**2**

votes

**0**answers

73 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-...

**0**

votes

**1**answer

132 views

### $\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...

**2**

votes

**1**answer

137 views

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= \...

**2**

votes

**1**answer

237 views

### How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be ...

**2**

votes

**0**answers

188 views

### Multiplication Map, Is it invariant?

Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree -1 vector bundle $F$ on $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ ...

**2**

votes

**1**answer

176 views

### Is this structure a Banach bundle?

Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology.
...

**2**

votes

**2**answers

214 views

### A line bundle over the manifold of singular matrices

According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we ...

**1**

vote

**1**answer

180 views

### Stable vector bundle Projective or Injective?

I have a very easy question, which I couldn't get in the literature. Please forgive me if it is so easy!!!
Question: Is a stable vector bundle over a curve $C$ is projective (as a $\mathcal O_C$-...

**14**

votes

**1**answer

387 views

### Anything between vector bundles and sphere bundles?

There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...

**1**

vote

**1**answer

320 views

### Stable Vector bundles

Let $C$ be smooth curve, and let $F$ be a stable rank 2 vector bundle of degree equal to $2c+1$, $c\in\mathbb{N}$, and fix a point $p\in C$:
Can one choose an epi-morphism $u:F\rightarrow \mathbb C_p$...

**1**

vote

**0**answers

148 views

### Hermitian metric on conic Kaehler-Einstein setting

I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler -...

**1**

vote

**1**answer

180 views

### Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998):
2.6 Spin Bundles and the Dirac Operator
...

**2**

votes

**0**answers

133 views

### Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...

**1**

vote

**1**answer

301 views

### A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $...

**2**

votes

**1**answer

275 views

### Degeneracy locus and flatness over local Artinian ring

Let $X$ be a projective scheme flat over a local Artinian ring $A$, the residue field of $A$ is algebraically closed, and the special fiber of $X$ (under the natural morphism from $X$ to $A$) is ...

**1**

vote

**0**answers

108 views

### Help understanding the proof of a theorem about Cohomology of vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states:
Let $\mathcal{E}$ be a vector bundle on $\...

**3**

votes

**0**answers

119 views

### Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface $...

**15**

votes

**1**answer

596 views

### Soft and hard part of geometry [duplicate]

While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" part....

**0**

votes

**1**answer

95 views

### Is the kernel of a map between finite dimensional vector bundles still of finite type?

I'm not sure whether the level of this question is suitable for Mathoverflow.
Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow ...

**0**

votes

**0**answers

110 views

### A question on tangent bundle (and second tangent bundle)

Let $M$ be a $n$ dimensional manifold and $p:TM\to M$ be the projection map. Then $\ker Dp$ is a $n$ dimensional vector bundle on $TM$, as a sub bundle of $TT(M)$.
For what type of manifolds, ...

**6**

votes

**0**answers

373 views

### Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective

$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...

**5**

votes

**1**answer

251 views

### Is the unit tangent bundle of $S^{n}$ parallelizable?

Is the unit tangent bundle of $S^{n}$ a parallelizable manifold. This is motivated by the fact that $TS^{n}$ is parallelizable?

**7**

votes

**1**answer

732 views

### semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...

**6**

votes

**0**answers

286 views

### Global sections for a locally free sheaf over curves

Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg \...

**1**

vote

**2**answers

210 views

### Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...

**11**

votes

**1**answer

534 views

### Is there any relationship between the Euler class and the Vandermonde determinant?

Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...

**13**

votes

**2**answers

756 views

### The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get $[\...

**1**

vote

**0**answers

118 views

### Pullback of the tautological vector bundle and the nubmer of trivializations

I've heard about the followign result: for each two natural numbers $d,n \in \mathbb{N}$ one can find a number $k \in \mathbb{N}$ with the following property: for each CW-complex $X$ with $\dim X \leq ...

**0**

votes

**1**answer

114 views

### Vector bundle on ruled surface $X\times \mathbb{P}^{1}$

let $E$ be a vector bundle of rank $r$ on $X\times \mathbb{P}^{1}$ where $X$ is a smooth projective curve. Assume now that $E|_{F_{p}} \cong \mathcal{O}_{\mathbb{P}^{1}}^{r}$ for any p-fiber where $p: ...

**1**

vote

**0**answers

57 views

### A question on smooth sections of a vector bundle

Let $E\to M$ be a real smooth vector bundle and $N_{1},N_{2},\ldots,N_{k}$ are disjoint compact (proper) submanifolds of $M$. Are there smooth sections $s_{1},s_{2},\ldots,s_{k}$ such ...

**1**

vote

**0**answers

269 views

### The “Rolle theorem” for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

**2**

votes

**1**answer

312 views

### On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...

**0**

votes

**0**answers

239 views

### “Spreading out” locally free sheaves

Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...

**3**

votes

**0**answers

237 views

### Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...

**6**

votes

**1**answer

346 views

### Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...

**0**

votes

**0**answers

212 views

### Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...

**4**

votes

**0**answers

124 views

### Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...

**1**

vote

**0**answers

223 views

### Two questions on canonical line bundle over $\mathbb{C}P^{n}$

The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...

**2**

votes

**1**answer

182 views

### deformations of vector bundles on curves

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of $...

**1**

vote

**0**answers

105 views

### DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...

**0**

votes

**1**answer

97 views

### Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.
(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)
Assume we ...

**1**

vote

**1**answer

361 views

### Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...

**8**

votes

**1**answer

435 views

### Clutching functions and Classifying maps

Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, $E|_{C_+...

**2**

votes

**1**answer

296 views

### FIltrations on a vector bundle on a curve

Let $X$ be a smooth connected projective curve of genus $g$ over an algebraically closed field.
Let $E$ be a vector bundle on $X$ of rank $n$.
Is it true that there exists a constand $N(g,n)$ such ...

**2**

votes

**1**answer

259 views

### vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$

Let $A = \mathbb{C}[x,y,z]/(x y - z^k)$. In fact $A$ is the ring of $\mu_k$ invariants: $A = \mathbb{C}[u,v]^{\mu_k}$ where $g \in \mu_k$ acts by $g(u,v) = (g u, g^{-1} v)$.
This allows one to ...

**0**

votes

**0**answers

384 views

### On Bott's Formula on projective Spaces

I'm following the book of Okonek, Schneider and Spindler, Vector Bundles on Complex Projective Spaces, and they say that is a useful exercise try to prove Bott's Formula that calculates the cohomology ...

**2**

votes

**2**answers

329 views

### Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...

**0**

votes

**1**answer

74 views

### A question on parallelizability

Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?

**-1**

votes

**1**answer

362 views

### Computing the Chern class of $S^6$ [closed]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...

**3**

votes

**0**answers

142 views

### Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
...