**-4**

votes

**0**answers

28 views

### Vector application [closed]

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

**0**answers

140 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 ...

**7**

votes

**1**answer

257 views

### 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

**0**answers

90 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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

65 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

**1**answer

155 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

**1**answer

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

**2**answers

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

**2**answers

274 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

**1**answer

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

**0**answers

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

**2**answers

266 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

**2**answers

465 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

**0**answers

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

**0**answers

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

**2**answers

231 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

**0**answers

192 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

**0**answers

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

**1**answer

285 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

**0**answers

61 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

**0**answers

90 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

**0**answers

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

**4**answers

510 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

**1**answer

52 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

**0**answers

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

**0**answers

79 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

**2**answers

327 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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

156 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

**1**answer

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

**1**answer

254 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

**1**answer

315 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

**1**answer

173 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

**1**answer

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

**2**answers

791 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

**1**answer

92 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

**1**answer

220 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

**1**answer

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

**0**answers

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

**1**answer

189 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

**3**answers

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

**0**answers

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

**2**answers

248 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

**1**answer

205 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 ...