
39
votes
0answers
594 views

Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
1
vote
0answers
78 views

Pull back of a semistable vector bundle to a product is semistable?

Let $X$ be a smooth projective surface 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 ...
6
votes
2answers
326 views

Pullback along Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
4
votes
1answer
132 views

covering map from spheres to projective spaces and the associated vector bundle

Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering $$ S^n\longrightarrow\mathbb{R}P^n. $$ We have an associated vector bundle $$ \xi: \mathbb{R}^2\longrightarrow ...
4
votes
0answers
108 views

Extend a vector bundle on a flat family

Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open ...
7
votes
1answer
153 views

classifying maps of Whitney sums of vector bundles

For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy $$ f(\xi): B\longrightarrow BG, $$ $f(\xi)\in [B;BG]$, ...
8
votes
0answers
154 views

Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?

The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, ...
4
votes
1answer
115 views

Vector bundles with symmetric perfect form

Let $X$ be a smooth projective curve, and $E$ a vector bundle on $X$ such that there exist a bilinear perfect symmetric form $$E\otimes E\rightarrow \mathcal O_X$$ When I see $E$ as a $GL_r$ ...
6
votes
1answer
253 views

Why do we need ampleness in the definition of stability/semistability

This is a general question that I have. Let $X$ be a projective variety over an algebraically closed field $k$. Let $L$ be an ample line bundle over $X$. Let $F$ be a vector bundle on $X$. We say that ...
7
votes
1answer
311 views

Zero scheme of global sections of vector bundles on affine varieties

I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties. Let $\mathbb{K}$ be an algebraically closed field, ...
14
votes
1answer
172 views

Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps. ...
4
votes
1answer
136 views

Semistability of principal bundle vs vector bundle

Ramanathan has defined the semistability of a principal $G-$bundle $E$ over a curve $X$ as follows: $E$ is semistable iff for any parabolic subgroup $P\subset G$, for any reduction of the ...
8
votes
2answers
789 views

Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?

In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle ...
2
votes
1answer
91 views

Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf

Given a locally free sheaf $E$ of rank $r$ on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero ...
4
votes
1answer
219 views

A question on complex line bundle over $S^{2}$

Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$. Assume that $\ell$ is a sub line bundle of ...
1
vote
1answer
116 views

triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle

Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle $$ \xi: ...
3
votes
0answers
96 views

line bundle on affine grassmannian and central extension

Let $G$ be a connected reductive group over $\mathbb{C}$, let $Gr$ be the affine grassmannian of $G$. On $Gr$, we know that there is a canonical line bundle $L$ (the generator of $Pic(Gr)$). Now ...
2
votes
1answer
187 views

Is this affine-subspace analogue of a Grassmannian a classifying space?

Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the ...
4
votes
3answers
192 views

A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural ...
1
vote
0answers
125 views

Twisting locally free sheaves in characteristic $p$

Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...
4
votes
2answers
246 views

triviality of Whitney sums of a vector bundle

Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by ...
6
votes
1answer
202 views

Vector bundles on open (affine) curves

It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$. As ...