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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Minimal non-klt center of asymptotic linear system
Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...
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Holomorphic line bundles on smooth points of a quotient
I am an amateur algebraic geometer, so maybe this question is trivial and if this is the case, then I apologize. This is a question that came up while working on something completely different.
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Moving curves and small transformations
Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...
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Cone of moving curves
Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...
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Number of conditions imposed by fat points to a linear system
Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...
user97096
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Line bundles with vanishing cohomology on Calabi-Yau manifold
Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...
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Are two line bundles with the same ramification type necessarily isomorphic?
I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...
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Hopf lemma for line bundles on curves in algebraic geometry
In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...
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canonical model of a reducible curve
Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
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How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
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pairing theta functions for different complex structures
I apologize for my previous attempt to ask this, which was very badly written.
Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...
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Theta functions and Fourier transforms
Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...
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Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]
Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
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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 $H^1 \neq 0$...
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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
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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 ...
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Restriction of a line bundle to a two-cycle
I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...
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Do line bundles with enough sections on surfaces have generic divisors which are irreducible?
Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...
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Trivial Line Bundle-Riemann surfaces
What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function $\...
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Holomorphic sections to anti-holomorphic sections
Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am trying to give a norm preserving isomorphism between the space of holomorphic ...
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Two questions about line bundles over Kuranishi families
i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...
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Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
user39066
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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 ...
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Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?
I try to understand the following version of the Kodaira embedding theorem:
Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample.
I have a problem with the '...
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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 ...
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Sections of vector bundles interpreted as sections of line bundles
Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...
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$H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?
Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.
Is there a constant $C=...
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When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...
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Big divisors and small transformations
Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:
$-K_Y$ is ...
user49214
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canonical bundle of the relative spectrum
maybe it is a very trivial quetion but:
suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum $Spec(Sym(\mathcal{A}))\...
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Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...
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