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.
782
questions
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Geometry of the complex Gauge group
Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$.
Is there a way to endow $\...
6
votes
2answers
248 views
1d TQFT minus connection =?
Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent.
Going into more detail (and ...
16
votes
4answers
987 views
Analogy between Stiefel-Whitney and Chern classes
There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading ...
5
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0answers
147 views
Hopf fibration extended to bundle over $\mathbb{C}^2$
Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
2
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0answers
105 views
Extending vector bundles on divisors
Let $X$ be a smooth variety. Let $Y$ be the union of the four sub-varieties of $X\times \mathbb{P}^1 \times \mathbb{P}^1$ determined by $X \times \mathbb{P}^1 \times \{0\}$, $X \times \mathbb{P}^1 \...
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0answers
78 views
Variation of the (Chern) curvature with respect to the metric
Let $E\rightarrow X$ be a holomorphic vector bundle, for any Hermitian metric $h$ on $E$ we denote by $F_h$ the curvature of the Chern connection associated to $h$. Fix a metric $h_0$ and consider a ...
5
votes
0answers
175 views
Derivative of the Bott-Chern forms
The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
3
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0answers
68 views
How big is the complement of stable locus $\operatorname{Bun}G$
Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type).
What is the ...
6
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1answer
197 views
Homogeneous vector bundles on Abelian varieties
I recently encountered a result about vector bundles on Abelian varieties, which I found interesting. It characterizes homogeneous (translation invariant) vector bundles on Abelian varieties. More ...
2
votes
1answer
124 views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
5
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0answers
208 views
Is this a stack?
A continuous map $f:X\to Y$ and a vector bundle $E\to X$ seem to give rise to a presheaf of groupoids on $Y$ along the following lines. For an open $U\subseteq Y$, each section of $f$ over $U$ (i. e. ...
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0answers
156 views
Splitting principle in algebraic geometry and ample line bundles
Splitting theorem in algebraic geometry claims that if we have a vector bundle $V$ on $X$ (we consider a smooth projective variety for this question), if we pull-back $V$ to $\mathbb{P}(V)$, we get a ...
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0answers
123 views
Splitting after pullback under finite flat morphisms
I recently became aware of an interesting result, which claims for smooth projective curve over a finite field, you can pull back any vector bundle, under combinations of Galois covers and Frobenius, ...
3
votes
0answers
118 views
Stably deforming vector bundles
Let $X$ be a smooth projective variety. $V_1$ and $V_2$ are two vector bundles on $X\times \mathbb{A}^1$ such that $V_1|_{X\times \{0\}}\cong V_2|_{X\times \{0\}}$ and $V_1|_{X\times \{1\}}\cong V_2|_{...
1
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2answers
215 views
Vector bundle over compact complex manifold which is not holomorphic?
A vector bundle over a complex manifold is said to be holomorphic if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex ...
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1answer
101 views
Connections on vector bundles over elliptic curves - concrete computations
This is linked to my question on math.Stackexchange for which I had no answer.
I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me ...
3
votes
1answer
216 views
Rank 3 Lagrangian vector bundles on an elliptic curve
Let $k$ be an algebraically closed field of characteristic zero (feel free to assume $k= \mathbb{C}$) and $E$ an elliptic curve over $k$ with identity $P \in E(k)$.
I am interested in certain ...
1
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1answer
87 views
Can the dimension of Hom space between vector bundles on an algebraic curve predicted by Riemann-Roch type formula be the minimal possible?
Let us study vector bundles $E$ and $F$ on a smooth projective curve $C$. There is a Riemann-Roch type formula for the Euler characteristic $\chi(E,F)=dim\, Hom(E,F)-dim\, Ext^1(E,F)$ in terms of ...
5
votes
1answer
217 views
Shrinking and stretching of vector bundles
Let $M$ be a manifold, $p:E\to M$ a rank $d$ vector bundle. Suppose that $U \subset E$ is an open subset such that $U \cap p^{-1}(x)$ is nonempty and convex for all $x \in M$. Is it true that $U \to M$...
1
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0answers
55 views
Minimal radius of a ball admitting a trivialization of a vector bundle
Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
2
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0answers
87 views
Locus of non-injective maps between vector bundles
Suppose $E,F$ are two vector bundles of ranks $e$ and $f$ with $e<f$ on a smooth projective surface $X$ over $\mathbb{C}$ and let $r:=f-e$. I am interested in estimating the (co)dimension of the ...
4
votes
1answer
247 views
Relative logarithmic cotangent bundle
Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the ...
23
votes
1answer
549 views
Vector bundles on $\mathbb{A}^n / G$
Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
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0answers
116 views
Find torsion classes using flat bundles
My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
2
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0answers
105 views
Vector bundles on complete smooth variety $X$ in char $p$ and Frobenius
In "Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe." Math. Z. 156 (1977), no. 1, 73–83. by Herbert and Ulrich, the authors consider a complete smooth variety $X$ defined ...
3
votes
1answer
263 views
Chern classes of complex vector bundle
I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
2
votes
0answers
69 views
Definition of Lie derivatives of sections of natural vector bundles - product preservation needed?
Section 6.15 of Natural Operations by Kolár, Michor, and Slovak defines the Lie derivative of a section of a natural vector bundle along a vector field. Set-theoretically, the definition is clear. ...
