For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.
A line bundle is a vector bundle of rank $1$ over a topological space.
Questions tagged [line-bundles]
299 questions
77
votes
1 answer
Divisor -- line bundle correspondence in algebraic geometry
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of Cartier divisor there is very confusing. It is certainly not…
user977
17
votes
2 answers
Can I comb unoriented hair on a ball?
I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball.
(I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every…
Joonas Ilmavirta
- 26,345
9
votes
2 answers
Remembering Riemann-Roch
Embarrassingly, I've always struggled to remember the form of the Riemann-Roch theorem for curves. Does anyone have any intuition to share about how to remember the some of the terms in the formula?
Recall that for $C$ a Riemann surface and $D$ a…
Maxim
- 488
9
votes
4 answers
Examples of non trivial vector bundles
Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles.
Here, I want to collect such examples with justification of one or two lines saying why this vector bundle is non trivial.
Please add…
user537667
7
votes
1 answer
Geometric interpretation of the isomorphism $\mathcal{N}_{Y/X} \cong \mathcal{O}_X(Y) \vert_Y$
Let $X$ be a smooth variety / manifold over $\mathbb{C}$ of dimension $n$ and suppose that $Y \subset X$ is a
smooth $n-1$-dimensional subvariety. The normal bundle $\mathcal{N}_{Y/X}$ comes from exact sequence
$$ 0 \to \mathcal{T}_Y \to …
user267839
- 9,217
7
votes
1 answer
Restriction of the exceptional divisor to itself as a line bundle
If we take a complex projective variety $X$ and blow it up at a point, we get an exceptional divisor $E\cong \mathbb{P}^{n-1}$, where $n=dim(X)$.
My question basically regards $\mathcal{O}_{\tilde X}(E)$ or to be more precise $\mathcal{O}_{\tilde…
Notone
- 2,460
7
votes
0 answers
How to write holomorphic line bundles on compact Riemann surfaces in coordinates
I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$.
For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let
\begin{align*}
U_0 &= \{ [x_0 : x_1] \mid x_0 \neq 0 \} = \{…
Earthliŋ
- 2,592
6
votes
1 answer
Removing zero sections of n-twisted strips
First of all, a note: This question is motivated by my friend, who is a physics PhD student (I'm a math PhD student) and arose when I was trying to explain to him what line bundles are. (I wanted to tell him that a wavefunction is really a global…
Cranium Clamp
- 3,423
- 1
- 10
- 30
6
votes
2 answers
Line subbundles of tangent bundle of a manifold is trivial?
I was pretty sure about this result but don't know how to prove it. I will state the question again:
Is any smooth line subbundle (or equivalently smooth 1-dimensional distribution) of the tangent bundle of a manifold is always trivial? Namely, once…
J. Yang
- 523
6
votes
1 answer
Geometric meaning of the degree of the normal bundle $\mathcal{N}_{C/X}$
Assume all varieties are projective and smooth over $\Bbb{C}$.
Let $X\subset\Bbb{P}^3$ be surface and $C\subset X$ a curve on it.
The normal bundle $\mathcal{N}_{C/X}$ is the cokernel of the map $T_C\hookrightarrow T_X\big|_C$, which intuitively…
rmdmc89
- 10,709
- 3
- 35
- 94
6
votes
1 answer
Degree of the Normal Bundle of a Line in a Projective Variety
Let $S\subset \mathbb{P}^3$ a smooth surface of degree $d$, and $L$ a line in $S$. I would like to compute the self intersection of $L$, which I know to be $L^2=deg(N_{L/S})$, where the last one is the normal bundle of $L$ in $S$. Also, I could…
Serser
- 189
5
votes
2 answers
Definition of base-point-free sheaf
Let $X$ be a scheme with an invertible sheaf $\mathcal{L}$. We call it base-point-free if for any $x\in X$ we can find an $s\in\Gamma(X; \mathcal{L})\equiv \mathcal{L}_X(X)$ such that $s(x)\neq 0$.
Question: What do we mean when we write $s(x)$? I…
A.D.
- 417
5
votes
0 answers
Which line bundle on $A$ is the pull-back of the Poincaré bundle via a given morphism $A \to A^*$?
Let $A$ be an Abelian variety (over an algebraically closed field of characteristic zero), and denote by $A^*$ its dual, which parameterizes degree zero line bundles on $A$. If $Q$ is a point of $A^*$, the corresponding degree zero line bundle on…
57Jimmy
- 6,408
5
votes
1 answer
Non trivial holomorphic section
Hello, let L be a holomorphic line bundle over a compact complex manifold of dimension 2. Suppose $\int_{X}c_{1}(L)^{2} > 0$ ($c_{1}$ means first Chern class). I would like to show $L^{\otimes m}$ or $L^{\otimes -m}$ admits a non vanishing…
Analyse300
- 121
5
votes
3 answers
Holomorphic line bundles with trivial Chern class are flat
Let $X$ be a complex, projective algebraic variety and let's work in the differential-complex setting.
Let $L$ be a non-trivial hermitian holomorphic line bundle and assume that $c_1(L)=0$. Can we always find a connection such that the associated…
Dubious
- 14,048