A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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5
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1answer
285 views

Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample? And just a small note here: For $n\leq 2$, we can ...
0
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0answers
88 views

Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\int_M\|df\|dvol_g\qquad f:(M,g)\to(N,h).$$ In many Books such as Calculus ...
1
vote
1answer
88 views

homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe: $\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...
13
votes
1answer
153 views

Vector bundles with exactly one nonzero SW-class

I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...
0
votes
0answers
30 views

A C(B)-module structure on the function algebra of the total space of a vector bunlde $\pi:V \to B$

For a continuous vector bundle $\pi:V \to B$ vector bundle over a compact Hausdorff space $B$, and $C(B)$, $C(V)$ the continuous complex valued functions on $B$ and $V$ respectively, we can give ...
3
votes
1answer
138 views

Definition of Levi-Civita connection map and a theorem about it?

Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem: Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and ...
3
votes
2answers
207 views

Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...
2
votes
1answer
208 views

Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$ ...
6
votes
2answers
213 views

The Thom space of a Whitney sum of vector bundles

Let $\xi$ and $\eta$ be vector bundles over the same base space $X$. Their Whitney sum is a bundle $\xi\oplus\eta$ over $X$. I read somewhere (without proof) that its Thom space is given by $$ ...
3
votes
0answers
108 views

“Parameterising” extensions $0\to E\to W\to F\to 0$ by $\mathbb P(H^1(E\otimes F^*))$?

Let $X$ be a complex projective manifold and $E$ and $F$ be holomorphic vector bundles on $X$. The extensions of $F$ by $E$ are classified by elements $e\in H^1(E\otimes F^*)$. On the other hand, for ...
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0answers
69 views

A family of rank two bundles on $\mathbb CP^1$ parameterized by $H^1(O(-2n))$

Let $n$ be a positive integer. Consider the family of extensions $$0\to O(-n)\to E\to O(n)\to 0,$$ parameterized by $H^1(O(-2n))$. For each element $e\in H^1(O(-2n))$ we get a rank two bundle $E$ ...
3
votes
1answer
210 views

Confusion surrounding the Koszul-Malgrange theorem

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem. According to nlab, the ...
7
votes
1answer
236 views

Closed formulas for topological K-theory?

Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line ...
0
votes
1answer
66 views

Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf?

Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact. Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. ...
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0answers
87 views

Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set ...
4
votes
1answer
74 views

Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
6
votes
1answer
186 views

Very stable vector bundles

Let $X$ be a smooth curve, and $E$ a rank $r$ vector bundle over $X$, $E$ is said very stable if every nilpotent map $$u:E\rightarrow E\otimes K_X$$ is zero (nilpotent means that the composition ...
8
votes
2answers
435 views

Moduli space of (all) vector bundles on $\mathbb{P}^1$

It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles. ...
1
vote
1answer
75 views

Triviality of a circle fibration induced by an almost complex structure

Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More ...
3
votes
2answers
239 views

Borel--Bott--Weil for the Grassmannians

The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians? More precisely, suppose $G(\mathbf ...
3
votes
0answers
70 views

How to visualize the dual objects of jets of functions?

I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...
2
votes
1answer
173 views

Tangent bundle of a homogeneous space and the euler exact sequence

Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$: $$0 \to \mathfrak{h} \to \mathfrak{g} \to ...
5
votes
1answer
97 views

Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ [closed]

Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}_k(S^q) \simeq \pi_{q - ...
3
votes
1answer
55 views

vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle $$ \xi(M,G): \mathbb{R}^n\longrightarrow ...
0
votes
0answers
97 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...
6
votes
1answer
134 views

non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle $$ \xi:\mathbb{R}^k\longrightarrow ...
9
votes
2answers
206 views

non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum ...
11
votes
1answer
223 views

Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group?

This question is a followup to two of my previous questions, see here and here. Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using ...
2
votes
0answers
410 views

On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) ‎‎\subseteq‎‎ TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. ‎L‎‎et $\bar{J} : TTM \longrightarrow ...
7
votes
2answers
137 views

Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?

Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?
13
votes
1answer
250 views

Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?

Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...
4
votes
0answers
154 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
7
votes
1answer
273 views

When is a general sheaf (on the projective plane) globally generated?

Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...
5
votes
0answers
99 views

Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that ...
1
vote
1answer
119 views

Transferring connection information to associated bundles and back

This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try. At the risk of repeating well known stuff I tried ...
4
votes
1answer
207 views

Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...
2
votes
1answer
149 views

the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
4
votes
0answers
161 views

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
0
votes
0answers
73 views

Horizontal vector fields and the push forward of the differential of the projection map

I am not very familiar with differential geometry but need to understand some aspects of it for my research. This includes in particular the notion of horizontal vector fields and I would like to ...
0
votes
1answer
162 views

Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...
1
vote
1answer
176 views

If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable?

Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$. Let $E$ be a vector ...
3
votes
2answers
476 views

Principal bundles that can't be detected by spheres

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are ...
6
votes
2answers
286 views

Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$

Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}_{\mathbb P^n}^{r} \longrightarrow ...
3
votes
1answer
90 views

self-Whitney sum of the canonical vector bundle on Grassmannians

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow ...
0
votes
0answers
93 views

Sheaf of Sections of Cone

Fulton's intersection theory book at Chapter 4 makes the following claim: If $\mathcal{E}$ is a locally free sheaf (on $X$), and $E:=Spec(Sym(\mathcal{E}))$ a total space of some cone/bundle on $X$, ...
8
votes
2answers
280 views

Maximal trivialising subspace for a vector bundle

Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...
8
votes
2answers
496 views

Vector bundles, finitely generated projective module?

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
2
votes
0answers
62 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
3
votes
0answers
173 views

Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type? [closed]

Using Stiefel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
5
votes
2answers
247 views

Relative Characteristic classes

A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...