For questions on vector bundles.

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7
votes
1answer
76 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
2
votes
0answers
35 views

Help needed in understanding a question to show that $M$ is a smooth manifold

Let $\rho : \mathbb{Z} \hookrightarrow GL(\mathbb{R}^r)$ be a representation. Consider $\mathbb{Z}$ as a subgroup of $(\mathbb{R},+)$ in the usual way. Define $M$ as the quotient of $\mathbb{R} \times ...
7
votes
3answers
186 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
0
votes
1answer
30 views

Definition of covariant derivative

Let $E \to M$ be a vector bundle (with fibres $V$) over a smooth manifold $M$. Define the covariant derivative $\nabla$ as a map $$\nabla : C^\infty(M,E) \to C^\infty(M,E \otimes T^*M),$$ where ...
4
votes
1answer
65 views
+50

Reference Request: Thom Spectrum of a virtual vector bundle

Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse. I unsuccessfully browsed through literature to find a reference for an explicit ...
3
votes
1answer
23 views

Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
1
vote
1answer
33 views

cohomology homomorphism induced by classifying map

Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle. Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map. Let $f^*: ...
1
vote
1answer
39 views

Reference request: second Chern class of P^2

I have heard that $c_2(T_{\mathbb{P}^2})=e(\mathbb{P}^2)$. What's the general result and where can I read about it? Thanks.
0
votes
0answers
28 views

Basic question: degree of normal bundle is not self-intersection number

For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$. Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular? Why do ...
2
votes
0answers
18 views

Contraction of second exterior covariant derivative with metric

Let $G \hookrightarrow P \to M$ be a principal $G$ bundle, $P \times_\rho V$ be a vector bundle associated to representation $\rho$ of $G$ on $V$. If $\omega$ is a connection $1$-form on $P$ then we ...
4
votes
0answers
39 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
2answers
28 views

The first Chern class of projective line $\mathbb{CP}^1$

I am studying the Chern class using by some textbooks and lecture notes. One day, I found an example of the first Chern class of $\mathbb{CP}^1$. Let $\xi$ be a tautological line bundle of ...
1
vote
1answer
33 views

Why a semi-stable non stable bundle $E$ is S-equivalent to $L_1\oplus L_2$

Let $M(2,d)$ be the set of all vector bundles of rank $2$ and degree $d$ over a smooth projecitve curve of genus $g\geq 3$. Let $M(2,0)^s$ and $M(2,0)^{ss}$ be the stable and semistable vector ...
1
vote
0answers
27 views

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 ...
0
votes
0answers
23 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant over a curve of genus $g$ [duplicate]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
1
vote
1answer
56 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
0
votes
0answers
13 views

If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
4
votes
1answer
35 views

How to prove line bundle L is trivial if and only if its dual bundle us trivial?

How to prove line bundle L is trivial if and only if its dual bundle us trivial ?
2
votes
1answer
42 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
2
votes
3answers
71 views

normal bundle on a submanifold

Can you give me an example of a nontrivial normal bundle of a submanifold (of any manifold)? There is standard example of the core circle of a mobius band, but can you give an example of a submanifold ...
0
votes
0answers
11 views

Reference: Differential operators and principal symbols

I am looking for good references about differential-/pseudodifferential operators and principal symbols. thanks
0
votes
1answer
33 views

0th-order differential operator vs. 1st-order differential operator on a vector bundle $(E, \pi,M)$

Consider a vector bundle $(E,\pi,M)$. A 0th-order differential operator on $E$ is a $C^\infty(M)$-linear endomorphism $\Gamma E\rightarrow\Gamma E$. $\Gamma E$ is the set of sections on $M$. A ...
1
vote
1answer
44 views

Complex structure on a real vector bundle

Let $M$ be a smooth manifold and $\pi:E \rightarrow M$ a real vector bundle and note $E_x:=\pi^{-1}(x), \forall x\in M$. We set a bundle $\text{End}(E)=E\otimes E^*$. Now suppose there exists a smooth ...
2
votes
0answers
19 views

Real vector bundles over $S^1$

Prove that for every real vector bundle over $S^1$ there exist open connected subsets $U_1,U_2 \subset S^1$ with $U_1 \cup U_2=S^1$ such that $E$ is trivial over $U_1$ and over $U_2$. I need an ...
3
votes
0answers
37 views

Do homotopic transition functions define isomorphic bundles (on smooth manifolds)?

