# Tagged Questions

For questions on vector bundles.

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### Simple exact sequences of vector bundles

I've come across some simple exact sequences of vector bundles that make manifest some basic confusions I have. These questions may be quite intertwined, in ways that my limited understanding obscures,...
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### Normal bundle of a point

Let $X$ be a projective variety over a field $k$. I am trying to understand the notion of the normal bundle of a closed immersion. Let $x$ be a closed point of $X$. What is the normal bundle of $x$ ...
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### Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$

On page $10$ of Hatcher's Vector Bundles and K Theory, he gives a proof that the Whitney sum of the trivial line bundle over $\mathbb{R}P^n$ and the tangent bundle is equal to the Whitney sum of ...
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### What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
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### parallelizable sphere product closed disk

From Wall's Surgery on Compact Manifolds, P9: Observe that $S^r \times D^{m−r}$ is parallelisable. If $m > r$, this is true, because spheres can be embedded in Euclidean space of one ...
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I wrote the following argument to prove that $S^1$ is parallelizable, that is, to show that the tangent bundle is trivial. It looks fairly reasonable to me. Let $\tau=2\pi$. We define a map $\... 0answers 26 views ### Vector bundle over an open set of$\mathbb{R}^n$I can't see or understand if it is true or not if all vector bundles on over an open set of$\mathbb{R}^n$are trivial or not. Is there an easy way to see it? The problem comes from the fact that we ... 1answer 43 views ### Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold This question is motivated by an observation of Milnor. Theorem: Let$M^m$,$N^n$be parallelizable smooth manifolds, and$i:M\to N$an embedding. If$n>2m$, the normal bundle is trivial. The ... 1answer 19 views ###$r$-jet of a smooth function and its fiber bundle. Let$M$be a smooth manifold of dimension$n$. Let$E$denote the bundle of germs of smooth functions on$M$. For every stalk$E_x$we can define the ideal $$I_x^k=\{ f \in\mathcal{C}^{\infty}(M) \... 2answers 71 views ### Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant? When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ... 1answer 26 views ### What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers? I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ... 1answer 71 views ### Is T\mathbb{C}\mathbb{P}^n globally generated? A vector bundle E\to X is globally generated if there exists global holomorphic sections s_1,\dots,s_n such that E_x is spanned by s_1(x),\dots,s_n(x) for all x\in X. Consider the ... 2answers 74 views ### Defining a Riemannian metric Let M be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product g_p:T_pM\times T_pM\to\mathbb{R} which is symmetric and positive-definite,... 0answers 53 views ### Motivation for equivalence of Tautological Line Bundle and {\operatorname{Bl} _0}\mathbb{A}_k^{n + 1} I am currently reading a book on Algebraic Geometry and both the Tautological Line Bundle L \to \mathbb{P}_k^n and the blow up {\operatorname{Bl} _0}\mathbb{A}_k^{n + 1} are defined set ... 0answers 71 views ### Definition of cotangent and conormal bundle I have read the following definition of cotangent bundle: Let X be a n-dimensional smooth algebraic variety. For any p\in X there exist a neighbourhood U_{p}\subseteq X and functions (... 1answer 55 views ### Pullback of global sections? Let X, Y be abelian varieties and let f:X\to Y be a morphism. They told me that we can define the pullback f^*s of a global section s\in\Gamma (L) where L is an ample line bundle on Y, ... 0answers 25 views ### Global holomorphic vector field on a two-sphere I'm sure this question has been asked before... Till today, I thought that one cannot define a global holomorphic vector field on a two-sphere due to the hairy ball theorem. However, here's an ... 1answer 34 views ### Degree of filtered vector bundle Suppose I have the sheaf \mathscr{M} defined by$$0\to \mathscr{M}\to \mathscr{O}_{\mathbf{P}^r}^{r+1}\to \mathscr{O}_{\mathbf{P}^r}(1)\to 0 $$that is, \mathscr{M}\simeq \Omega_{\mathbf{P}^r}^1(1)... 1answer 41 views ### Cohomology class of zero set of a section Say we have a rank r smooth vector bundle E\to X and a smooth section s:X\to E of it. Shortly after the 30 min mark in this video Joe Harris defines the rth Chern class of this bundle to be ... 