For questions on vector bundles.

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Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
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Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
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$H^1$ of some vector bundle on a cubic 3-fold

This question is a sequel to the following one Dimension of moduli space of some stable vector bundles on a cubic 3-fold. Let $E$ be a stable rank 2 vector bundle on a cubic 3-fold, say $X$, with ...
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Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
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What is the pull back of this line bundle to the effective divisor defining it?

Let $S$ be a surface and $C$ be an effective divisor on $S$. Let $L=\mathcal{O}_X(C)$ be the line bundle corresponding to $C$. Let $i:C\longrightarrow S$ be the inclusion morphism. Then what is ...
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2answers
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Vector bundle and its definition

Take this definition of vector bundle: its a triple $(V,\pi,M)$ where $V$ is a set, $\pi:V \rightarrow M$ a sutjective map such that for every $m \in M$ the set $\pi^{-1}(m)$ has the structure of real ...
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1answer
62 views

Dimension of moduli space of some stable vector bundles on a cubic 3-fold.

I'm trying to understand the claim that the moduli space of stable rank 2 vector bundles on a (general?) cubic 3-fold, say $X$, with $deg c_2=6$ and $det=O_X(2)$ is of dimension 9. I belive what I ...
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2answers
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Dual Bundles' Local Trivializations Confusion

$$\newcommand{\R}{\mathbb R}$$ Let $\pi:E\to M$ be a rank $k$ smooth vector bundle over a smooth manifold $M$. I will below describe how to form the dual bundle, wherein lies my question. Let ...
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(basic?) isomorphism of topological K-theory and reduced K-theory

first I want to state which definitions I use for K-theory and reduced K-theory: Let $X$ be a compact topological Hausdorffspace and $V(X):=\{\text{Isomorphism classes of (complex) vector bundles ...
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24 views

Pontryagin classes of a tensor product of bundles

This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$. Specifically, given two real vector bundles $E$, ...
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26 views

Why is this scheme $Y$ an affine bundle?

Suppose you have $F$ a subbundle of a vector bundle $E\to X$ over a scheme $X$. Recall that there is a scheme $\varphi\colon Y\to X$ with the points of $Y$ over the point $x\colon\mathrm{spec}(A)\to ...
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Isomorphism of the Clifford bundle of a Riemannain manifold

Let $(M,g)$ be an oriented Riemannian manifold and $Cl(M):=\bigcup_{x\in M}Cl(T_xM,g_x)$ be the clifford bundle of $(M,g)$. (Here $Cl(T_xM,g_x)$ denotes the clifford algebra of the vector space ...
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1answer
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Riemann Roch Meromorphic section on a line bundle.

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha w)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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32 views

Divisors corresponding to hypersurfaces in Projective space

I'm looking at hypersurfaces on $\mathbb{C}\mathbb{P}^2$. That is, the zero set of an irreducible homogeneous polynomial $f(x_0, x_1, x_2) = 0$. This corresponds to a divisor, $D_f$, let's say which ...
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1answer
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Why is the evaluation map of sheaves injective

Let $E$ be a globally generated vector bundle of rank $r$. Let $V$ be a subspace of $H^0(S,E)$ of dimension $r$. We have the evaluation map $ev:V\otimes \mathcal{O}_S\longrightarrow E$. Why is this ...
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When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
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1answer
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Curvature for tautological bundle of projectivation

I'm trying to compute locally the curvature of $\mathscr{O}_{\mathbb{P}(E)}(-1)$, where $E\to X$ is a holomorphic bundle. The covering is given by $U_i\times \mathbb{P}(E)_j$, where $\{U_i\}$ is a ...
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1answer
29 views

Direct sum of nontrivial vector bundles?

