All Questions
Tagged with line-bundles dg.differential-geometry
18 questions
3
votes
0
answers
218
views
Line bundles with meromorphic transition functions
I have the following situation: let $X$ be a projective complex manifold and let $f \in H^1(X,\mathcal{M}^{\times})$. So $f$ defines something like a line bundle with meromorphic transition functions.
...
- 153
12
votes
3
answers
711
views
Modern treatment of Dirac monopoles and related topics
I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s)...
- 5,215
1
vote
1
answer
285
views
Riemannian vector bundle [closed]
I'm trying to show the curvature of a one dimensional vector bundle with a Riemannian metric vanishes, no matter what the connection is. I found this can be done for orientable bundles, because an ...
- 21
4
votes
0
answers
142
views
Classification of square roots of line bundles and metalinear/metaplectic structures
Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
9
votes
2
answers
405
views
Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
- 5,824
4
votes
2
answers
307
views
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
- 311
3
votes
1
answer
963
views
Isomorphism classes of line bundles with connections
Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...
- 587
1
vote
1
answer
210
views
Hermitic connections on complex line bundles with imaginary curvature form
It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
- 5,407
2
votes
2
answers
526
views
Holomorphic Line Bundles over a Homogeneous Space
Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...
0
votes
4
answers
474
views
An isomorphism on space of smooth sections
Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to \mathbb{...
user21574
0
votes
2
answers
435
views
Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
- 1,329
2
votes
2
answers
1k
views
Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
user21574
9
votes
1
answer
1k
views
Non-compact Kähler manifolds which admit a positive line bundle
A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...
- 19.4k
9
votes
1
answer
975
views
Is there a mathematical explanation for the Aharonov-Casher effect?
Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows.
Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
- 37.8k
0
votes
2
answers
3k
views
Line bundles with complex connection
Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
- 1,025
15
votes
1
answer
1k
views
Are bundle gerbes bundles of algebras?
The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(...
- 37.8k
3
votes
1
answer
700
views
Pulling back a line bundle on the Jacobian to a spin bundle on the curve
I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta ...
7
votes
5
answers
3k
views
Indexing the line bundles over a Grassmannian.
As is well known, the line bundles over *CP*$^1$ are indexed by the integers. My question is how are the line bundles over *CP*$^n$, $n > 1$, and *Gr*$(n,k)$ indexed? Moreover, do there exist any ...
- 3,409