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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Homology of Hirzebruch surfaces
Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\...
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51 views
The heat kernel in Hermitian bundles over Riemannian manifolds
In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
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53 views
Vector Bundle Structure of Conformal Block Bundle
Apologies in advance if this question is too elementary but I wasn't expecting any luck asking on math.stackexchange. Anyway, my question is about the conformal block bundle, which (following Kohno's "...
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1answer
140 views
Dual of a stable locally free subsheaf is a locally free quotient sheaf
Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 1.2:
A line bundle $L$ ...
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208 views
Motivation for classifying vector bundles
The statement I am familiar with regarding classification of vector bundles is :
Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective ...
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Harder-Narasimhan over arbitrary coefficients
Let $X$ be an $n$ dimensional smooth projective variety over $k$. Let $H$ be a hyperplane section. Define the slope $\mu(E)=\frac{c_1(E).H^{n-1}}{rank(E)}$ for vector bundles on $X$. Does the Harder-...
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71 views
First cohomology class of co-sphere bundle of $\mathbb{C}P^n$ [closed]
Pick a Riemannian metric on $\mathbb{C}P^n$ (say the one coming from Kähler structure). That gives us a Riemannian metric on $T^*\mathbb{C}P^n,$ so we define the co-sphere bundle $S^*\mathbb{C}P^n$ as ...
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400 views
What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
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44 views
different reductions for a vector bundle
Given a vector bundle with structure group G that has two proper Lie subgroups $G_1,G_2\subset G$, to both of which the bundle can be reduced; Is there possibilities to deduce a further reduction (e.g....
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1answer
108 views
Sections of vector bundles with exactly one zero
In this question asking whether a connected manifold with Euler characteristic zero has a vector field without zeroes, there is the following comment by Tom Goodwillie (see here):
"Two zeroes of ...
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1answer
214 views
Confusion in known result about Moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2
Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The ...
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271 views
Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
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216 views
Fourth obstruction, Pontryagin and Euler class
Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first ...
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134 views
When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?
Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...
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234 views
Harder Narasimhan filtration for the endomorphism bundle
Let $E$ be a vector bundle over a compact Riemann surface $X$, and let $$0=E_0\subsetneq E_1\subsetneq \ldots \subsetneq E_n=E$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ ...
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1answer
79 views
Gluing locally defined continous functions over complex domain
This is a cross-post to the question I asked at MSE over almost a month ago.
Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...
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114 views
semistability of an extension of bundles
Is there a non-semistable bundle of rank $3$ and degree $1$ which is an extension of a stable bundle of rank $2$ degree $1$ by a stable bundle of rank $1$ degree $0$?
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209 views
About universal Ext space
Let $X$ be a smooth projective curve of genus $g$. Consider the Ext space over $J_{d_1}\times J_{d_2}$ i.e., the vector space of isomorphism classes of extensions of line bundles of degree $d_2$ by ...
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67 views
Stable torsion free sheaf on smooth projective surface
Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \...
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69 views
Existence of sections of a fibre bundle which are covariantly constant along certain directions
Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...
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92 views
When is the moduli of generalized parabolic bundles with fixed determinant smooth?
Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
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1answer
51 views
Set of sections whose zeroes avoid a given divisor is (Zariski) dense?
Let $X$ be a smooth complex projective variety of dimension $n$, and let $\mathcal{F}$ be a globally generated rank $n$ vector bundle on $X$. Let $D$ be a smooth divisor on $X$.
Is it true that ...
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1answer
117 views
Restriction of vector bundles
I am trying to compute the Chern classes of the restriction of a rank two vector bundle on $\mathbb{P}^3$, denoted by $E$, with fixed Chern classes, $c_1(E) = c_1$ and $c_2(E) = c_2$, to a hyperplane $...
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47 views
Singularities of quotient of a vector bundle by a lattice
Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....
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141 views
Extension of a rational section of a projective bundle
Let us assume that we work over the complex field and let $X$ be a smooth projective variety and $\pi: P \to X$ a projective bundle (i.e. a fibration in projective spaces of constant dimension). Let $...
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3answers
216 views
Do $\mathbb{HP}^2\#\overline{\mathbb{HP}^2}$ and $\mathbb{OP}^2\#\overline{\mathbb{OP}^2}$ arise as sphere bundles over spheres?
Recall that $\mathbb{RP}^2\#\mathbb{RP}^2$ is the Klein bottle and can be seen as a non-trivial $S^1$-bundle over $S^1$. In particular, it is the total space of the sphere bundle of $\gamma\oplus\...
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2answers
192 views
Elements of graded algebra associated with the algebra of differential operators as smooth sections
Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential ...
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1answer
119 views
Connection 1-form of the frame bundle associated to a vector bundle with a connection
Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.
We ...
