**0**

votes

**1**answer

321 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

**0**answers

71 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

**1**answer

397 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

**1**answer

381 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

**2**answers

393 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

**1**answer

411 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

**1**answer

498 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

**1**answer

305 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

**6**answers

933 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 ...

**6**

votes

**3**answers

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

**2**answers

610 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

**4**answers

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

**3**answers

966 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

**1**answer

259 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

**1**answer

213 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

**1**answer

404 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

**0**answers

205 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

**2**answers

433 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

**1**answer

248 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

**2**answers

546 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

**1**answer

400 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

**1**answer

335 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

**1**answer

668 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

**1**answer

248 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

**1**answer

372 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

**1**answer

317 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

**1**answer

463 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

**1**answer

243 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

**1**answer

238 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

**2**answers

385 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

**1**answer

732 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

**2**answers

408 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

**1**answer

264 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

**1**answer

204 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

**0**answers

180 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

**1**answer

724 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

**0**answers

331 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

**2**answers

807 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

**0**answers

301 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

**1**answer

210 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 ...

**17**

votes

**0**answers

501 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

**1**answer

367 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

**1**answer

625 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

**2**answers

313 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 ...

**29**

votes

**3**answers

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

**1**answer

304 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

**1**answer

168 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.
...

**2**

votes

**0**answers

110 views

### Bundles of (co)chain complexes

A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic ...

**2**

votes

**1**answer

532 views

### Extending a vector bundle to a torsion free sheaf

Let's say that $X$ is an integral scheme of finite type over a field and $Y \subset X$ is a closed subscheme. Given a vector bundle $E$ on $Y$, is $E$ the restriction to $Y$ of a vector bundle on a ...

**5**

votes

**1**answer

414 views

### K-Theory as a special $\lambda$-ring

I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...