It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow ...
3
votes
2answers
58 views

Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and ...
3
votes
0answers
28 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
2
votes
2answers
38 views

Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
0
votes
0answers
23 views

vector bundles and cocycles

I need a detailed solution to a self-study book's exercise: "Show that two vector bundles on M are isomorphic iff their cocycles relative to some open cover are equivalent" I can show it in one ...
1
vote
0answers
26 views

Global Trivialization of $M\oplus M$

$\mathbb S^1$'s $\mathbb R^1$-bundle is $$\{\mathbb S^1\times\mathbb R^1\text{, open Mobius strip}\}$$ and its $\mathbb R^2$-bundle is $$\{\mathbb S^1\times\mathbb R^2\text{, open solid Klein ...
1
vote
1answer
23 views

“Restriction” of a Differential Operator on a Vector Bundle to the Space of Local Sections?

I'm trying to understand the definition of differential operators on vector bundles. The material I'm following http://www.mat.univie.ac.at/~stein/research/talks/nhops.pdf starts with the ...
0
votes
0answers
13 views

Reference request: Order of a vector bundle

Please could you link me to an accessible reference/set of lecture notes on the definition of the order of an algebraic vector bundle? Google just shows some very complicated definition which is ...
1
vote
0answers
28 views

homotopy via isotopy

Is it possible to realize a homotopy between two vector bundles (which are sub-bundles of the tangent bundle) over the same base $B$ as induced by an ambient isotpy? In other words, assume $X_t $ is a ...
2
votes
0answers
55 views

Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
0
votes
1answer
25 views

Defintion of Generalized Conic Bundle

Can some one help me understanding why the Definition of Generalized Conic Bundle is generalization of the Conic Bundle definition. This is the definition of a conic bundle from "Comparison theorems ...
0
votes
0answers
17 views

trivialization of a bundle

suppose we have a $D^2$-bundle $X$ over a surface with boundary $F$. It is said that we can always trivialize the bundle: X is diffeomorphic to $F\times D^2$ but I do not see why this is true.
2
votes
1answer
24 views

Holonomy representation: is it actually a class of representations?

In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb ...
0
votes
1answer
14 views

Multiple points in the parallel transport equation

Let $E \to M$ be a vector bundle with connection $\nabla$, $\gamma \colon [0,1] \to M$ a smooth curve. A section $s = \gamma^* \tilde s$, $\tilde s \in \Gamma(E)$ of $\gamma^* E$ is called parallel if ...
1
vote
0answers
38 views

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 ...
4
votes
2answers
68 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
0
votes
0answers
5 views

Elements in the zero subbundle

I am in the following situation: There is a vector bundle $W$ with basis $\Omega$ and a flow on $W$, i.e. a function $\Psi\colon W\times\mathbb{R}\to W, (w,r)\mapsto w\cdot r$ that fullfills the ...
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vote
0answers
43 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
2
votes
1answer
38 views

Homotopy groups of infinite Grassmannians

Let $G_k$ be the infinite rank $k$ Grassmannian. For $n>k$, is $\pi_n(G_k)$ trivial? Phrased differently, is every rank $k$ vector bundle on an $n$-sphere, for $n>k$, trivial? (This is motivated ...
0
votes
0answers
43 views

Finding global sections of a vector bundle over a compact manifold which generate each fiber

The following is an exercise I am doing for review for a midterm exam in differential geometry. Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is ...
1
vote
0answers
9 views

Vector bundle base space map

Is it true, and if it is, is there some easy way to see the following? Suppose that $\xi = (\pi, E, B)$ is an $n$-vector bundle with $B$ paracompact but not necessarily compact. Is there a base ...
0
votes
0answers
14 views

Embedding into the projective space - tangent level injection

I am reading Griffiths and Harris, the section of embedding a manifold $M$ into projective space. Let $\mathcal{L}$ be a line bundle over $M$, with dim $H^0(M,\mathcal{L})=N$, and let $s_0,...,s_N$ be ...
3
votes
1answer
54 views

Canonical isomorphism between vector bundle and dual?

So, we've been asked to show, given a real vector bundle equipped with a metric, that there is a canonical isomorphism from the vector bundle and its dual. Now, there's a theorem that says two vector ...
0
votes
1answer
40 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and ...
1
vote
1answer
26 views

Smooth structure on a quotient vector space

How do I know if $$f:\mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})\to \mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})$$ is smooth? Can't find the definition of the canonical smooth ...
1
vote
1answer
43 views

A Question on Vector Bundles

I am having trouble proving the following about vector bundles. I would think it would be rather easy, but I can't think of how. This isn't homework, but something I want to be true so that I can ...