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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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
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I'm looking for the NLab page on particle species
This is just a reference request.
I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently.
If someone can point ...
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Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
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Are coherent modules with integrable log-connections locally free?
Let $X$ be a smooth Noetherian scheme over a field $K$. It is known that every coherent module with integrable connection on $X$ is locally free.
Is the same true for coherent modules with log-...
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A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
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Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates
Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\...
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
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Stable equivalence and stability theorem of vector bundles
I am going through this paper by Tanaka. In the proof of Proposition 3.2(1) given below
The author says that by the stability theorem as $\dim (B)\le m$ we have $\alpha\oplus1\cong m\oplus1$. But I ...
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Derived flat bundles
I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
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category of vector bundles with connections and its K-theory
For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
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Flat Maurer-Cartan connection iff flat Berry connection
I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
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Extending topological vector bundles and obstruction theory
This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
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Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism
Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle
$$
\mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
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Slopes of the Harder-Narasimhan filtration of a subbundle
Suppose $(p_1, ..., p_n)$ and $(q_1, ..., q_n)$ are two (non-strictly) decreasing sequences of positive real numbers. We say that $(q_1, ..., q_n)$ dominates $(p_1, ..., p_n)$ if $q_n \geq p_n$, $q_n+...
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Rank and determinant of the image of a vector bundle after applying a Schur functor?
Let $\mathcal{E}$ be a vector bundle of rank $r$ and degree $d$ over some smooth projective variety $X$. Furthermore, let $\lambda$ be a partition of $n$. We apply the $\lambda$-th Schur functor to $\...
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Reference for numerically non-negative polynomials for nef vector bundles
Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
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Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
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Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$
I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well.
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:...
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Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
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Obstructions to the existence of a flat connection on a vector bundle
Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
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A question on the proof of pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes
Here is a question about proving the pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes Bundles, Homotopy, and Manifolds. The proof is in page 73 of the notes. For me, ...
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Pushforwards in vector bundles over a topological spaces
I have been reading the discussion from Pushforward and pullback..
I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
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A question about vector bundle isomorphisms
Let $E\to X\times[0, 1]$ be a vector bundle. Define a map $i_t:X\to X\times[0, 1]$ by $i_t(x)=(x, t)$. Then we know the vector bundles $i_0^*E$ and $i_1^*E$ are isomorphic, and probably there are many ...
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The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds
$\def\sO{\mathcal{O}}
\def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
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Differential graded modules and the Serre-Swan theorem
I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules.
If $E \rightarrow B$ is a vector bundle, or even just any ...
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Homology groups of moduli of parabolic bundles with fixed determinant
I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant.
In particular, what is the second homology group of such ...
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Non-homogeneous line bundles over a homogeneous space
Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...
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The curvature of the induced connection on the antidual bundle
Let $E\to M$ be a complex vector bundle over a (real, smooth) manifold and $\nabla$ a connection on $E\to M$ whose curvature is $R$. From Section 1.5 of "Differential Geometry of Complex Vector ...
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Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question ...
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Equality of topologies in the spaces of section of a vector bundle
In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has
Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...
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The definition of the determinant of a coherent sheaf
Let $ X $ be a smooth (projective) variety and $ \mathcal{F} $ a torsion-free coherent sheaf of rank $ r $ on $ X $. The determinant $ \det \mathcal{F} $ can be defined by
(1) $ \det \mathcal F := ( \...
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Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?
Let $M$ be a smooth manifold and let $A=C^\infty(M).$
We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
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Rank of a tangent map related to holomorphic line bundles
Let $L,\,\,J$ be two holomorphic line bundles over a compact Riemann surface $X$ of genus $g_X>0$ such that
(1) $d_1:=\dim H^0\big(\operatorname{Hom}(L,J)\big)>0$ and $d_2:=\dim H^0\big(\...
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Normal bundle of veronese as iteration extension of symmetric powers
In this post, an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But ...
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Unicity of modifications of vector bundle on a regular base
I think that I have overheard the following statement, and would be grateful for either a reference or an explanation about why hearing must be slightly faulty and a clarification about what must ...
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Algebraizable image of a morphism of Galois cohomology stacks
Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
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Exterior differentiation of foliations
Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
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Equivariant sheaves on $\mathbb P^1$
Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
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Pullback of a vector bundle extension class
Let $ X $ be a variety with an automorphism $ \phi : X \rightarrow X $. Suppose there is a short exact sequence of vector bundles $ 0 \rightarrow F \rightarrow E \rightarrow G \rightarrow 0 $ on $ X $ ...
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Stability of cotangent bundle of hypersurface
Let $ X $ be a smooth hypersurface of degree $ d > 1 $ in $ \mathbb{P}^{n+1} $. What can be said about the stability (Slope/Gieseker) of the cotangent bundle of $ X $?
The closest reference I could ...
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A smooth family of lattices on the tangent bundle?
I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...
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$N$th-order approximation of point stabilizing diffeomorphisms by $N$th-order jet group?
NOTE: migrated from math SE.
I was wondering if ever higher jet groups of frames on a (possibly pseudo) Riemannian manifold $M$ approximate the point stabilizing subgroup of diffeomorphisms on $M$ as ...
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Pushforward of locally free sheaf by open immersion
Say $X$ is a smooth variety (even just $\mathbb{A}^n$) and $j\colon U\hookrightarrow X$ is an open immersion with $X - U$ of codimension 2 such that $E$ is a locally free sheaf on $U$. Since $X$ is ...
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Tensor product of vector bundles
The Whitney sum (where fibre dimensions add) of two real, or two complex, vector bundles $\pi : E \to X$ and $\pi' : E' \to X$ over a topological space $X$ is not hard to get an intuitive grasp of. ...
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The definition of a homogeneous vector bundle
For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
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Singularities of fibrations in conics
Consider a rank two vector bundle $E = \mathcal{O}(a)\oplus \mathcal{O}(b)\oplus \mathcal{O}(c)$ over $\mathbb{P}^1$. Fix coordinates $u_0,u_1$ on the base $\mathbb{P}^1$ and $v_0,v_1,v_2$ on the ...
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Integral mean value property
Let $V$ be the space of all continuous functions $f$ on the real line with $f(x)=\frac12\big(f(x-1)+f(x+1)\big)$.
It contains the space of periodic functions. The latter equals the space of ...