When is it true that the direct sum (or whitney sum) of two nontrivial vector bundles is nontrivial? Also, if you have a direct sum of vector bundles, with $a$ and $b$ global sections respectively, ...
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1answer
25 views

Finding Eigenvectors when we have lots of zeroes

\begin{array}{cc} 0 & 0 \\ 0 & 8 \\ \end{array} I have $\lambda_1=8$ and $\lambda_2=0$ but cannot find $V_1$ or $V_2$ I try \begin{array}{cc} \lambda & 0 \\ 0 & 8-\lambda \\ ...
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45 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
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1answer
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Stiefel-Whitney Numbers of $\mathbb{R}P^2\times \mathbb{R}P^2$

I'd like to calculate the Stiefel-Whitney numbers of $\mathbb{R}P^2\times\mathbb{R}P^2,$ but don't know how to. My first instinct was to say that the tangent bundle is isomorphic to the product of ...
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1answer
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Direct proof of decomposition of real vector bundle of odd degree into the direct sum of a trivial bundle and another of even degree

The real splitting principle tells us that when taking a real, oriented vector bundle of odd dimension $\zeta$ over a manifold $M$ you can always write $\zeta$ as $\tilde{\zeta} \oplus \varepsilon^1$, ...
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1answer
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Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings?

A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?
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1answer
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The exact sequence $0 \rightarrow L(-1) \rightarrow L(0)^2 \rightarrow L(1) \rightarrow 0$

I am trying to show that there is an exact sequence: The exact sequence $0 \rightarrow L(-1) \rightarrow L(0)^2 \rightarrow L(1) \rightarrow 0$, where $L(-1)$ is the tautological line bundle of ...
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1answer
51 views

Tangent bundle of manifold with no odd dimensional sub-bundles

First, a preliminary remark: The Whitney sum of two vector bundles is orientable. I saw this statement somewhere and was wondering if it's true. In particular, it's easy to show that ...
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Thom space for complex bundles

It is well known that for real vector bundle with Riemannian metric we can construct Thom space using associated disk and sphere bundles. Can we do it for complex bundle with hermitian metric ?
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Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
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1answer
41 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
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1answer
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Vector Bundles:differential geometry vs algebraic geometry

I am in trouble about the vector bundle part in the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle, I know what is a vector bundle(or a fibre bundle) in the differential geometry, ...
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What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to X$ a ...
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1answer
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Why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$?

I have a very brief question. If you have a bunch of line bundles $L_1,\dots,L_p$ over a scheme $S$, why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$, and can't find ...
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27 views

Lifting of Commuting Maps of Vector Bundles

Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
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1answer
33 views

Stiefel-Whitney Classes of a submanifold

Suppose we have a manifold $M$ that is the product of $k$ copies of real projective space, say $$ M = \prod_{i=1}^k \mathbb{R} P^{n_i}.$$ Then $H^*(M; \mathbb{Z}/2) = \mathbb{Z}/2 [ \alpha_1, \dots, ...
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Quick question: Pull back under double cover of tangent space on the projective plane is stable?

Let $f:\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^2$ be the double cover branched along some conic $C\subset \mathbb{P}^2$. Is $f^*T_{\mathbb{P}^2}(-1)$ $\mu$-stable/semistable? Is there any ...
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Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
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1answer
27 views

Vector space operations on fibres of associated bundles.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let ...
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1answer
70 views

Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
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1answer
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Decomposition of $\pi\colon E\to\mathbb{P}^1_k$ as a direct sum of tensor powers of the tautological line bundle?

Suppose you have a vector bundle $\pi\colon E\to\mathbb{P}^1_k$, where $k$ is some field. Is it always possible to decompose the vector bundle into a direct sum of tensor powers of the tautological ...
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1answer
102 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
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Morphism of modules of sections of pullback bundles

Suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles ...
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1answer
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Section of pullback bundle

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the ...
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Trivial sections of tautological line bundle for $\mathbb{P}_F(V)$.

I have a brief question which has been bothering me. Suppose you have a finite dimensional $F$-vector space, call it $V$. Is there a nice proof of why there only exist trivial sections of the ...
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Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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1answer
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Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
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Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
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2answers
85 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
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1answer
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Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
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Reduction of the structure group

Given a vector bundle $V$ of rank $r$ over a curve. What proprieties should $V$ satisfies in order to admet a reduction of the structur group to $O_r(k)$ (resp. $SL_r(k)$, $Sp(r,k)$) Could you give ...
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1answer
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Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
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1answer
56 views

Is the tensor product of a complex line bundle with itself trivial?

Let $\xi$ be a complex line bundle over a manifold $M$. Then $\xi\otimes \xi$ is a trivial complex line bundle. Is my statement right?