1answer 37 views ### Vector subbundle and frame field relation Question: Let E \to M be a vector bundle of rank k. Suppose that for each p \in M we are given a subspace E'_p of E_p and consider the set \displaystyle E' = \bigcup_{p \in M} E'_p .... 1answer 88 views ### Cohomologies with line bundle vs. coherent coefficients I recently learned in a lecture that the derived category of a smooth variety is generated/spanned by (complexes of) locally free sheaves. (Unfortunately I haven't been able to find a more precise ... 1answer 56 views ### Relation between projective equivalence and linear equivalence of divisors For the whole question I'll be working in \mathbb{P}^n_{\mathbb{C}} and assume that everything is smooth. We say that two sets U,V\subseteq \mathbb{P}^n are projectively equivalent if there ... 0answers 71 views ### How to show Hom(V,V)\rightarrow Hom(V_x,V_x) is injective, V being semi-stable Let V be a semi-stable vector bundle over a smooth irreducible projective curve of genus g\geq 2. Let x\in X. How do we show that the canonical map Hom(V,V)\rightarrow Hom(V_x,V_x) which ... 0answers 61 views ### Chern classes of a double cover Let X be a compact complex surface and let D be a double cover of X. Let \pi:D\to X be the double cover map (a 2:1) map. If E is a vector bundle (rank at least 2) on X with c_1(E) = A ... 1answer 65 views ### The non-existense of the fine moduli scheme of vector bundles. Why? The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let C a ... 1answer 90 views ### Show that \Omega^*(M)\otimes \Omega^*(N) is isomorphic to \Omega^*(M\times N) Let M, N be compact manifolds and \Omega^* its algebra exterior. How to prove that \Omega^*(M)\otimes \Omega^*(N) is isomorphic to \Omega^*(M\times N)? I thought about the function f(\omega,... 1answer 53 views ### Are there characteristic classes for symplectic vector bundles? Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ... 0answers 39 views ### What does “factoring out an (group) action \tau of a group G acting on some set E” mean? I am reading a survey article where they define the following objects: \Gamma:=\mathbb{Z}^{n} seen as a group of translations. \mathbb{T}:=\mathbb{R}^{n}/\Gamma is the n-dimensional ... 0answers 82 views ### Bundle isomorphism \Phi : TM \oplus T^*M \to T(T^*M). Prove that there is a bundle isomorphism \Phi : TM \oplus T^*M \to T(T^*M) which identifies the summand T^*M with the vertical vectors. If \omega_{can} is the canonical symplectic ... 1answer 70 views ### Whitney sum bundle vs. direct product bundle [closed] The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different? PS: I have realized my mistake: let E_1 \to B and E_2 \to B be two bundles, their ... 1answer 21 views ### On the zero in the fibre of a vector bundle. Let X be a differentiable manifold, let \{U_i \mid i\in I\} be an open cover of X, let \{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\} be a set of differentiable maps satisfying the cocycle condition, ... 0answers 60 views ### Roots of canonical line bundles that are not necessarily square roots I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are 2^{2g} choices of such a root. But what about further roots? ... 1answer 80 views ### Computing Curvature of a Connection (Dirac Monopole) I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ... 1answer 21 views ### Extending a section to a basis of sections in a trivial bundle. Suppose we have a manifold M of dimension m and let E be the rank n trivial vector bundle on M. Let \Gamma be a section of the trivial bundle E. My question is, are there some conditions ... 1answer 52 views ### Exact sequences of bundles and orientations If we have an exact sequence of finite-dimensional vector spaces$$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$then an orientation of any two induces an orientation of the third. I have ... 0answers 20 views ### Linear connection in a vector bundle in terms of the vertical projection Let$\pi:E \rightarrow M$a vector bundle. Can we define a linear connection as a connection$\Phi \in \Omega^1(E,VE)$such that the horizontal projection$p_H = \text{id}_{TE} - \Phi$is a vector ... 2answers 64 views ### Why does a tangent bundle have dimension 2n instead of n? Let$n=dim(T_pM)$for every$p\in M$, where$M$is a smooth manifold. I understand that specifying$p$is not enough to determine an element of$TM$, but what if do we specify only$v\in T_pM\subset ...
I am trying to prove that there exists a formal immersion from a manifold $M$ of dim $m$ into $\mathbb{R}^{2m}$. Formal immersion is just an injective bundle map from $TM$ into $T\mathbb{R}^{2m}$ that ...