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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
112 views

global algebraic functions $\Gamma(T^{*}M)$ on the cotangent bundle of moduli space

Let $X\colon$ smooth projective curve, $\mathcal{M}\colon $ moduli space of semistable higgs bundle of rank $r$ and with fixed determinat $\xi$, and $H\colon \mathcal{M}\rightarrow W=\oplus_{i=2}^{r} ...
2 votes
1 answer
274 views

Varieties satisfying the extension of vector bundles property

We know if we have a regular variety $X$ with $U$ an open sub-scheme such that $codim(X\setminus U)\geq 2$, then any reflexive sheaf has a unique extension from $U$ to $X$. My question is when a ...
9 votes
2 answers
634 views

How to recognize a vector bundle?

Given a connected topological space $E$, under which conditions is it possible to find a subspace $B$ such that $E$ can be regarded as a (rank $n$) vector bundle over $B$? Is it possible to find the ...
3 votes
1 answer
380 views

Pull-back a section of a vector bundle

Let $M$ be a manifold of dimension $n$ and $\mathcal D$ be a distribution of dimension $n-1$. We consider the quotient bundle $TM/\mathcal D = \bigsqcup_{p \in M} T_pM/\mathcal D_p$ with the ...
5 votes
1 answer
596 views

Characteristic classes in term of cocycles

Giving a vector (principal) bundle is equivalent to give a family of cocycles ${g_{\beta \alpha}: U_\alpha\cap U_\beta \to G}$ where $G$ is the structure group of the bundle. Chern classes are ...
0 votes
1 answer
131 views

Local triviality condition in vector bundles [closed]

Let $E$ and $M$ be smooth manifolds (of finite dimension, Hausdorff and second countable). Let $\pi:E\longrightarrow M$ be a surjective submersion such that: $E_p:=\pi^{-1}(p)$ is a real vector space ...
2 votes
1 answer
314 views

Dual of a specific coherent sheaf that is a vector bundle

Let's assume we have two short exact sequence of vector bundles on a smooth variety $X$ over a field. $l_1: 0\rightarrow V_1 \rightarrow V \rightarrow V_2 \rightarrow 0$ and $l_2: 0\rightarrow V_3 \...
7 votes
1 answer
319 views

Can non-split extension be isomorphic to the split one as objects

Is it possible to have a non-split short exact sequences of vector bundles (on some smooth variety) $0\rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$. Such that $V_2\cong V_1\oplus V_3$ ...
1 vote
0 answers
79 views

Symplectic structure on the space of complexes of holomorphic vector bundles

Let $E\rightarrow X$ be a holomorphic vector bundle over a complex manifold. Denote by $Dol(E)$ the space of holomorphic structures on $E$. Fix any Hermitian metric $h$ on $E$ and denote by $\mathcal{...
3 votes
0 answers
194 views

$2$-vector spaces and algebraic $2$-stacks

I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases: $\mathfrak{X}_{1}=$ Higher linear stack classifying (...
1 vote
0 answers
85 views

Curvature of a superconnection

Let $E\rightarrow X$ be a $\mathbb{Z}_2$-vector bundle (or superbundle for connoisseurs) and consider the superconnection $$A=\nabla + B$$ where $\nabla$ is a connection on $E$ and $B\in\Gamma(End(E))^...
6 votes
1 answer
272 views

Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom

This is a sort of continuation of this question. In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that ...
1 vote
0 answers
120 views

Vector bundle defined by using divisors of very ample line bundle

Let $X$ be a smooth projective curve. Suppose that $L_1$ and $L_2$ are line bundles on $X$, and $L_1$ is very ample. $\operatorname{Div}(s)$ denotes a divisor defined by a global section $s\in H^0(X,L)...
2 votes
0 answers
96 views

The openness the set of $s\in \bigoplus H^0(C,K_{C}^{\otimes i})$ for which the spectral curve is irreducible and reduced

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...
9 votes
1 answer
440 views

K-theory on finite-dimensional (possibly not finite) CW complexes

I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
13 votes
1 answer
463 views

Vector bundles on $\mathbf{P}^1$ and the Iwasawa decomposition

As everyone knows, every vector bundle on $\mathbf{P}^1$ splits as a direct sum of line bundles $\mathcal{O}(a_1)\oplus\cdots\oplus\mathcal{O}(a_n)$. This means that in the Weil-uniformisation ...
4 votes
0 answers
129 views

Global algebraic function over the moduli space of semistable higgs bundles $\mathcal{M}$

Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the ...
2 votes
0 answers
62 views

Singularity of reproducing kernel for elliptic operator

Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly ...
2 votes
0 answers
257 views

Fibers of Hitchin fibration are equidimensional

Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of ...
7 votes
0 answers
267 views

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 ...
2 votes
0 answers
326 views

Differences between induced vector fields on a smooth manifold and on a principal bundle

In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I'm having some problems with the ...
3 votes
1 answer
405 views

Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$ In many Books such ...
2 votes
1 answer
232 views

Extension of a holomorphic vector bundle on a nodal curve

I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve. Let $C$ be a nodal curve without closed componets and $E$ a ...
1 vote
0 answers
129 views

The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and $\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...
4 votes
0 answers
139 views

