Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...