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103 views
Extending vector bundles over a regular divisor in a regular affine scheme
This is more-or-less question (3) on page 170 of Quillen's "Projective Modules over Polynomial Rings" (link):
Let $A$ be a regular Noetherian ring and let $f \in A$ be an element of $A$ such that $...
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210 views
Endomorphism of involution invariant vector bundles
Let $C$ be an hyperelliptic curve with an involution $\sigma$, and let $E$ be a rank $2$ stable involution invariant vector bundle on $C$, that is $\sigma^{*}E \cong E$. Let $(End(E) \otimes K_C)_{+}$ ...
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114 views
Which object related to families of algebraic varieties over a scheme $ S $ corresponds to the tensor product of vector bundles?
I asked yesterday on math.stackexchange.com that if the fiber product of two vector bundles seen in general as the fiber product of two families of special algebraic varieties over a scheme $ S $ ...
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109 views
Endomorphism sheaves of vector bundles
Let $X$ be a hyperelliptic curve, $\pi: X \to \mathbb{P}^{1}$ denote the ramified covering and $W$ the set of Weirstrass points. Let $F,G$ be two involution invariant (with respect to the ...
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128 views
Exercise in the book “Lectures on Kähler geometry”
I am currently studying the book "Lectures on Kähler geometry" by Andrei Moroianu and am looking for help concerning Exercise 5.8 (3) which is to prove the following Lemma 5.11
Let $f: M \...
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55 views
Existence of very Stable bundle on a compact Riemann surface
Let $C$ be a compact Riemann surface of genus $g>1$ and $L$ be a fixed line bundle of degree $d>2g-3$. Does there exists a very stable bundle $E$ on $C ?$ ( $E$ very stable bundle means ...
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229 views
Positive cones in K-groups
Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just ...
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124 views
semistability of parabolic bundles
Let $X$ be a rational curve and $E$ a stable parabolic vector bundle on $X$. Is the sheaf of parabolic endomorphisms (i.e the endomorphisms preserving the flag) of the sheaf $E$ also stable (or ...
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63 views
Principal $G$-bundles on affine toric varieties
Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ ...
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2answers
242 views
Interesting examples of vector bundles on hyperkahler varieties
I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already:
For $X$= the Hilbert scheme of points $S^{[n]}$ on a ...
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1answer
78 views
Extend a gauge transformation
Suppose $M$ is a smooth manifold and $P$ is a principal bundle on $M$. Let $U^\prime\Subset U\Subset M$ be strictly contained precompact open subsets. Let $g\in C^\infty(U, \hbox{Ad}P|_U)$ be a ...
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54 views
Vector bundles on space of germs
Let $X$ be the diffeological space of germs of paths $c: \mathbb{R} \rightarrow \mathbb{R}^n$, where two paths $c_1, c_2$ are equivalent if $c_1(t) = c_2(t)$ for all $t$ in some interval $(-\...
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1answer
226 views
Role of stably free modules in algebraic geometry
For any ring $R$, a non-zero module $S$ is stably free if $S\oplus R^a$ is free ($a\geq 1$). This may be an overly vague question, but I am wondering in what contexts do stably free modules arise in ...
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46 views
Tangent space and a subset of a tame Lie group
I am curious if the set of all orientation-preserving diffeomorphisms with a given rotation number is a tame Lie subgroup or a tame submanifold of all orientation-preserving diffeomorphisms on the ...
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64 views
A refined comparison of isomorphic vector bundles on a compact space
Let $X,Y$ be $2$ topological spaces such that the compact open topology on $Homeo(X)$ makes it as a topological group where $Homeo(X)$ is the group of homeomorphisms of $X$.
We define an ...
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1answer
586 views
Holomorphic line bundles on $\mathbb{P}^1$ from gluing data
Let $g$ be an $n \times n$ matrix of functions $g_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is ...
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1answer
325 views
Isomorphism classes of line bundles with connections
Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...
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174 views
Vector fields on quasi-spheres
In 1962, Adams proved that there do not exist $\rho(n)$ linearly independent vector fields on the sphere $S^{n-1}$, where $\rho(n)$ is the Hurwitz-Radon number. I wonder if this is still true in the ...
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83 views
Spinor bundle tensored with certain line bundle gives the dual spinor bundle
Let $E$ be a $spin^c$ bundle and $L_E$ be a (complex) line bundle defined using transition functions $\nu \circ g_{U,V}$ where $\nu:spin^c(n) \to \mathbb{T}$ is map such that $\ker \nu=spin(n)$ and $...
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1answer
348 views
First Chern class of a specific line bundle
Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider ...
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1answer
91 views
On sections into Banach bundles over a compact manifold
Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...
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126 views
theta bundle on Moduli of vector bundles
Let $\mathcal{M}$ be the moduli stacks of vector bundles of degree $0$ and rank $n$ over smooth curve $X$, $\mathcal{M}_{\mathcal{O}}$ be the moduli stack of vector bundles with fixed determinant $\...
