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.
1,251 questions
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Restriction of curve semistable Hodge bundles to generic fibre of an elliptic fibration
Let $\pi\colon X\to B$ be a proper, flat morphism; where $X$ is a smooth, irreducibile surface, and $B$ is a smooth, irreducibile curve; both are complex and projective. I assume that the generic ...
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Positivity in excess Porteous formula on $\mathbb P^n$
Crossposted from math.SE after two weeks of no reponse.
In the following, everything happens in/on $\mathbb P^n_{\mathbb k}$, $\mathbb k$ algebraically closed (no harm in assuming $\mathbb k = \mathbb ...
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Homology of Quotient of sobolev spaces and the spectral theorem for line bundles
Let $(M,g)$ be a finite dimensional, closed, smooth manifold. I'm interested in computing the homology of quotients of real/complex valued sobolev spaces (with $0$ removed). It is known (say via the ...
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Definition of Thom isomorphism in topological K-theory
I have a confusion regarding the definition of the thom isomorphism map in topological K-theory.
Suppose $\pi: F \rightarrow X$ is a complex vector bundle. Let $i:X \rightarrow F$ denote the zero ...
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Do vector bundles of sufficiently high rank over compact metric spaces with finite cohomological dimension admit sections
If $E$ is a $k$-dimensional complex vector bundle over an $n$-dimensional CW-complex $X$, such that $n\leq 2k-1$, then $E$ has a one-dimensional trivial subbundle (or equivalently, a global non-...
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Tannakian eqivalence of flat principal G--bundles
In paper: Moduli of representation of the fundamental group of a smooth projective variety II, in page 55, C.Simpson gives the statement:
The category of principal $G$ bundles with flat connection ...
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Why principal symbols are defined such that?
I am a master's student in mathematics with a focus on algebra and geometry, currently taking a course on vector bundles and Hodge theory. During the course we have defined the principal symbol as ...
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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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