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0
votes
1answer
199 views

eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...
6
votes
2answers
399 views

Homotopy invariance of vector bundles by parallel transport: reference needed for my students.

Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are ...
1
vote
0answers
105 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
0
votes
1answer
302 views

Cotetrad, spin connection and Dirac operator

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...
2
votes
0answers
69 views

Properties of special Cayley-Bacharach bundles on a K3-surface

Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$. Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We ...
4
votes
1answer
382 views

Representations of infinite dimensional Lie algebras as vector fields on manifolds

Suppose I have e.g. the Witt algebra, $\left[l_n,l_m \right] = -(n-m)l_{n+m}$. I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the $l_n$ span the Lie ...
1
vote
1answer
356 views

Schemes associated to vector spaces

Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. ...
7
votes
2answers
353 views

References/surveys concerning characteristic classes of flat vector bundles

I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}_n(\mathbb{R})$ with the discrete topology, orientable flat $n$-dimensional real vector ...
0
votes
1answer
398 views

Zero locus of a generic smooth section

Let $V$ be a smooth manifold, $E \rightarrow V$ a vector bundle over $V$ and $\Gamma$ be a finite group acting nontrivially on $V$ and $E$. Let $s \in C^\infty(E)$ be a generic $\Gamma$-equivariant ...
4
votes
1answer
477 views

etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...
1
vote
1answer
278 views

dual of locally free sheaf

For simplicity, assume everythings occur on a smooth projective variaty $X$. Dual bundle of the given line bundle $\mathcal L$ is determined by $\mathcal L$ and $c_1(\mathcal L)$. $\mathcal L^*= ...
12
votes
6answers
861 views

Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
4
votes
3answers
2k views

Vector bundles vs principal $G$-bundles

It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres $$F=\pi^{-1}(x), \ \ \ x\in B $$ over any $x\in B$, are ...
9
votes
2answers
574 views

Configuration spaces and non homeomorphic vector bundles

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign. ...
37
votes
4answers
2k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
8
votes
3answers
803 views

Relationship between monodromy representations and isomorphism of flat vector bundles

This question is somehow related to this one. Let $M$ be a smooth (compact, if you wish) connected manifold. Then, it is well known that there is an equivalence between the isomorphism classes of ...
7
votes
1answer
256 views

Does direct limit commute with functor of smooth sections?

Consider a countable family of finite-rank vector bundles $V_k$ over a finite-dimensional smooth manifold $M$. The direct limit of such a family is still a topological vector bundle even though it may ...
1
vote
1answer
204 views

Criteria for acyclicity

Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally, a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is, for the property $H^i(X,E)=0$ ...
3
votes
1answer
399 views

non-trivial locus of a holomorphic vector bundle

Let $X$ be a holomorphic vector bundle over $Y$ (where $Y$ is an arbitrary complex manifold, not necessary projective). Does there exist an analytic subset $Z$ of $Y$ such that the restriction of $X$ ...
0
votes
0answers
197 views

Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?
1
vote
2answers
413 views

complex vector bundles and curvature

Let us suppose that $X$, with a 2-form $\omega$. Suppose $J$ is an element of $su(2)$ such that $J^2=-e$ for $e$ the identity. Is there a necessary and sufficient condition on $\omega$ which will give ...
7
votes
1answer
222 views

Triviality of direct multiples of flat complex vector bundles

Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
3
votes
2answers
536 views

Stiefel-Whitney classes of a projective space bundle

Hi! Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...
4
votes
1answer
380 views

Associated vector bundles of infinite rank and induced connections

Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...
5
votes
1answer
318 views

A^1-invariant for Vector Bundles?

We know that if $X$ is a smooth connected variety over a field $k$, then any line bundle on $X\times_k\mathbb{A}^1$ is from a line bundle on $X$. This is simply because they have the same Picard ...
12
votes
1answer
617 views

Splitting principle for holomorphic vector bundles

Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a ...
2
votes
1answer
240 views

Do cohomologically trivial line bundles affect morphisms?