Linearly dependent vector fields not spanned by fewer continuous vector fields

This is a follow-up to the question https://math.stackexchange.com/questions/3845080/linearly-dependent-vector-fields-that-are-not-spanned-by-fewer-continuous-vect, which attracted very little ...
7 votes
0 answers
407 views

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 ...
1 vote
0 answers
214 views

Fiber of the Hitchin map

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...
2 votes
0 answers
91 views

lie algebra bundle and underlying vector bundle

Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$. As a vector bundle it is trivial, ...
10 votes
1 answer
511 views

Chern classes of a mapping torus vector bundle in terms of the construction data

Let $\pi:E\to X$ be a complex vector bundle*, and $f:E\to E$ a bundle isomorphism. Consider the mapping torus $$E(f) := \frac{E\times [0,1]}{E \times \{0\}\sim_f E \times \{1\}}$$ where the ...
5 votes
0 answers
182 views

Non-isomorphic short exact sequences on isomorphic objects

$\DeclareMathOperator\Ext{Ext}\newcommand\SE{\mathrm{SE}}\newcommand\Def{\mathrm{Def}}$Assume $X$ is a smooth projective variety over a field. There are two non-isomorphic short exact sequence of ...
4 votes
0 answers
172 views

Deformations of vector bundles and tubular neighborhood

I had a number of questions that are somewhat related to each other. I decided to post them altogether instead of separately. I'd appreciate any kinds of answers, ideas or sources regarding any of ...
4 votes
1 answer
189 views

Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE]

Note: This question came from MSE, but since I've received some useful observations I posted it here. Post on MSE Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}...
3 votes
1 answer
193 views

Vanishing locus generic section $(\mathrm{sym}^2 \mathcal{R})(1)$

Let $n = 2m$ be an even integer and let $\mathcal{R}$ the tautological bundle on the Grassmannian $\mathrm{Gr}(2,n)$. I am looking for an explicit description of the degener The bundle $(\mathrm{Sym}^...
0 votes
0 answers
354 views

Definition of Hitchin map

This may be a dumb question. $\mathcal{M}(r,d)$ is a coarse moduli scheme for semistable pairs $(E,\phi:E \rightarrow K_X \otimes E)$ of rank $r$, degree $d$ on a smooth projective curve $X$ over $\...
5 votes
0 answers
166 views

Hitchin map and vector bundles

I've been learning a bit about automorphisms of moduli spaces of vector bundles and the Hitchin map. I'm reading this paper of Indranil Biswas, Tomas L. Gomez, V. Munoz, and I have a problem about ...
5 votes
1 answer
403 views

Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
5 votes
0 answers
156 views

Relation between Bott-Chern forms and Second fundamental form

Given a short exact sequence of holomorphic Hermitian vector bundles $$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$ the second fundamental form measures the obstruction of $E\simeq F\oplus ...
0 votes
0 answers
96 views

On the locality of the Lefschetz conditions

Given a smooth projective variety $X$ of dimension $\geq 3$ and an ample divisor $Y$. The pair satisfy the effective Lefschetz condition $Leff(X,Y)$. This implies that the category of vector bundles ...
10 votes
2 answers
1k views

If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic

I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", Annals of Mathematics, and I am trying to prove (or disprove) the following result. The more elementary the ...
1 vote
0 answers
311 views

On definition of stable vector/Higgs bundle

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as $\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
6 votes
0 answers
100 views

$K_0$ 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 ...
4 votes
0 answers
100 views

Serre vanishing on one-point blow-ups

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
5 votes
0 answers
130 views

Riemannian version of topological $K$-theory

Let $X$ be a compact Hausdorff space.Put $Vec(X)$, the space of all real (or complex) vector bundles over $X$.We put also $Vec_g(X)$, the space of all Riemannian vector bundles over $X$, that is the ...
5 votes
1 answer
226 views

A question on moduli space of Hitchin's equations

I am reading Hitchin's Self-Duality paper. In section 5 (page 85), he is trying to prove that $Dim H^1=12(g-1)$. In doing so, he defines an operator $d^*_2+d_1$, where $d^*_2$ and $d_1$ are given by $...
5 votes
1 answer
235 views

Koszul-Malgrange Holomorphic structure on a pullback bundle

I'm finding myself a little confused about Koszul-Malgrange holomorphic structures in a certain context. Suppose $M$ is a complex manifold, $N$ is a smooth manifold with a smooth complex vector bundle ...
4 votes
2 answers
832 views

Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
2 votes
1 answer
1k views

Fubini-Study metric induced by submersion

The Fubini-Study metric $g:=g_{FS}$ is the unique $U(n+1)$-invariant Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space which by $U(n+1)$-invariance can ...
1 vote
0 answers
98 views

Serre-Swan theorem for infinite rank Frechet or Hilbert bundles?

Is there a version of Serre-Swan theorem for infinite rank Frechet or Hilbert bundles? If it existed, it would imply if $E\to X$ and $F\to X$ are infinite rank Frechet or Hilbert bundles over a smooth ...
2 votes
1 answer
170 views

Vector field along an immersion whose covariant derivative is the differential

Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
58 votes
4 answers
6k views

Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

Suppose for some reason one would be expecting a formula of the kind $$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$ valid in $H^*(Y)$ where $f:X\to Y$ is a ...

1
6 7
8
9 10
24