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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39
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595 views

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 ...
18
votes
0answers
548 views

Characteristic Classes for $E_8$ Bundles

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 associated vector bundle $V=P\times_{\rho}\mathbb ...
16
votes
0answers
192 views

“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 ...
12
votes
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402 views

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 ...
11
votes
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1k views

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 ...
11
votes
0answers
355 views

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 ...
10
votes
0answers
446 views

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 ...
8
votes
0answers
156 views

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, ...
8
votes
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400 views

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 ...
7
votes
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100 views

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 ...
6
votes
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160 views

Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$. Studying some branched covers of $A$, I was led to consider some rank $2$ holomorphic ...
6
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160 views

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 ...
6
votes
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314 views

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 ...
6
votes
0answers
262 views

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 ...
6
votes
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497 views

Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from this question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective curve $X$ is ample if and ...
5
votes
0answers
90 views

Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that ...
5
votes
0answers
134 views

Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
5
votes
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192 views

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 ...
4
votes
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142 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
4
votes
0answers
46 views

Isometry theorem, exists homeomorphism that carries each fiber isomorphically onto itself

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically onto ...
4
votes
0answers
90 views

Higher tangent bundles of manifolds with non integer dimension

One way to define the tangent space of a manifold at a point $p\in M$ is the following: We define an equivalent relation on the space of curves passing $p$ as follows: Two curves $\alpha, \beta$ are ...
4
votes
0answers
108 views

Extend a vector bundle on a flat family

Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open ...
4
votes
0answers
83 views

Dual involution on the $Ext^1$

Let $X$ be a smooth algebraic curve over $\mathbb C$, and let $F$ be a vector bundle on it of degree $1$, take the dual of an extention $$0\rightarrow F^*\rightarrow E\rightarrow F\rightarrow0$$ is ...
4
votes
0answers
111 views

No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...
4
votes
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117 views

Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up. I have a sequence of (smooth, complex, rationally connected) ...
4
votes
0answers
304 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
4
votes
0answers
347 views

Harder-Narasimhan filtration of rank 4 vector bundles on $P^2$

Given a non-semistable vector bundle on $P^2$ of rank 4, are explicit conditions known for when the ranks in its Harder-Narasimhan filtration are (3,1), (2,2) and (1,3) respectively? I would be very ...
4
votes
0answers
426 views

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 ...
3
votes
0answers
113 views

Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...
3
votes
0answers
154 views

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
3
votes
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97 views

line bundle on affine grassmannian and central extension

Let $G$ be a connected reductive group over $\mathbb{C}$, let $Gr$ be the affine grassmannian of $G$. On $Gr$, we know that there is a canonical line bundle $L$ (the generator of $Pic(Gr)$). Now ...
3
votes
0answers
36 views

Is the unit bundle of a Finsler vector bundle a sphere bundle?

I asked this at mathstackexchange but got no answer, so I am trying here. Let $E$ be a Finsler vector bundle* of rank $k$ over a manifold $M$. Does the unit "bundle" $UE$ admits a structure of a ...
3
votes
0answers
108 views

Dimension of the singular locus of $\mathcal M_X(r,d)$

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with ...
3
votes
0answers
48 views

Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
3
votes
0answers
124 views

Deformation of vector bundle on projective space with same Hilbert polynomial as multiple of structure sheaf

Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$. Does there exist a vector bundle over ${\bf P}^n \times {\rm ...
3
votes
0answers
210 views

Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...
3
votes
0answers
132 views

Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles. ...
3
votes
0answers
96 views

Proof of Lemma in “Harmonic maps and the self-duality equations” by Donaldson

I am referring to this paper by S. K. Donaldson. I could not find a freely available version, hence I feel uncomfortable to copy & paste parts of his paper, and won't do so. Nevertheless, I will ...
3
votes
0answers
180 views

Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$. ...
3
votes
0answers
144 views

Orthonormal frame bundle orthogonal to a curve

This is a duplicate of this question on math.stackexchange, since I got there not a single answer. Let $M$ be a $n$-dimensional smooth riemannian manifold and ...
3
votes
0answers
142 views

Paracompactness and inner product on vector bundles

Let $X$ be a Hausdorff topological space. It is well-known that every vector bundle on $X$ possesses a continuous inner product, provided that $X$ is paracompact. Is the converse true? -- Namely ...
3
votes
0answers
190 views

extending a vector bundle

given a vector bundle $V \rightarrow N$ over a manifold $N$ and let's assume $N \hookrightarrow M$ is embedded into a manifold $M$ is there a way to extend $V$ to a bundle over $M$, i.e. is there a ...
3
votes
0answers
257 views

Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...
3
votes
0answers
184 views

Extending intersection bundles

Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that ...
3
votes
0answers
238 views

Ricci flat metrics on holomorphic vector bundles over Riemann surfaces

I am interested in the local geometry of holomorphic curves in Calabi-Yau threefolds. The setup and question are then the following: Consider a $\mathbb{C}^2$ bundle over a compact Riemann surface ...
2
votes
0answers
53 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
2
votes
0answers
61 views

Are “vector spaces” over a smooth scheme with constant fiber dimension locally free?

I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism $p:V\to S$ of schemes of finite type over some base field. Assume that $p$ has all the ...
2
votes
0answers
130 views

Doubt on elementary transformations in the paper - On a family of algebraic vector bundles by Maruyama

Let $S$ be a surface. $C$ be a curve on $S$ and $i:C\hookrightarrow S$ is the inclusion. Let $E$ be a rank 2 vector bundle on $S$ and $F$ a line bundle on $C$. Suppose we have a surjection ...
2
votes
0answers
174 views

Fiber Bundle with a perfect bilinear map

Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$ ...
2
votes
0answers
185 views

Geometric proof of Borel-Weil theorem

I am curious if there is any geometric proof of Borel-Weil theorem. Borel-Weil is a geometric realization of irreducible unitary representation. The proofs I found, however, all use Weyl unitarian ...