Assume we have a locally free sheaf $R$ of associative $O_S$-algebras of rank $r^2$, possibly noncommutative, where $S=\mathbb{P}^2$ over $\mathbb{C}$. Furthermore given a locally free $O_S$-module ...
2
votes
1answer
353 views

fano moduli varieties of vector bundles

Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is ...
3
votes
1answer
305 views

A simple question about the degree of some vector bundle over rational curve.

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, ...
8
votes
1answer
458 views

Weak Vector Bundles

The following notion has arisen in a paper I'm writing. Definition. A map $p: E\to B$ of spaces is said to be weak vector bundle if for all compact subspaces $K \subset B$ the restriction of $p$ to ...
2
votes
1answer
232 views

family of torsors and family of vector bundles

Suppose $X$ and $Y$ are smooth connected schemes over a field $k=\bar{k}$, $f: X\times_kY\to X$ is the first projection. You may assume $Y$ is proper if you like, my question is if $P\to X\times_kY$ ...
1
vote
1answer
231 views

Intersection of subvector bundles

Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi_1: E_1 \rightarrow B_1$ and $\pi_2: E_2 \rightarrow B_2$ such that the bases $B_1$ and $B_2$ are ...
0
votes
2answers
361 views

Normal bundle of $CP^1$ in $CP^2$ [closed]

I'm studying the book "Differential forms of algebraic geometry" of Bott, Tu. At page 75 there is an exercise about the normal bundle of $CP^1$ in $CP^2$, and there is written that the transition ...
3
votes
1answer
662 views

Terminology of “covariant derivative” and various “connections”

I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to ...
2
votes
2answers
402 views

degree 0 vector bundles

If $X$ is a smooth projective curve over a field $k$, $V$ is a subbundle of some finite direct sum of $O_X$, then is it true that $V$ is of degree 0?
2
votes
1answer
259 views

When is restriction an equivalence of categories of equivariant vector bundles?

Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor $i^*: Vect^G(X) ...
0
votes
1answer
203 views

Restriction of the tautological class to a subbundle

Let $E$ a rank $r\geq 3$ vector bundle over a curve $C$ and let $E'$ a rank $r-1$ subbundle of $E$. Thus we have $\mathbb{P} (E') \subset \mathbb{P} (E)$; what can be said about $ ...
3
votes
0answers
178 views

Extending intersection bundles

Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that ...
2
votes
1answer
645 views

Linearization of a vector field

In a paper that I was reading, I stumbled across the following theorem: Let $X$ be a vector field with $$X= > a^ix^i\partial_{x^i} + > \mathcal{O}(|x|^2),$$ where $x$ is some chart and ...
11
votes
0answers
313 views

Atiyah-Bott from Beauville-Laszlo

This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
9
votes
2answers
719 views

Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
1
vote
0answers
287 views

Splitting of vector bundles on a complex torus

Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is ...
1
vote
1answer
208 views

A specific degeneration of a rank 2 bundle

I wish to know if there is a rank 2 vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $\mathbb{P}(E)$ when restricted to $\mathbb{P}^1 \times [0:1]$ is the $n$th Hirzebruch surface and ...
15
votes
0answers
476 views

Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb ...
4
votes
1answer
354 views

A vector bundle with a given jumping line

I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to ...
2
votes
1answer
560 views

Spectral sequence of symmetric or exterior algebras?

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads: Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence ...
1
vote
2answers
301 views

Top self-intersection of the tautological line bundle

Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection ...
28
votes
3answers
2k views

When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
4
votes
1answer
300 views

Reference for nef coherent sheaves?

The definition and basic properties of nef locally free sheaves appear for instance in the second volume of Lazarsfeld's book "Positivity in Algebraic Geometry" (beginning of chapter 6). However, I ...
2
votes
1answer
165 views

Closed immersion into (relative) projective bundle.

Hello, I have a question about certain immersions. If $X$ is a base scheme, and $E$ is a loccally free quasi coherent sheaf over X, then we can build $P(E)$ the projective bundle associated to E. ...