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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6 votes
1 answer
377 views

Does any projective bundle on a compact complex manifold have an associated holomorphic vector bundle?

Let $X$ be a compact complex manifold, and $f: Y\to X$ a proper surjective holomorphic map with fiber $\mathbb{CP}^n$. Is there always a holomorphic vector bundle $E$ of rank $n+1$ such that $Y$ is ...
1 vote
0 answers
102 views

Tangent space of moduli of stable vector bundles

I'm new to this area, so it may very well be possible that I may be missing something easy here. Let $E$ be a stable complex vector bundle over $X$ of degree $d$ and rank $n$. Then the moduli space $\...
3 votes
1 answer
470 views

Trivial subbundle of universal bundle on the Grassmannian $\mathbb{G}(k,n)$

I was looking at the following paper by Tango: https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-14/issue-3/On-n-1-dimensional-projectlve-spaces-contained-in-the-...
2 votes
1 answer
206 views

Picard group of moduli of principal bundles

I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$. Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...
4 votes
1 answer
362 views

$Ext$-algebra of stable vector bundles

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$. Question: What can we say about the algebra structure of ...
4 votes
2 answers
640 views

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 $...
1 vote
1 answer
387 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)...
1 vote
0 answers
74 views

Harmonic forms on holomorphic line bundle

Let $L$ be a holomorphic line bundle on a Kähler manifold $X^n$. Let $h$ be a hermitian metric on $L$ which gives us a $\mathbb{C}$-antilinear isomorphism $h:L\cong L^*.$ Next we have \begin{align*} \...
4 votes
2 answers
706 views

Vector bundles on Grassmannians

Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$. Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and ...
2 votes
1 answer
210 views

Relative affine schemes

I was reading these notes by D. Gaitsgory, and I don't understand a claim he makes about relative affine schemes. Namely, on page 3 he says that if $f: Y \rightarrow X$ is an affine scheme over $X$, ...
3 votes
0 answers
235 views

Tensor product of associated vector bundles

Let $(P, X, \pi, G)$ and $(P', X, \pi', G')$ be two principal bundles (with Lie groups $G$, $G'$ respectively). Given a vector space $V$ and representations $\rho, \rho'$ of the Lie groups in this ...
4 votes
0 answers
103 views

Singularities of hypersurfaces in projective bundles

I am doing some calculation on a toy example from the question here. Let $\mathbb P(E) \rightarrow \mathbb P^1$ be the projectization of the vector bundle $E = \mathcal O \oplus \mathcal O \oplus \...
4 votes
0 answers
215 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
1 answer
195 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
0 answers
224 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 ...
24 votes
0 answers
780 views

Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if the ...
5 votes
1 answer
293 views

Exact sequence involving spectral data for Higgs bundles

In Beauville, Narasimhan, Ramanan's Spectral curves and the generalized theta divisor, Remark 3.7, the following exact sequence is presented: $0 \rightarrow M(-\Delta) \rightarrow \pi^* E \...
2 votes
1 answer
101 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
1 answer
107 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 ...
12 votes
3 answers
1k 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 ...
6 votes
0 answers
480 views

Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks

The following is, amongst others, a Hartshorne exercise: Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...
4 votes
1 answer
455 views

Frobenius rank of a manifold

The rank of an smooth manifold M is defined by Milnor, as follows: "The maximum number of independent commuting vector fields on M" For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...
10 votes
2 answers
2k views

Maslov index of a pullback bundle

This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology. Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex ...
16 votes
0 answers
218 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 $M$...
1 vote
1 answer
179 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 ...
15 votes
1 answer
649 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 $C(X)$...
-4 votes
1 answer
470 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!
5 votes
1 answer
417 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
0 answers
57 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 ...
3 votes
0 answers
151 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
0 answers
93 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
0 answers
198 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, \...
4 votes
1 answer
473 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
1 answer
108 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 : $X=\sqcup_{i=1}^{n}U_i$ ...
4 votes
0 answers
211 views

Real part of a holomorphic section of a vector bundle

Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
3 votes
1 answer
519 views

Product rule for vector bundle (Leibniz rule)

Let $\pi:E\to Y$ be a vector bundle. Write $\mathrm{T}(E/Y)\subset \mathrm TE$ for its vertical bundle. Write $\delta:\mathrm{T}(E/Y)\cong \pi^\ast E:\ell$ for the vector bundle isomorphisms over $E$ ...
1 vote
0 answers
125 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, ...
9 votes
1 answer
381 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
0 answers
93 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
0 answers
112 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 ...
4 votes
1 answer
384 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 ...
2 votes
0 answers
63 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
0 answers
157 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
0 answers
181 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 ...
1 vote
1 answer
177 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 $Z$, ...
0 votes
1 answer
226 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
0 answers
149 views

Functorial lift of certain vector bundles to the ambient projective space

Given an very ample line bundle $L$ on a projective variety we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. Consider the full sub-...
7 votes
1 answer
1k views

Chern classes via degeneracy loci

According to book of Eisenbud-Harris Page 332 and the following summary http://pbelmans.ncag.info/blog/2014/10/09/what-are-chern-classes/ one can describe Chern classes in terms of degeneracy loci. ...
1 vote
0 answers
132 views

A possible kind of $K$ theory via comparison of sphere bundles associated to given vector bundles

Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner ...
3 votes
0 answers
187 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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