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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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 ...
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Characteristic classes for $E_8$ bundles
$\DeclareMathOperator\B{B}\DeclareMathOperator\SU{SU}$Given a principal $E_8$ bundle $P\rightarrow X$ one can take the
adjoint representation $\rho :E_8\rightarrow \SU(\mathbb C^{248})$
and form the ...
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Is there a useful theory of D-modules on smooth (non-analytic) manifolds?
D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
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"High-concept" explanation for proof of a theorem of Ochanine?
See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here.
Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
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Trying to reconcile two facts about the Appell-Lerch sum learned from Polishchuk and Zwegers
One of the key characters in the thesis of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau)=\frac{e^{\pi i u}}{\vartheta(v;\tau)...
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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 $M$...
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What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
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conformal blocks for beginners
I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
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Any progress on Strominger-Yau and Zaslow conjecture?
In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and ...
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Exotic smoothness and Parallelizability
Regarding the parallelizability of the Milnor's seven dimensional exotic spheres:
Parallelizability of the Milnor's exotic spheres in dimension 7
The following question naturally arises:
Suppose ...
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Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
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Big tangent bundle
Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
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Atiyah-Bott from Beauville-Laszlo
This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
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Which differential forms commute with the curvature form?
Consider a vector bundle, $E \to M$, with connection, $\nabla$, and curvature $2$-form, $F$ on $M$. For $E$-valued differential forms on $M$, $\Omega(M, E)$, we have an exterior covariant derivative, ...
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Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?
The following question is related to Singular Yang-Mills theory
The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-...
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Deformations of some simple quotient stacks.
I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...
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Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
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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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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 ...
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Lifting $\mathbb C^*$ actions to holomorphic bundles
Let $X$ be a smooth complex projective variety and $V$ be a holomorphic bundle on $X$. Suppose we have an algebraic $\mathbb C^*$-action on $X$. Is it true that the bundle $V$ can always be deformed ...
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Can we use sheaf cohomology to say anything interesting for vector bundles with non-flat connections?
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...
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Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?
The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, $\...
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semi flat connections
Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
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Triviality of holomorphic vector bundles over $\mathbb{C}$
Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle.
I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
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407
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A question about pullbacks of $C^\infty_M$-modules
First, let me state a definition: Let $M$ be a smooth manifold and suppose $\mathcal{E}$ is a sheaf of $C^\infty_M$-modules. Given a point $x \in M$ let $I_x$ denote the vanishing ideal at $x$. We ...
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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 ...
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Different definitions of Stiefel-Whitney classes
It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...
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The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
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Obstructions for existence of a fiber wise covering space structure( A bundle of covering spaces)
Let $S^n \times \mathbb{T}^n$ be the trivial Torus bundle over $S^n$.
Assume that we have a continuous fiber preserving map $\phi :TS^n \to S^n \times \mathbb{T}^n$ which restriction to each fiber ...
7
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Affine covering whose intersections are distinguished affine open
Let $X$ be a quasi-compact scheme. Say that $X$ has property $\mathbf{P}_{n}$ if $X$ admits an open cover $X = \bigcup_{i=1}^{n} U_{i}$ such that each $U_{i}$ is affine and each pairwise intersection $...
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Why study Bogomolov's T-Stability
Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-...
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Hartshorne conjecture on vector bundles on projective spaces
A conjecture of Hartshorne states that any vector bundle of a small rank on a projective space of large dimension is split (i.e. isomorphic to a direct sum of line bundles). I would like to know the ...
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Bundles over Grassmanian with given top Chern class
So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
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Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
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Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?
If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension $\widetilde{SA}...
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Topological obstructions to extending algebraic vector bundles
Ariyan and Kevin Lin have asked about the problem of extending vector bundles defined on an open subvariety across the rest of the variety. There can be subtle commutative algebra obstructions, as in ...
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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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Bundles equivariant with respect to a transitive Lie algebra action
Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...
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The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
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Sections of the sum of two copies of the tangent bundle of the $2$-sphere
If $T$ is the tangent bundle of $S^2$, then $T\oplus T$ is trivial of rank $4$. Indeed, $T\oplus\mathbb R=\mathbb R^3$, so $T\oplus T\oplus\mathbb R^2=\mathbb R^6$ and $T\oplus T$ is stably trivial: ...
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Is the space of analytic sections of a vector bundle a Fréchet space?
Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of ...
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Surfaces of general type with globally generated cotangent bundle
There is a lot of work about compact complex surfaces of general type $X$ having ample cotangent bundle $\Omega_X$: for instance, one can read the recent works of Damian Brotbeck and collaborators in ...
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$K_0$ of configuration of hyperplanes
Let $\ell_n$ where $n\geq 3$ be the configuration of $n$ lines in a plane, such that $n-1$ of them pass through a single point and the last one does not and it intersects rest of the $n-1$ lines. I'm ...
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What’s the limit of a vector bundle?
In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current.
In the geometric ...
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Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
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Can we "Curve" a manifold, as much as possible?
Assume that $M$ is a $k$ dimensional manifold which is embedded in $\mathbb{R}^n$. We define the map $\phi_{M,n}: M \to G(k,n)$ with $\phi_{M,n} (x)= T_x M$, the tangent space to $M$ at point $x\in M$....
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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 ...
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Global sections for a locally free sheaf over curves
Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg \...
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How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
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Line bundle whose pushforward is a complex of vector bundles
If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...