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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4
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0answers
83 views

Pseudo-tensor- and tensor-densities: Sections of what bundle?

Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle $$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$ ...
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The tangent map of the multiplication map of a vector bundle

If $\beta: \mathbf{U}\times \mathbf{V}\to \mathbf{W}$ is a bilinear map between real linear spaces then its derivative at a point $(u,v)$ is given by the Leibniz rule $$D\beta(u,v)(u_0,v_0)=\beta(u,...
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127 views

Extending an embedding with trivial normal bundle

I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
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1answer
88 views

Finding a local normal form regarding distribution rank properties

I am working in geometry control field, fall last week on this exercice and I can't figure it out. I have a distribution $\mathscr{D}$ with $rank(\mathscr{D})=m+1$ in $\mathbb{R}^n$ with $n\leq 2m+1$. ...
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1answer
93 views

Intersection Grassmanian planes

I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
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1answer
167 views

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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Reference request: Milnor rank of spheres

Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
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3answers
445 views

Extending group actions to vector bundles

Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$. Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
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Definition of tautological vector bundle [closed]

Could you please give a detailed definition (or construction)of tautological vector bundle of Grassmannian over arbitrary base scheme? Thank you in advance!
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1answer
386 views

Jumping conics in Grassmannians

Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
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Obstructions to existence of global extension realising fibrewise extensions

We work throughout with spaces (say varieties or maybe DM stacks) over $\mathbb{C}$. Take a smooth map $\pi:X\rightarrow B$ and let $X_{b}$ denote the fibres over closed points $b\in B(\mathbb{C})$. ...
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Vertical bundles of higher order tangent bundles

Let $M$ be a smooth (finite dimensional, Hausdorff and second countable) manifold. Let $T^kM$ be the manifold of equivalence class of curves that their derivates (in charts) agree up to order $k$. Let ...
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557 views

Reference for the Swan-Serre theorem as a monoidal equivalence

Let $X$ be a compact Hausdorff The well-known Swan--Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$...
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1answer
127 views

Dual of stable vector bundle on a Fano threefold

Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$. Question. Is it true that $E(-1)=E^*$? What I am able to show is that ...
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Semistability of restrictions of a semistable vector bundle over a reducible nodal curve

Let $C$ be a reducible nodal curve over complex numbers with two smooth components $C_1$ and $C_2$ intersecting at the only node $P$. Let $E$ be a $\omega$ semistable vector bundle over $C$ of rank $r$...
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Exterior product of one-forms

Let $\mathcal M$ be a compact $3D$ differentiable manifold and let $\alpha, \beta, \gamma$ be three one-forms on $\mathcal M$. I want to consider the scalar quantity $$ F(\alpha, \beta, \gamma)=\int_\...
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Geometric interpretation of $\mathbb{C}^{\times}$-gerbes

Let $X$ be a (nice enough, e.g. smooth etc.) variety over the complex numbers, and let $\mathcal{G}$ be a gerbe on $X$. Then $\mathcal{G}$ is classified by a cohomology class in $\alpha \in H^2(X, \...
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190 views

$H^1(X, GL(n, \mathcal{O}_X))$ 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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1answer
72 views

Disjoint union of clopen sets such that the fibers has constant cardinality [closed]

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that : $X=\sqcup_{i=1}^{n}U_i$ ...
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94 views

Schur's lemma for sheaves with different reduced Hilbert polynomials

Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement: Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
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1answer
181 views

Set theoretic equation for Veronese varieties

Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
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first segre class of a line bundle

I am reading the proof of proposition 3.1(e) in Fulton's Intersection theory book: If $E$ is a line bundle and $\alpha \in A_*X$, then $s_1(E) \cap \alpha = -c_1(E) \cap \alpha$ The proof says $P(E) = ...
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Splitting the relative tangent bundle of $\mathbb{P}(E\oplus O_X)$

Suppose $X$ is a smooth scheme and $E$ is a vector bundle on $X$. We have an exact sequence $$0\to T_{\mathbb{P}(E)/X}\to T_{\mathbb{P}(E\oplus O_X)/X}|_{\mathbb{P}(E)}\to O_E(1)\to 0.$$ Does this ...
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40 views

Hypoellipticity or parabolic regularity for vector bundles

Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients ...
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143 views

Coherent sheaves and space filling curves

This paper constructs smooth space filling curves for smooth varieties over finite fields. Let's say we are working in char $p$ on the variety $X$ then this means that there is smooth curve $C_i$ in $...
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Splitting vector bundles on $\mathbb{P}^n$

There are results by Kleiman and Sumihiro that claim you can split an algebraic vector bundles (in the sense that it admits filtration that quotients are given by line bundles) by applying ...
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1answer
177 views

Is there a version of Quillen-Suslin-Lindel for power series?

