For questions on vector bundles.

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3
votes
1answer
40 views

When is the symmetric algebra of a vector bundle finitely-generated?

Let $X$ be a projective variety over a field $k$, and $\mathcal L$ a vector bundle on $X$, i.e. a locally free $\mathcal O_X$-module of finite rank. For each $n\geq 0$, $\text{Sym}^n \mathcal L$ is a ...
7
votes
2answers
105 views
+50

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
1
vote
1answer
69 views

Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
4
votes
0answers
47 views

Pullback of indecomposable bundles on an elliptic curve

I consider an elliptic curve $\mathcal C$ over $\mathbb{C}$ and the multiplication by $[n]$ map on the curve. Then I consider an indecomposable vector bundle $E$ on $C$. What can I say of the ...
1
vote
0answers
46 views

Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
0
votes
1answer
30 views

Quadratic form on Vector Bundle

A quadratic form of a vector space $V$ over a field $\mathbb{F}$ is a bilinear symmetric map $V\times V \rightarrow F$. How does one define a quadratic form over a vector bundle.
5
votes
1answer
77 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...
7
votes
1answer
102 views

What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
2
votes
2answers
68 views

Elements of the zero-th Čech cohomology group versus global holomorphic sections

Something that is confusing (well, to me) has come up in the course of asking other questions. Let $\pi:V\to X$ be a holomorphic vector bundle of rank $r$ on a complex manifold $X$, such that $V$ is ...
0
votes
0answers
35 views

Non-isomorphism of topological line bundles on a Riemann surface, from first principles only

Although this question is in the same vein as my previous query, Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves, it is nonetheless ...
3
votes
1answer
99 views

Maps between two line bundles versus sections of their tensor product

There is a point about maps betwen line bundles (continuous or smooth or holomorphic --- I don't think it matters for this question) that is glossed over in many texts. A map from one line bundle ...
2
votes
1answer
57 views

Prove the holomorphic line bundle $\lambda(p+q)$ is the dual of the natural projective bundle

Let $M=\mathbb{C}P^1$ be the complex projective space, $U_0=\{[z_0,z_1]:z_0\ne 0\}$, $U_1=\{[z_0,z_1]:z_1\ne 0\}$ be the coordinate charts and define ...
5
votes
1answer
70 views

Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line ...
5
votes
0answers
45 views

Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...
0
votes
1answer
48 views

Smallest angle between two vectors?

I have a robot and I am going to turn it clockwise (negative degrees) or counterclockwise (positive degrees). If I turn the robot -270 degrees, that is the same as turning +90 degrees. Is there a way ...
5
votes
0answers
102 views

Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
1
vote
0answers
38 views

Spin structures, frame bundles, and trivializations over the 2-skeleton

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition: Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW ...
3
votes
1answer
93 views

Degree of a torsion-free subsheaf

Suppose that $R$ is a torsion-free subsheaf (of positive rank) in another torsion-free sheaf $S$, on a smooth complex projective variety $X$. If $S$ is (slope) semistable, is it true that the degree ...
1
vote
1answer
46 views

First Chern Class of divisors on compact Riemann surfaces

let $X$ be a compact Riemann surface and $D$ a divisor on $X$. I'm looking for a argument for the statement $c_1(\mathcal{O}_X(D)) = \deg(D)$, where $\mathcal{O}_X(D)$ is the associated line bundle to ...
7
votes
1answer
92 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 ...
8
votes
3answers
242 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
37 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
86 views

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
49 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
37 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
45 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
35 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
23 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
50 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
32 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
36 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
38 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
26 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 $GIT$ ...
2
votes
1answer
66 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
16 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
38 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
48 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
87 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
1answer
35 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
47 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
22 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
42 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
65 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
29 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
40 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 ...
1
vote
1answer
46 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
29 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
25 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
15 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
30 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 ...