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.
925
questions
4
votes
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
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
1
vote
1answer
67 views
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,...
2
votes
0answers
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 ...
2
votes
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$. ...
0
votes
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 ...
4
votes
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 ...
16
votes
0answers
190 views
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$...
13
votes
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 ...
-4
votes
1answer
97 views
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!
10
votes
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)$ ...
1
vote
0answers
58 views
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})$. ...
1
vote
0answers
34 views
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 ...
14
votes
1answer
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)$...
1
vote
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 ...
2
votes
0answers
76 views
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$...
2
votes
0answers
66 views
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_\...
5
votes
0answers
128 views
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, \...
3
votes
1answer
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 ...
1
vote
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 :
$X=\sqcup_{i=1}^{n}U_i$
...
1
vote
0answers
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, ...
10
votes
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 ...
1
vote
0answers
67 views
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) = ...
0
votes
0answers
92 views
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 ...
2
votes
0answers
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 ...
4
votes
0answers
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 $...
3
votes
0answers
124 views
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 ...
0
votes
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 ...
3
votes
0answers
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 ...
1
vote
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 $Z$, ...
1
vote
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 $E$...
3
votes
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 ...
4
votes
0answers
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}^...
1
vote
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 ...
2
votes
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
votes
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
votes
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$ ...
1
vote
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 \...
4
votes
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 ...
2
votes
0answers
81 views
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 ...
0
votes
0answers
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,
$$
...
2
votes
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 ...
7
votes
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 ...
3
votes
0answers
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 ...
4
votes
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 $G$ ...
5
votes
0answers
88 views
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 ...
8
votes
0answers
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 ...
2
votes
0answers
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 ...
3
votes
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 ...
1
vote
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 $(n-1)$-...
1
vote
0answers
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 $...