Is there an analogue of Quillen-Suslin theorem for power series? Let $A$ be a regular noetherian ring over a field. Consider the power series ring $A[[T]]$. Are projective modules on $A[[T]]$ extended ...
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122 views

Generalization of the Kobayashi-Hitchin correspondence to almost-complex manifolds

Let $(X, \omega)$ be a compact Kahler manifold of dimension $n$. Then the so-called Kobayashi-Hitchin correspondence in this case says that Theorem Let $E\rightarrow (X, \omega)$ be a holomorphic ...
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1answer
77 views

Real part of the Ward correspondence

I am currently very confused about the real side of the Ward correspondence. Recall that the Ward correspondence gives a one-to-one correspondence between: $M$-trivial holomorphic bundles $E$ on $Z$, ...
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1answer
117 views

Embedding extension of sheaves in direct sum

Let $X$ be a smooth projective curve of genus $g\geq2$, we can construct rank $2$ vector bundles on $X$ with determinant $\omega_X$ by extension $$0\to \mathcal{O}\to E\to\omega_X\to 0,$$ does such $E$...
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1answer
128 views

Extension of stable vector bundles on curves

Let $X$ be a smooth projective curve, let $E',E''$ be stable vector bundles on $X$, with $\mathrm{slope} (E'')>\mathrm{slope} (E')$. Let $0\neq[E]\in \mathrm{Ext}^1(E'',E')$ be an extension, $$0\to ...
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112 views

Relative Thom isomorphism

Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
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1answer
139 views

Flat familiy of coherent sheaves over a scheme

I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
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1answer
105 views

Locally free sheaves and vector bundles over smooth connected projective curve

Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
4
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1answer
240 views

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 ...
3
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1answer
186 views

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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1answer
142 views

Vector bundles admitting resolution by ample line bundles

Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \...
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1answer
300 views

(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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Intuition behind Nakano positivity

I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly ...
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140 views

Generalization of universal sequence over Grassmannians

Setup: Let $V$ be a $(n+1)$-dimensional vector space. We define $\mathbb{P}V=\mathbb{P}^n$ as follows: points of $\mathbb{P}V$ correspond to 1-dimensional vector subspaces of $V^\vee$. Moreover, $$ ...
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1answer
144 views

Are splitting vector bundles closed under kernel or cokernels?

Given an ample line bundle $L$ on a smooth projective variety of dimension $\geq 2$, let $C$ be the category of vector bundles that are direct sums of powers of $L$. Two related questions: Given a ...
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2answers
284 views

tangent bundle on noncommutative manifold

Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold $(A,H,D)$, by replacing vector bundle by finitely generated projectve module $M$. For the construction of ...
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159 views

Cofinality theorem for derived categories

For a projective variety $X$ and an ample line bundle $L$ on it, we consider the family of line bundles $L^{\otimes i}$ for $i\in \mathbb{Z}$. Let $\mathfrak{C}$ be the category generated by the ...
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1answer
146 views

Locally trivializing a G vector bundle?

In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
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What is known about this conjectured symmetry in the generalized Radon-Hurwitz numbers?

The generalized Radon-Hurwitz number $\rho(m, n)$ is defined as the maximal dimension of a subspace contained in $Q_{m,n }$, the subset of all real $m\times n$ matrices $A$ which satisfy $AA^T=\lambda ...
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128 views

Elementary proof of an inequality for the Radon-Hurwitz numbers

Edit: In all likelihood, the original question does not have a positive answer (see comment by abx). Modified question: Let $\rho_H(n)$ be the maximal dimension of a space of symmetric real matrices ...
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64 views

Splitting of short exact sequence of strongly-semistable sheaves

Does short exact sequence of strongly semi-stable bundles (torsion-free sheaves) of the same slope split after applying few Frobenius pullbacks? Strongly semi-stable means that pullback under ...
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1answer
174 views

Stability of syzygy bundles of smooth curves

For a very ample line bundle $L$, the kernel of the surjection $H^0(L)\otimes \mathcal{O}_X\rightarrow L \rightarrow 0$ is denoted by $M_L$ and is called syzygy bundle. In this paper authors claim in ...
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1answer
207 views

Vector field tangent to a submanifold and transverse to the zero section

In Hirsch's Differential Topology there's the following : Suppose a compact $n$-manifold can be expressed as $A\cup B$ where $A,B$ are compact $n$-dimensional submanifolds and $A\cap B$ is an $(n-1)$-...
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80 views

Examples of effective Lefschetz condition

When a closed subvariety $Y$ of $X$ satisfy effective Lefschetz condition $Leff(X,Y)$, it implies there is an equivalence if categories between the vector bundles on the formal neighborhood of $Y$ in $...

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