Questions tagged [line-bundles]
A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.
231 questions
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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
(...
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Cohomology theory for symplectic manifolds
Suppose I have a symplectic manifold $(M,\omega)$ and a line bundle $\mathcal L$ with a connection with curvature $\omega$ (or perhaps it's more standard to say $\frac i{2\pi}\omega$; anyway, the ...
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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 ...
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Is there an algebraic construction of the Quillen (determinant) Line Bundle?
Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let $Y=\operatorname{Hom}(\pi,G)/\\!/G$...
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1
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Followup; Strict Transform of a Line Bundle
This is a follow up to my previous question, and I have lowered my demands to a situation as follows:
Let $X$ be an algebraic variety, $\mathcal{I}$ a coherent sheaf of ideals and $\mathcal{L}$ a ...
CommunityBot
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1
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3
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On the Clifford index of a curve
Let X be an algebraic curve and c be the Clifford index of X.
When c is small (e.g c=1), what is the classification of the line bundle who computes c?
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4
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2
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Vague question on $Pic^0$
For a smooth variety $X$ when $Pic^0(X)$ is trivial, we get an isomorphism between $N^1(X)$ and Picard group and life become easier.
My question is whether in general there are theorems, criteria ... ...
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Pullback along the Torelli map is an isomorphism
I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard ...
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0
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Deformations of some simple quotient stacks.
I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...
- 1,180
6
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1
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A line bundle not big but with good intersection numbers
Let $X$ be a complex projective manifold of complex dimension $n$ and $A\to X$ an ample line bundle. Let $L\to X$ be a line bundle such that
$$
c_1(L)^k\cdot c_1(A)^{n-k}>0,\quad k=1,\dots,n.
$$
Is ...
- 28.9k
2
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1
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Different ways to construct maps and the tensor products of line bundles
Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective ...
- 66.3k
3
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1
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Can the class of the canonical bundle be recovered from the total space of the cotangent bundle if one forgets that it is a cotangent bundle?
This is a somewhat speculative question, so bear with that (or not, as is your preference).
Let $X$ be a smooth projective variety, and let $\omega_X$ be its canonical sheaf. The Euler class of ...
- 44.7k
5
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1
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graded ring associated to a line bundle in a tensor category
Let $\mathcal{A}$ be an abelian tensor category with unit $\mathcal{O}$. An object $\mathcal{L}$ is called invertible or a line bundle if there is some $\mathcal{L}^{-1}$ such that $\mathcal{L} \...
- 52.7k
5
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Line bundles in abelian $\otimes$-categories
By an abelian $\otimes$-category I mean a symmetric monoidal category $(\mathcal{A},\otimes,\mathcal{O})$, such that $\mathcal{A}$ also is an abelian category and for every $M \in \mathcal{A}$ the ...
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4
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2
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finite-dimensionality of cohomology groups on compact riemann surfaces
does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...
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Total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}^n$
It is well known that total space of the tautological line bundle $\mathcal{O}(-1)$ over projective space $\mathbb{P}^n$ is closed subvariety of $\mathbb{P}^n\times\mathbb{A}^{n+1}$. My question is ...
- 39.3k
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Technique to prove basepoint-freeness
Let $X$ be a smooth projective variety over $\mathbb{C}$.
And let $L$ be a big and nef line bundle on $X$.
I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).
The only way ...
CommunityBot
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1
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Tensor product of a line bundle with a large multiple of another positive line bundle also positive?
Let $X$ be a complex manifold and $\mathcal{L}$ be a positive line bundle on $X$. If $E$ is any other line bundle on $X$, then is it true that for all sufficiently large $m$, $\mathcal{L}^m \otimes E$ ...
- 28.9k
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volume of big line bundles under finite morphisms
Let $X$, $Y$ be complex projective varieties of dimension $n$, let $f:X \rightarrow Y$ be a surjective finite morphism of degree $d$ and let $B$ be a big line bundle on $Y$.
Is that true that vol($f^*...
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5
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What do gerbes and complex powers of line bundles have to do with each other?
We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...
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2
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What kind of line bundles have Chern class of Hodge type (2,0) or (0,2)?
If $L$ is a complex line bundle on a topological space $X$, let $c_1(L)$ denote the image of its Chern class in $H^2(X;C)$. A complex manifold structure on $X$ [ok which is also compact and say ...
- 13.2k
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What does the ample cone look like?
For a variety $X/k$, consider the monoid $A$ of classes of ample line bundles in $NS(X)$. What does $A \otimes_\mathbf{Z} \mathbf{R} \subset NS(X)_\mathbf{R}$ look like?
- 1,366
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Line Bundles on Torus Quotient
Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can ...
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For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)
Hi everyone,
Let $X$ be a smooth projective variety over a field $k$ and let $L$ be a line bundle on $X$. I'm reading the article Heights for line bundles on arithmetic varieties and there one ...
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1
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Line bundles: from transition functions to divisors
Recently I was thinking about how local systems are the same thing as vector bundles with flat connection, and how representations of the fundamental group gave rise to vector bundles. This got me ...
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Relationship between Line Bundles with isomorphic ring of sections
Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\...
- 44.7k
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3
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Holomorphic and antiholomorphic forms of projective space
For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\...
- 21k
4
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1
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Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces
I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally ...
- 30.6k
3
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3
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Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
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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 ...
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4
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Sections of a divisor on elliptic curve
I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...
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