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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.

1,244 questions
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3 votes
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Let $n \in \mathbb{Z}_{>0}$. It is not hard to prove that there exists a CW complex $X$ with a single cell in each dimension 0, $2n$, $4n$, $6n$, $\dots$, and no further cells, and a $2n$-...
user171227's user avatar
  • 819
6 votes
2 answers
304 views

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 ...
Jacques Holstein's user avatar
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1 vote
1 answer
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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}...
prochet's user avatar
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1 vote
0 answers
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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 $\exists$ ...
Sandipan Das's user avatar
  • 39
0 votes
1 answer
95 views

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\...
Bence Racskó's user avatar
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1 vote
0 answers
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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 $\sigma_X^*\overline{V}$...
Sandipan Das's user avatar
  • 39
4 votes
1 answer
250 views

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 ...
Anubhab Pahari's user avatar
  • 193
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0 answers
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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 ...
Roch's user avatar
  • 515
2 votes
0 answers
180 views

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 ...
Roch's user avatar
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1 vote
0 answers
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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 ...
user577413's user avatar
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3 votes
1 answer
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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 ${\mathbb P}^1$ ...
adrian's user avatar
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4 votes
0 answers
268 views

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 ...
Springeer's user avatar
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1 vote
1 answer
202 views

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 $k$-...
Sandipan Das's user avatar
  • 39
20 votes
4 answers
1k views

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 ---...
ccriscitiello's user avatar
  • 903
2 votes
0 answers
80 views

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:...
user1234567890's user avatar
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