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.
4
votes
1answer
178 views
Stable extensions by line bundles on Riemann surfaces
Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension
$$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$
$E$ ...
5
votes
0answers
94 views
The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
0
votes
0answers
65 views
Semistable vector bundles over the complex projective line are trivial bundles
I currently found some argument about semi-stable vector bundle over the complex projective line, which says that
"Over the complex projective line, a (holomorphic) vector bundle is semi-stable if ...
1
vote
1answer
33 views
Conditions for a pushforward of a involutive vector bundle to be involutive
I know that the following statement is true, but I would like to find a reference for it so I don't have to write the proof. Do you guys have a reference?
Let $\Omega$ and $\Omega'$ be smooth ...
4
votes
0answers
160 views
Is a Procesi bundle equipped with a hyperholomorphic connection?
Haiman has constructed in the paper the unusual tautological bundle $P$, called Procesi bundle, of rank $n!$ over the Hilbert schemes of points on the affine plane in the following way.
Let $H _ { n }...
1
vote
0answers
71 views
example of rank 2 torsion free sheaf with no global sections that is not stable
Ìn the book "Vector bundles on complex projective spaces" the authors prove in Chapter 2 Lemma 1.2.5 that, if $E$ is a rank $2$ reflexive sheaf on $\mathbb{P}^n$, then $E$ is stable if, and only if, $...
2
votes
0answers
44 views
Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)
Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...
6
votes
1answer
204 views
Why the Thom spectrum of $-\xi$ (or more generally of a virtual bundle) is defined as it is?
As the title suggests, I'm trying to find motivation on the definition of the Thom spectrum of $-\xi$, or more generally on the definition of the Thom spectrum of a virtual bundle.
In this paper by S....
0
votes
1answer
199 views
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\...
3
votes
0answers
60 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....
1
vote
0answers
58 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 "...
0
votes
1answer
146 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 $L$ ...
3
votes
1answer
218 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 ...
4
votes
0answers
131 views
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-...
1
vote
0answers
77 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 ...
15
votes
0answers
422 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 ...
0
votes
0answers
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....
2
votes
1answer
115 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 ...
4
votes
1answer
224 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 ...
3
votes
1answer
284 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 ...
9
votes
1answer
219 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 ...
3
votes
0answers
158 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}...
6
votes
2answers
242 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 $V_i:=E_i/E_{i-1}$ ...
1
vote
1answer
86 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 ...
0
votes
1answer
117 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 $0$?
0
votes
1answer
281 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 ...
3
votes
0answers
70 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 \...
4
votes
0answers
71 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 ...
2
votes
1answer
94 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 ...
2
votes
1answer
52 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 ...
1
vote
1answer
130 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 $...
2
votes
0answers
49 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 $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....
2
votes
0answers
146 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 $...
10
votes
3answers
224 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\...
5
votes
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 ...
1
vote
1answer
135 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 ...
3
votes
0answers
112 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 $...
8
votes
0answers
214 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 $(End(E) \otimes K_C)_{+}$ ...
1
vote
0answers
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 $ ...
2
votes
0answers
111 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 ...
4
votes
0answers
129 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 \...
3
votes
0answers
59 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 ...
9
votes
1answer
236 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 ...
2
votes
0answers
129 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 ...
5
votes
0answers
66 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 $k$ ...
5
votes
2answers
251 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 ...
2
votes
1answer
83 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 ...
4
votes
0answers
55 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 $(-\...
6
votes
1answer
228 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 ...
1
vote
0answers
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 ...
