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.
1,245 questions
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Barton - Kleiman Criterion for the ampleness is sharp
Does there exist a smooth complex projective variety $X$ of dimension $n\geq2$, a rank $r\geq2$ vector bundle $E$ over $X$ such that
$\det(E)$ is ample;
for any smooth complex projective curve $C$, ...
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Minimal CW complex detecting all powers of euler class
Let $n \in \mathbb{Z}_{>0}$. I'm wondering about pairs $(X,V)$ where $V$ is a $2n$-dimensional vector bundle on the CW complex $X$, such that
The ring homomorphism $\mathbb{Q}[e] \to H^*(X;\...
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Trivialising homogeneous vector bundles over a homogeneous space
Take the sphere $S^2$ with the standard indexing of its line bundles $E_k$, for $k$ an integer. Is it true that $E_{k} \oplus E_{-k}$ is a trivial vector bundle? If so, what is the easiest way to see ...
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Family of vector bundles over a relative A^1
Let $A=\mathbb{C}[[s]]$, I am looking for an example of a non-trivial vector bundle (or $\operatorname{SL}_n$-torsor) on $\mathbb{P}^{1}_{A[t,t^{-1}]}$ that is trivial over $\mathbb{P}^{1}_{\mathbb{C}...
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Real structure of dual vector bundle
Suppose $V$ is a holomorphic vector bundle on a Riemann surface X which is endowed with an anti-holomorphic involution $\sigma_X: X\longrightarrow X$. $V$ is said to have real structure if $\exists$ ...
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Is every submanifold the zero of a sufficiently regular section of a vector bundle on some neighborhood of the submanifold?
Let $M$ be an $m$-manifold and $N\subseteq M$ an (embedded) $n$-submanifold $(0<n<m)$. Everything in this question is assumed smooth if relevant and not stated explicitly otherwise.
Let $\pi:E\...
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Connection induced by pull-back of conjugate bundle
Suppose X be a Riemann surface endowed with anti-holomorphic involution $\sigma_X$. $V$ is a holomorphic vector bundle on $X$ with holomorphic connection $D$. It is a fact that $\sigma_X^*\overline{V}$...
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Isomorphism between Moduli spaces of vector bundles with fixed determinants, over a curve
Let C be a smooth projective curve, and let $L_1$ and $L_2$ be two line bundles with degree $d_1$ and $d_2$ respectively, where $gcd(r,d_i)=1$ for $i=1,2$. Suppose $M(r, L_1)$ and $M(r, L_2)$ are two ...
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Notion of equivalence of vector bundles
I have a question:
Let $V$ be a holomorphic vector bundle of rank $n$ on a Riemann surface $X$ given by cocycles $\Phi_{ij}$ for a trivializing cover $U_i$ which on $U_i\cap U_j\cap U_k$ satisfy the ...
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Hybrid of a flat vector bundle and a line bundle
I have a simple queation:
Suppose $V$ is a vector bundle that is isomorphic to a tensor product of a flat vector bundle and a line bundle $V=F\otimes L$. Where $F$ is a flat vector bundle and $L$ is ...
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Descent of vector bundles on Quot schemes?
Let $S$ be a scheme and $T : M \to N$ a morphism of locally free rank $n$ sheaves on $S$. Let $p : Q(T , r) \to S$ where $Q(T , r) := \mathrm{Quot}(\mathrm{CoKer}\,T , r)$ for $0 \leq r \leq$ minimal ...
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Additive-equivariant vector bundles on the projective line
Let ${\mathbb G}_a = ({\mathbb C},+)$ act on ${\mathbb P}^1$ by $a \cdot [X:Y] = [X+aY:Y]$.
Question. Is the classification of ${\mathbb G}_a$-equivariant (algebraic) vector bundles on ${\mathbb P}^1$ ...
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Is there any Galois correspondence about fiber bundle (vector bundle)?
I have learned Galois correspondence about universal covering space over a topological manifold $M$. Since a covering space over $M$ can be viewed as a fiber space over $M$ with discrete fibers, and a ...
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An exact sequence of jet space associated to a vector bundle
Suppose $L$ be a line bundle over Riemann surface $X$. Then show that $ 0 \longrightarrow J^2(L) \longrightarrow J^1(J^1(L)) \longrightarrow L\otimes K_X \longrightarrow 0 ,$ where $J^k(L)$ is the $k$-...
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Explicit proof that tangent bundle of the 2-sphere, TS^2, is parallelizable
Let $T S^2$ denote the tangent bundle of the 2-sphere $S^2 = \{x \in \mathbb{R}^3 : \|x\|=1\}$. In this paper, Fodor proves that $T S^2$ is parallelizable, using machinery that I do not understand ---...
