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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Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
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A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
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Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
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What is the adjoint bundle of groups $P\times_{G}G$?
It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
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Question on notation for definition of symbol of differential operator
I was looking at this definition of the symbol of a differential operator, and am unsure what "$T^*X\otimes_XE$" means. I couldn’t find an explanation anywhere on nlab either. My main ...
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Stable homotopy of vector bundles
Consider the category $\mathsf{VectBun}$ of real vector bundles over topological spaces, where
the morphisms are bundle maps that are fiberwise isomorphisms.
This category has a stabilization functor
...
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Holomorphic fibre bundles over noncompact Riemann surfaces
Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...
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Can we get a connection on the principal bundle from a connection on the associated vector bundle?
Assume $G$ is a Lie group, $P \to M$ is a smooth principal $G$-bundle, and $\rho \colon G \to GL(V)$ is a smooth representation of $G$. We can define a connection on the associated vector bundle $E := ...
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A question related to the local family index theorem without the kernel bundle assumption
Let $\pi:X\to B$ be a submersion with closed, oriented and spin fibers of even dimension and $E\to X$ a Hermitian bundle with a unitary connection $\nabla^E$. By putting metrics and connections on ...
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Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
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Extending the natural thom form of a vector bundle from the boundary of a manifold
(Edited after taking into account Tom Goodwillie's answer.)
Let $E \rightarrow X$ be an orientable vector bundle.
In this MO answer it is explained how to obtain a representative of the Thom class (...
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Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?
Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$
It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
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Converging paths implies converging parallel transports along those paths?
Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...
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Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)
I'm familiar with the definition of the category of vector bundles, but I'm trying to derive it from some first principles about general fiber bundles. My intuition is that vector bundles should be ...
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