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
9
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1
answer
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Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
- 137
1
vote
0
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81
views
Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
2
votes
0
answers
128
views
Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
1
vote
0
answers
105
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Isomorphism between spectrum of $\mathcal{O}_{\mathbb{P}^1_{[y_0:y_1]}}[y_0^2t,y_0y_1t,y_1^2t]$ and the line bundle $\mathcal{O}_{\mathbb{P}^1}(2)$
Let $\mathbb{P}^1$ be the projective line over a base field $k$, with homogeneous coordinates $[y_0 : y_1]$. Consider the sheaf of $\mathcal{O}_{\mathbb{P}^1}$-algebras $\mathcal{A} = \mathcal{O}_{\...
- 11
2
votes
0
answers
68
views
On spin structure for Kähler manifolds and square roots of $\det (TX)$
I'm stuck on the proof that for a (compact) Kähler manifold $X$ (of complex dimension $n$), a spin structure on the tangent bundle $TX$ is equivalent to a line bundle $L$ together with an isomorphism $...
- 220
0
votes
1
answer
193
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Is the square of a special line bundle also special?
Suppose $C$ is a smooth projective curve over, say, $\mathbb{C}$. I'm interested in knowing whether the following is true.
Let $\mathcal{L} \in Pic^d(C)$ be a special line bundle, i.e. its $H^1 \neq 0$...
- 129
4
votes
1
answer
288
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Characterizing principal polarizations of abelian surfaces
Suppose $X$ is a complex abelian variety of dimension 2. Then I believe the ring of endomorphisms $\mathrm{End}(X)$, tensored with $\mathbb{C}$, is isomorphic to a subalgebra $M_2(\mathbb{C})$ of $2 \...
- 22.3k
1
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0
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146
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 275
1
vote
0
answers
47
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Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
- 111
4
votes
1
answer
333
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Question regarding the definition of linearization of line bundles
I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...
- 737
3
votes
1
answer
248
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A question on "Ample subvarieties of algebraic varieties"
Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...
- 8,998
4
votes
1
answer
362
views
One-point compactification of ample line bundle
Given a smooth complex projective variety with an ample line bundle $L$, it seems to be folklore that one can get a one-point compactification of the total space $\mathbb{V}(L)$ of $L$ such that ...
- 151
12
votes
3
answers
711
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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
0
votes
1
answer
274
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Universal covering of symmetric product
Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
- 153
1
vote
1
answer
141
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Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?
Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= ...
- 1,286
7
votes
0
answers
242
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Mumford's definition of an abelian variety's $Pic^0$
I'm not sure whether this is a research-level question, but upon skimming through Mumford book of Abelian Varieties I noticed he gives this definition
$$
\begin{equation}
\label{eq}
\text{Pic}^0(A)=\{\...
- 71
3
votes
1
answer
493
views
What are meromorphic line bundles?
Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it ...
- 18.4k
2
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0
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134
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Symmetric group-cocycle descends to symmetric product
Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
- 153
1
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0
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277
views
How to define Cartier divisor and Weil divisor on algebraic stack?
How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler ...
- 523
3
votes
0
answers
218
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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
6
votes
1
answer
340
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Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor
Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...
- 5,966
1
vote
0
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330
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Computing Picard groups of arbitrary quadric hyperplane
I know the Picard group of a smooth two dimensional quadric surface is $\mathbb Z^2$, but I am wondering if the computation can be generalized to higher dimension? In particular, is the Picard group ...
- 231
3
votes
0
answers
208
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Metric on line bundle defined fiberwise
Let $(X,\mathcal{O}_X)$ be an analytic space, and let $L$ be a line bundle on $X$. Intuitively, a metric $||\cdot||$ is a continuous choice of a metric for each fiber of the line bundle, which is a ...
- 131
5
votes
1
answer
287
views
Extension of first order deformations of a line bundle
Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
1
vote
0
answers
200
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Bundles vs. line bundles
Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...
user30211
5
votes
1
answer
390
views
Compact complex non-Kähler manifolds with nef canonical bundle
Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...
- 275
2
votes
0
answers
242
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Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
- 8,998
2
votes
1
answer
193
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Linear system giving the projective embedding of the tangential variety
I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...
- 1,343
1
vote
1
answer
190
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Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve
We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
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3
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1
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234
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Global choice of eigenvectors on an open surface
Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...
- 2,095
3
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0
answers
265
views
Ample divisor of degree two on a blow-up of $\mathbb P^2$ at nine points
Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position.
I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$).
Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \...
- 1,841
2
votes
1
answer
204
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Morphism attached to a big and globally generated line bundle
Let $X$ be a smooth projective variety over a field $k$. Let $L$ be an invertible sheaf on $X$. Suppose that $L$ is big and globally generated. Can one conclude that the associated morphism $\phi_L : ...
- 63
1
vote
1
answer
163
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Arithmetic ampleness and scalings of the metric
Let $\overline L= (L, h)$ be a hermitian $C^
\infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & ...
- 321
4
votes
1
answer
222
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Existence of non-trivial "line-symplectic" manifolds
One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...
- 183
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0
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Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
- 4,164
1
vote
0
answers
86
views
Representatives of line bundle cohomology over tori
Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
4
votes
1
answer
362
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Type vs degree of a polarized abelian variety
Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that
$d = \chi(L) = \dim H^0(A,L)$
since $L$ is ample.
I've read in a lot ...
3
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1
answer
339
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Galois invariant line bundle and base change
Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
- 321
3
votes
1
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When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.
Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
- 1,338
18
votes
1
answer
2k
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Embedding abelian varieties into projective spaces of small dimension
Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...
- 4,164
1
vote
0
answers
151
views
Extension of a section of a line bundle on a family of curves to the central fibre
I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line ...
- 341
8
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1
answer
2k
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
- 1,155
5
votes
0
answers
321
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Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$
A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
- 103
6
votes
3
answers
3k
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Classification of line bundles by second cohomology of a manifold
In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...
- 6,023
3
votes
0
answers
136
views
Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification
Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better.
In Liu-Lorenzini-Raynaud (...
- 1,142
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
3
votes
2
answers
399
views
Character which defines canonical bundle on flag variety
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
- 473
1
vote
1
answer
483
views
Making a vector bundle ample by twisting with ample line bundle
Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is ample if the relative twisting sheaf $\mathcal{O}_{\mathbf{P}(E)...
user279422
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
21
votes
3
answers
4k
views
How many flat connections has a line bundle in algebraic geometry?
Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
- 54.6k