For questions on vector bundles.

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On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
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Obstruction to the splitness of an exact sequence of holomorphic vector bundles

In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is Every exact sequence of holomorphic vector ...
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Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
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Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
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Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
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spin structures definition

One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of ...
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Tangent Bundle of Product Manifold

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that: $T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $ Can we conclude that $T(M\times N) \cong T(M) ...
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Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
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Reduction to the special orthogonal group

It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle However, I can't see why, because the isomorphism $E\cong E^*$ means ...
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Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
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Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
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Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
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1answer
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Trivialization of vector bundles

Is it true that any $R^n$-bundle over a space (say a simplicial space) of dimension $k<n$ is trivial? It seems to me any $R^n$-bundle ,for $n>1$, over $S^1$ is trivial. But I cannot figure out ...
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To Reconcile Two Different Descriptions of the Dual Bundle

$\newcommand{\mc}{\mathcal}$ Let $\pi:E\to M$ be a smooth vector bundle with typical fibre a $k$-dimensional vector space $\mc V$. There are (at least) two ways to construct the dual bundle of $E$. ...
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The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
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Rank 2 vector bundle V and its k$_V$

This question is from the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle ,Ex4.6 Suppose o$\le$a$\le$b$\le$c are three integers, show that there exist vector bundle V which fit ...
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Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
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1answer
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Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...
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What does the notation $\mathcal{O}_{\mathbb{P}^n}(1)$ mean?

I have tried looking at my sheaves notes but couldn't find anything.
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1answer
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Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
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1answer
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Orientability of $\gamma^n\oplus \gamma^n$ WITHOUT characteristic classes

I was curious to find an argument to show orientability of the $2n$-bundle $$\gamma^n\oplus \gamma^n$$ where $\gamma^n$ is the canonical $n$-bundle over the infinite grassmannians ...
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Prove that $w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$

Let $\gamma^n$ be the canonical $n$-plane bundle over the infinite Grassmann manifold $G_n(\mathbb{R}^{\infty})$. I'm asked to prove that $$w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$$ (exercise 9-A ...
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Exact sequence of vector bundles

I'am working on a shorter proof of a theorem but to manage it I need to know if a lemma is true. Conjecture: Given a manifold $M$ and an short exact sequence of vector bundles $$ 0 \rightarrow E' ...
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Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
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(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
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question on a vector bundle

let be $E=S^{1}\times \mathbb{R}$ be a trivial vector bundle on $M=S^{1}$ and $\nabla$ be a connection on it defined by $$\nabla:\chi(M)\times \Gamma(E) \to \Gamma(E)$$ and ...
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underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
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2-form as a curvature form

Let $M$ be a manifold and let $F$ be a global-defined $2$-form over $M$. Are there any conditions of $M,F$ such that $F$ is a curvature form of a connection on some line bundle ?
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Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
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commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
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Trivial bundle on projective space?

On $P^n(\mathbb R)$ I consider the open sets $U_j$ (with $x_j \neq 0$) and the transiction functions of a linear vector bundle $E_d$, $f_{hk}:p\to(x_k /x_h)^d$. I have to demonstrate that, if $d$ is ...
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Vector Bundle Structure on $\sqcup_{p\in M}\mathcal L(T_pM, T_{f(p)}N)$.

Let $f:M\to N$ be a smooth map between smooth manifolds. Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$. where $\mathcal L(V, ...
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1answer
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Projection map of a vector bundle induce isomorphism on top cohomology.

I'm reading a passage in Milnor-Stasheff about Euler class, and I noticed that he states that the projection map $$\pi \colon E \to B $$ where $(E,\pi,B)$ is a n-dim vector bundle, induces a canonical ...
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1answer
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vector bundle associated to the representation of a lattice

Suppose that $G$ is a connected semisimple Lie group and that $\Gamma$ is a cocompact lattice in $G$. Given a complex-linear representation $\chi$ of $\Gamma$ on a finite-dimensional complex vector ...
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Connection and reduction of the structure group

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections ...
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1answer
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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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Classifying all $E$-torsors up to isomorphism?

If $0\to E\to F\xrightarrow{\phi}\mathbb{A}^1_X\to 0$ is an exact sequence of bundles over the scheme $X$, then it is known that if $s\colon X\to \mathbb{A}^1_X$ is the constant section sending ...
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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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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
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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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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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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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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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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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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 ...