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Pullbacks of Lie algebroids 2.0
Do Lie algebroids pull back (along submersions)?
Regarding the linked question, I am interested in pulling back only the anchor map, ignoring the bracket. Formally, let
$p_E:E\rightarrow M$ and $p_F:...
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Spherical orientation order of the complex tautological bundle over loop SU(n+1)
$\textbf{Background:}$ The $k$-sum of a tautological line bundle $\mathbb{L}^{(n)}$ over $\mathbb{CP}^n$ is nontrivial for all natural number $k$. This can be deduced from noting that the first chern ...
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Fiber equivalence of a fibration
I am reading this paper, Lemma 1.1 to be precise. I am not able to understand exactly how the equivalence of the fibre $q^{-1}(\psi)$ is shown with the loop space. And after that, how exactly does the ...
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On minimal varieties of general type with semistable cotangent bundle
Let $X$ be a smooth complex projective variety of dimension $n\geq3$ such that:
$K_X$ is big and nef;
$\Omega^1_X$ is semistable with respect to some polarization of $X$;
$2(n+1)c_2(X)=nc_1(X)^2\in H^...
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Distinguished basis of global sections of vector bundles over projective toric varieties
Over a projective toric variety, say over $\mathbb{C}$, the vector space of global sections of any line bundle has a distinguished basis arising from characters. Explicitly, let $X$ be a projective ...
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Taylor expansion for section in geodesic coordinates (clarification / reference request)
I'm looking at the proofs in a handful of papers, and noticing the use of a Taylor expansion for a section of a smooth vector bundle over a compact manifold in geodesic coordinates. However, the ...
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Is pulling back vector bundles to the projective space bundle conservative and essentially injective?
To prove the splitting principle, we often consider pulling back a vector bundle $p: E \to M$ to the total space of the projective space bundle $q: P E \to M$, whose fiber at $x \in M$ is the ...
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Definitions of twisted cotangent bundle
What are the possible definitions of the twisted cotangent bundle? The twisted cotangent bundle is used in some papers (for instance Crooks) but it is not very easy to understand the definition of it ...
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Measurable fields of Hilbert spaces with Serre-Swan duality
The ncatlab article measurable field of Hilbert spaces uses Takesaki's definition of a measurable field of Hilbert spaces (MFoHS). Then it claims that MFoHSs has a Serre-Swan Duality. That is, the ...
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an absolute minimizer of Omega-Yang-Mills functional
I'm reading the paper Compactness for $Ω$-Yang-Mills connections by Chen and Wentworth. In Lemma 1.1, they state that an absolute minimizer of the $\Omega$-Yang-Mills functional $\textrm{YM}_{\Omega}$ ...
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Explicit generators for filtrations in Klyachko's classification of toric vector bundles
According to Klyachko's classification of toric vector bundles over toric varieties (see A. A. Klyachko "Equivariant bundles on toral varieties", Math. USSR, Izv. 35, No. 2, 337-375 (1990); ...
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(op)Cartesian morphisms of Vector Bundles
Considering the forgetful functor $U \colon \mbox{VBun} \to \mbox{Top}$ from the category of topological vector bundles into the category of topological spaces. Is this a fibration and(or) opfibration?...
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Degree of adjoint vector bundle $ad(E_G)$ of a principal $G$--bundle $E_G$ on a smooth projective curve
Let $G$ be a reductive linear algebraic group over $\mathbb{C}$. Let $E_G$ be a holomorphic principal $G$--bundle on a smooth projective curve $X$ over $\mathbb{C}$. Let ${\rm ad}(E_G) = E_G\times^G \...
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Explicit classification of real rank-2 vector bundles over compact surfaces via twisted Euler classes
There is an apparently 'well-known' complete classification of real rank-2 vector bundles (or equivalently, of principal $O(2)$-bundles) over a paracompact space $B$ in terms of characteristic classes....
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Chern classes in Čech cohomology
Let $X$ be a nice topological space with a finite open covering $\{U_i\}$. Let $V$ be a complex vector bundle over $X$.
Let us fix its trivialization over each $U_i$ and construct a matrix valued 1-...
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Extend Automorphic Vector Bundle on Connected Shimura Variety
Let $(G,X)$ be a Shimura datum and $K \subset G(\mathbb{A}_f)$ an open compact subgroup. For simplicity, assume that the largest anisotropic subtorus of $Z(G)$ remains anisotropic over $\mathbb{R}$. ...
