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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133 views

Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
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58 views

Reference request for a well-known lemma in Parabolic Vector Bundle

In the paper- "Moduli Space of parabolic vector bundles on a curve" - Usha N Bhosle, Indranil Biswas-Beitr Algebra Geom (2012), 53:437-449, DOI: 10.1007/s13366-011-0053-7, Lemma $2.1$ is being ...
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31 views

Moduli space of Parabolic Vector Bundles with arbitrary parabolic weights

I have just posted another question related to Moduli Space of Parabolic Vector Bundles on Curve. The questions came up when I was trying to read the paper (Desingularisation of the Moduli Varieties ...
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78 views

Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...
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103 views

Quaternionic projective bundle in complex Grassmann bundle

"What is the fundamental class of the projective bundle of lines of a quaternionic bundle in the Grassmann bundle of 2-planes of the underlying complex bundle?" In Quaternionic projective space in ...
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151 views

Symplectic structures on the total space of vector bundles

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure? To what extent all ...
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1answer
103 views

A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$

According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows: Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$. Is there a (...
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732 views

semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity. For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
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1answer
182 views

Scheme of Higgs reductions

I'm reading the Bruzzo and Graña Otero's paper Semistable and Numerically Effective Principal (Higgs) Bundles; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, ...
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339 views

Are homology spheres stably trivial?

A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...
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323 views

The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. 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 projective curve. Let $S$ ...
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148 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
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1answer
191 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: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$ In many Books such ...
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0answers
71 views

group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
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2answers
258 views

Global section of universal bundle on Grassmanian

Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (...
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1answer
153 views

Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\...
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1answer
131 views

Transitivity of “being the zero locus of a section of a vector bundle”

Let $Z\subset Y\subset X$ be smooth projective $k$-varieties. Suppose $Z$ (resp. $Y$) is defined in $X$ by the vanishing of a section of a vector bundle $E$ (resp. $F$). Is it true that $Z\subset Y$ ...
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110 views

Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
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1answer
405 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 (wrong): For $n\leq 2$, ...
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269 views

The “Rolle theorem” for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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1answer
110 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) ...
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217 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, ...
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33 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 $C(...
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1answer
241 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 $PT_{X}.$ ...
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149 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 $...
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2answers
2k views

First Chern class of a flat line bundle

A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one? Let $X$ be a nice space (...
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2answers
253 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 ...
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2answers
1k views

Riemannian metrics as sections of a vector bundle

Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle $(S^...
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9answers
9k views

What are some open problems in algebraic geometry?

What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
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226 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 $$ T(\xi\...
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1answer
151 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" ...
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112 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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70 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 $E$ ...
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1answer
233 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 ...
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3answers
1k views

How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best possible:...
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1answer
240 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 ...
6
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1answer
199 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 $E\...
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1answer
209 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 ...
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1answer
68 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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90 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 $U=Y\...
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2answers
510 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. ...
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1answer
81 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 ...
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1answer
76 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 ...
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262 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 ...
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415 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 ...
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1answer
198 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 \...
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73 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 ...
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104 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 - 1}(\...
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1answer
57 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 M\...
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2answers
214 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 $\xi\...