Questions tagged [vector-bundles]
A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
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Vanishing of Self-Ext groups of vector bundles
Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?
Edit: ...
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Existence of connections making a bundle endomorphism parallel
Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?
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Holomorphic bundles and maps to the Grassmannian ?
Hello,
In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I ...
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Modules, Sheaves and Vector bundles
Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an ...
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Intersection form of $2n$-manifold for odd $n$
Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\...
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Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?
Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...
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Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
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Chern classes of a vector bundle
Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...
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Connectedness of moduli of vector bundles
Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...
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Totally non parallelizable manifold
Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...
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A question on a projective bundle $\mathbb{P}(L\oplus \mathcal{O}_X)$
Let $X$ be a complex manifold and $L$ a line bundle on it. Define $Y:=\mathbb{P}(L\oplus \mathcal{O}_X)$ be the projective bundle over $X$. Here is a statement I don't understand:
The summands $L$ ...
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Is the Hitchin fibration proper?
By Hitchin fibration I mean the usual morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base (i.e. the direct sum of spaces of global sections of powers of the ...
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Vector bundles on quotient variety
Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V_{ss}$ is a variety of semi-stable ...
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If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable?
Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$.
Let $E$ be a vector ...
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Two ways to differentiate a section of vector bundle
Let $\pi:E\rightarrow M$ be a vector bundle, and $D$ a connection on it. Suppose $\sigma_1,\sigma_2\in\Gamma(E)$, $p\in M$, $V\in T_pM$ such that $\sigma_1(p)=\sigma_2(p)$. Are the following two ...
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$H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles
Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}_X^*)$.
Can it be generalized to higher rankal ...
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Torelli theorem for stable vector bundle
Let $X, X^{\prime}\colon$ smooth projective curve on $\mathbb{C}$ (genus $\geq 3$),
$M(r,d)\colon$ coarse moduli of stable vector bundles with rank $r\geq2$, and degree $d$ , and
$M(r,\xi)\colon$ ...
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Borel--Bott--Weil for the Grassmannians
The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians?
More precisely, suppose $G(\mathbf ...
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Canonical n plane bundle over Lagrangian Grassmanian
Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...
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Elementary transformations of ruled surfaces as maps of vector bundles
This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.
All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as $...
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Normal bundle to Veronese varieties $v_d(\mathbb{P}^n)$ into $\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d)))$
I was searching for a response on the internet but I was not able to find out an explicit answer.
It is known that if $\mathbb{P}^n \subset \mathbb{P}^N$ is embedded linearly then the normal bundle $...
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Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
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Endomorphism of globally generated sheaves on curves
Let $X$ be a smooth, projective curve (over $\mathbb{C}$) of genus at least $2$ and $E$ be a globally generated sheaf on $X$. I am looking for conditions/examples such that there exists a closed point ...
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analytic vector bundles
Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
Is $E$ a trivial analytic vector bundle?
I need to the ...
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Pushforwards of Line Bundles and Stability
I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
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Why Gateaux derivative is a distribution?
Thanks to Jan Bohr answer and comment I edited this question.
Let $E$ be a vector bundle , $E^*$ the dual bundle and $D$ a density bundle. Denote by $\Gamma(E)$ the space of section of the bundle $E$....
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A text about Schwartz distributions in vector bundles
If $M$ is a smooth manifold, one may talk about the space of test functions $\mathcal D (M)$ and its topological dual $\mathcal D ' (M)$ - the space of Schwartz distributions on $M$.
Now, if $E \to M$ ...
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Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
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(Contradiction) All symplectic manifolds are holomorphic
I’m studying symplectic manifolds and almost complex structures. This lead to two propositions:
Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...
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Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE]
Note: This question came from MSE, but since I've received some useful observations I posted it here. Post on MSE
Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}...
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Koszul resolution with wrong dimension
Let $X$ be the zero locus of $e_1, \dots, e_n$ sections of a vector bundle $\mathcal{E}$ of rank $r$ on $Y$. Assume that the codimension of $X$ is strictly less than $n$, then the Koszul complex ...
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Geometric Motivation for Hilbert $C^*$-Bimodules
I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
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Is the kernel of the coderivative infinite-dimensional?
$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (...
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Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ [closed]
Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}_k(S^q) \simeq \pi_{q - 1}(\...
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What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...
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Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?
I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states;
For $n\ge2$, the spectra E(n)
represent periodic homology theories which at present have ...
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Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...
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How to compute the index of such operator?
Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of $\...
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Quick ways to calculate cohomology of vector bundle/local system from transition functions?
Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The ...
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Tangent bundle of a compact two-dimensional manifold
For which (connected) two dimensional compact manifold M, oriented or not, the tangent bundle TM is trivial?
For which of these manifolds the complexified tangent bundle $T^\mathbb{C}M = TM\otimes \...
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Degeneration of vector bundles on an algebraic curve
Let $X$ be a smooth projective irreducible curve over an algebraically closed field $k$. Let $\mathcal E$ be a vector bundle (say, of rank $n$), and let $\mathcal F$ be another vector bundle of rank $...
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Blowing up vector bundles in the zero section
Assume we are given a scheme $X$ (feel free to add all the needed hypotheses, at this point I’m working with smooth schemes, but the fewer is needed, the better) and a vector bundle $E$ over $X$. I ...
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Analogue of Quillen-Suslin theorem for affine varieties
Phrased in the language of vector bundles, the Quillen-Suslin theorem states that vector bundles on $\mathbb C^n$ are algebraically trivial (for any algebraic vector bundle there exists an algebraic ...
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Does every disc bundle come from a vector bundle?
Kosinski in his book "Differential Manifolds" states:
"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc ...
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Vector bundles on punctured disc around isolated surface singularity
Let $X/k$ be a surface (over some field), smooth except for an isolated (closed) point $x$. One may look at the punctured local ring
$X:=\mathrm{Spec}(\mathcal{O}_{X,x}) - x$.
Are there non-trivial ...
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A question on complex line bundle over $S^{2}$
Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...
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Orientation bundle and its flat connection
Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
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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\...
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Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
4
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Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...