11
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3answers
528 views
A binary operation on vector bundles that adds Chern classes?
Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern ...
1
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1answer
115 views
A kind of isomorphicity of vector bundles
Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))...
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0answers
159 views
A vector bundle analogy of the Nash embedding theorem
Let $E$ be a Riemannian vector bundle over a manifold. Can $E$ be considered as a subbundle of trivial bundle $M\times \mathbb{R}^n$ for some $n$ such that the metric of each fiber is the restriction ...
3
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1answer
227 views
vector bundles over projective line over an affine line
Let $k$ be a field and $E$ be a vector bundle over $\mathbb{P}_{k}^{1}\times\mathbb{A}_{k}^{1}$, does it extend to
$\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
2
votes
0answers
72 views
Integral kernels for geometric langlands
My apologies for the imprecise question(s), it should be clear enough that I´m a complete beginner in this subject.
The (de Rham) Geometric Langlands Conjecture over $\mathbb{C}$ takes as input a ...
4
votes
0answers
113 views
Is the determinant map $det:\mathcal{M}(r,d)\rightarrow Pic^d(X)$ on moduli space of semistable vector bundles a fibration?
Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It ...
9
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1answer
254 views
Computing K-theory for cellular vector bundles
One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech ...
7
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0answers
233 views
Can every functor $F: \bf{Vect} \to \bf{Top}$ be lifted to a functor $\tilde F: \bf{VectBun} \to \bf{Bun}$? And if $F: \bf{Vect} \to \bf{Man}$?
My question is: if I have a functor from the category of vector spaces Vect to the category of topological spaces Top (or differentiable manifolds Man) can I lift it to a functor from the category of ...
7
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3answers
565 views
$\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?
Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in analtytic topology. It is well known that there exists a rank $k+1$ complex vector ...
0
votes
1answer
32 views
Keeping the covariant divergence intact under changes of frame
In Eulcidean 3-space with coordinates $(r, \theta, \phi)$ where $\theta$ is the polar angle and $\phi$ the azimuthal angle, we may write the covariant divergence of a vector $E = E^\mu e_\mu$ as
$$E^{...
0
votes
1answer
122 views
What happens to the metric when we normalize the basis? [closed]
Here is an Example in Euclidean 3-space: When using spherical coordinates $(r, \theta, \phi)$ with $\theta$ and $\phi$ the polar and azimuthal angles, respectively: a natural basis for these ...
2
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0answers
138 views
Bounded holomorphic section
Is it possible that there exists a multivalued holomorphic
function $\phi\not \equiv 0$ from $\mathbb {CP}^1$ to $\mathbb C^k$ that has monodromies
$A_1, \dots A_n \in GL_k(\mathbb C)$
around points $...
1
vote
0answers
90 views
Hodge theorem for cohomology group on holomorphic vector bundles and harmonic forms
Consider the holomorphic vector bundle $\pi: E \rightarrow M $ where $M$ is a complex manifold of dimension $\dim_{\mathbb{C}}M = m$. Denote by $\Omega^{p,q} (M)$ the bundle for bidegree $p,q$ ...
1
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0answers
68 views
Poset of degree zero bundles
Let’s assume we are working on a smooth projective curve $X$. For any vector bundle $E$ on $X$, the poset of non-trivial proper sub-bundles of $E$ is in bijection with the poset of non-zero proper sub-...
5
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1answer
192 views
Group of parallelizations of $M^3$ finitely generated?
Let $M^3$ be a compact orientable 3-manifold. Then $TM$ is trivial and let's go ahead and fix a trivialization $\tau : M \times \mathbb{R}^3 \to TM$. Then given a map $g : (M, \partial M) \to (SO(3),...
3
votes
1answer
171 views
Homogeneous vector bundles with zero chern classes
We know that a line bundle $L$ on the complex flag variety $G/P$ is trivial iff $c_1(E) = 0$. But if we have a homogeneous vector bundle $E$ of higher rank, then is it true that $c_i(E) = 0$ $ \...
1
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0answers
151 views
Degree of the direct image of a line bundle
Consider a $n$-branched cover $\pi:S\rightarrow M$, where $S$ and $M$ are both algebraic curves. If $\pi_{0}: L\rightarrow S$ is a line bundle over $S$, we can define the bundle $\pi_{*}L$ on $M$ ...
2
votes
0answers
255 views
Flat connections and global sections of vector bundles
Let $X$ be a (non-singular) complex surface and $(V,\nabla)$ be a vector bundle $V$ equipped with a flat connection $\nabla$ on $X$. Fix a point $x \in X$ and $v_0 \in V_x$ an element in the fiber ...
3
votes
1answer
157 views
The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space
Let $\mathcal{M}_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}_{d}(G)$ have a non-trivial $\mathbb{C}^{*}$-holomorphic action by multiplication of the Higgs field,
$$
z\cdot (E, \...
1
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1answer
170 views
Stability of holomorphic vector bundles
I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling ...
2
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0answers
99 views
Zero section of quasi-coherent bundle
Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...
7
votes
2answers
562 views
Beauville-Laszlo for schemes
For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
4
votes
1answer
187 views
Nowhere vanishing section implies reduction of structure group
Description
I noticed a repeating theme in vector bundle theory, and wonder if there's a theorem that describes this kind of phenomenon.
Given a vector bundle $E$ over a manifold $X$. If there is ...