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Gaps in second Chern numbers for anti-self-dual connections on compact, orientable 4-manifolds
Let's consider principal bundles $P$ with a compact, connected, semi-simple structure group $G$ (a Lie group) on a compact, orientable, Riemannian 4-manifold. Let's also assume $c_1(P)=0$ and $c_2(P) =...
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"Heat Kernels and Dirac Operators" existence and uniqueness proof
I'm trying to better understand the heat equation for vector bundles over Riemannian manifolds, so I've been reading "Heat Kernels and Dirac Operators" by Berlin, Getzler, and Vergne. I'm ...
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Element of cohomology group of locally free sheaf on projective scheme restricts to a closed subscheme as a nonzero element
I have the following question.
Let $X$ be some irreducible projective scheme over a field $\Bbbk$ (which can be ${\it any}$ field). Let $\mathcal{F}$ be a locally free sheaf on $X$, and let $a\in H^i(...
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Hilbert Bundles and Kuiper’s Theorem
Hilbert bundles are coming up in my research, and I’m trying to better understand them. Since there are multiple definitions of Hilbert bundles, I will clarify that I’m working with the definition in ...
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Is the cotangent bundle of a ball quotient surface globally generated?
Related question:
Surfaces of general type with globally generated cotangent bundle. Computing the invariants of ball quotient surfaces. How far is ample from globally-generated. When is a general ...
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Flat bundles on hyperbolic manifolds
Let $G$ be a discrete and torsion-free subgroup of $\mathrm{O}^+(n,1)$ and let $\rho\colon G \to \mathrm O(k)$ be an orthogonal representation. As I understand, we can construct a flat vector bundle $\...
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Shifted tangent bundle transition functions
I'm currently working with (graded)supermanifolds with applications to another fields. My main example is $T[1]M$ the shifted tangent bundle of a manifold. I understand how gluing functions work for ...
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Vertical and horizontal covariant derivatives on $TM$
Let $(M,g)$ be a (pseudo-)Riemannian manifold and $\pi:TM\to M$ the tangent bundle over $M$. Now, in the following, I am considering the space of sections of the pull-back bundle $\pi^{\ast}TM$. In ...
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Degeneration of extensions and the Shatz stratification
Consider $C$ a curve and let $V$, $V'$ be unstable vector bundles of rank $2$. We have two short exact sequences (corresponding to the Harder–Narasimhan-filtration):
$$0 \to L_1 \to V \to L_2 \to 0$$
$...
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Weights of the cotangent bundle of an Hermitian symmetric variety
Let $X=G/P$ be a rational homogeneous variety, where $G$ is a semisimple complex Lie group and $P\subset G$ is a parabolic subgroup. Let $\{\alpha_1,\dots,\alpha_n\}$ be a system of simple roots for $\...
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Proof of the Chern–Weil Formula
In Constructing Variations of Hodge Structure Using Yang–Mills Theory and Applications to Uniformization, Simpson proposes a formula (Lemma 3.2) for the degree of a non-trivial coherent subsheaf $\...
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strange duality and factorization of generalied theta functions
Recall the celebrated rank-level strange duality isomorphism:
$$H^0(SU_C(r),L^l) \to H^0(U_C(l), \mathcal{O}(r\Theta))$$
between level $l$ generalized theta functions on the moduli space of rank $r$ ...
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What is known about sub-bundles of the Hodge bundles over the moduli of curves?
E.g. what is known about the Harder-Narasimhan filtration of the Hodge bundle over this moduli ?
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Does an extension of a big vector bundle by the trivial line bundle give a big vector bundle?
I adopt Grothendieck's convention for projective bundles. Let $X$ be a smooth complex projective variety and $\mathcal{E}$ a big vector bundle (i.e. $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is big). ...
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When is a holomorphic fibration a vector bundle?
Suppose $B\subset\mathbb{C}$ is a contractible open set, and there is a complex manifold $X$ and a holomorphic map $\pi: X \to B$ such that each fiber $\pi^{-1}\{b\}$ isomorphic to $\mathbb{C}$, (then ...
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Undergrad searching for papers related to this research area
Im an undergrad looking to begin a Phd next year and I have been searching the literature but I cannot find any papers about the following. I can find papers that individually treat foliations, ...
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Connected sum of vector bundles?
The title of this post is both misleading and not.
Let say we have smooth complex vector bundles $E\to X$ and $F\to X$ over a smooth $n$-manifold $X$ of the same rank, say $k$. Assume $X$ is compact ...
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Deductions from the pushforward of the structure sheaf being the structure sheaf
Note: I originally asked this on MSE without any success.
Let $f\colon X\to Y$ be a morphism of schemes for which $f_*(\mathcal{O}_X)=\mathcal{O_Y}$. When this condition arises has been discussed in ...
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Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
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