3
votes
0answers
46 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
4
votes
2answers
105 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
5
votes
1answer
95 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
1
vote
0answers
77 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
3
votes
1answer
58 views

Partial Differential Operators on Vector Bundles

can anyone suggest me a nice reference for partial differential operators on vector bundles? Thanks..
2
votes
0answers
40 views

Why can the group of isomorphism classes of line bundles be identified with $H^1(C,\mathbb O_C^*)$?

This is a reference request to the fact in the title. Is there a book at most as advanced as Hartshorn's which explains this result?
3
votes
1answer
289 views

Determinant of a tensor product of two vector bundles

Let $X$ be a smooth variety over a field, $V_1$ and $V_2$ are two vector bundles over $X$ of ranks $r_1$ and $r_2$ respectively. Determinant of a vector bundle is the top exterior power of the vector ...
6
votes
2answers
96 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
3
votes
0answers
95 views

Triviality of (equivariant) holomorphic vector bundles

Let $G$ the 1-dimensional diagonalisable linear complex analytic group $\mathbb C^*$. We suppose that $G$ acts linearly on $\mathbb C^n$ with positive weights. Set $X=\mathbb C^n -\lbrace 0 \rbrace$. ...
3
votes
1answer
132 views

What is curvature, in terms of holonomy functors?

It is well known and understood that linear connections, as holonomy functors, are composition-preserving mappings from the path groupoid to a structure group $G$. This extends the idea of a 1-form ...
6
votes
3answers
132 views

introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
8
votes
1answer
165 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
9
votes
5answers
565 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
4
votes
1answer
84 views

Two questions on jet bundles

I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a ...
4
votes
0answers
173 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...