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0
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1answer
117 views

A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section ...
1
vote
0answers
145 views

Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X ...
0
votes
2answers
400 views

Recommended books/lecture notes for vector bundle on algebraic curve

I am going to enroll in a ceminar with the topic "vector bundle on algebraic curve". Except Algebraic Geometry(which I think GTM 52 by Hartshone is the main source), which topic I should prepare in ...
1
vote
2answers
203 views

Deforming to decompose vector bundles

After edit: How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)? Before edit: I am not sure ...
1
vote
1answer
125 views

Non-(stable)-triviality of the tautological bundles

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/ The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold ...
2
votes
2answers
174 views

Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 ...
7
votes
0answers
348 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 ...
1
vote
0answers
478 views

Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$? I know that such a bundle must ...
0
votes
0answers
67 views

sections of vector bundles transversal to a divisor

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$. $E$ a vector bundle over $X$ with a divisor $D$. We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough. ...
1
vote
1answer
251 views

non-trivial topological line bundles over cartesian product of manifolds not coming from a pullback

Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first ...
16
votes
1answer
449 views

Existence of non-split vector bundles on smooth projective varieties

Question. Is it known/easy to see that every smooth projective variety $X$ (over an algebraically closed field), except for the point and $\mathbb{P}^1$, has a vector bundle which is not a direct ...
5
votes
1answer
537 views

Coherent Sheaves and Holomorphic Vector Bundles

For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...
14
votes
3answers
879 views

what is a spinor structure?

There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$, a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$ ...
6
votes
1answer
348 views

Classifying Globally Generated Holomorphic Line Bundles over a Flag Manifold

I was recently looking back at an old question of mine, where I asked about the classification of the line bundles over a general complex flag manifold. Pavel Etingof gave the following excellent ...
5
votes
1answer
306 views

Alternate definition of vector bundle?

Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category): A $k$-dimensional vector bundle is a triple ...
0
votes
0answers
168 views

sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...
5
votes
1answer
233 views

When are the Smooth Sections of a Bundle Generated as a Module (over Smooth Functions) by the Holomorohic Sections

For a holomorphic vector bundle $E$ over a complex manifold $M$, we denote its space of smooth sections by $\Gamma^{\infty}(E)$, and its space of holomorphic sections by $\Gamma^{hol}(E)$. Now I've ...
0
votes
0answers
74 views

on trivialisation of T-torsors

Let $X$ a smooth connected projective curve over an algebraically closed field $k$ and $F$ its function field. $T$ a $X$-torus. Let $R$ be any ring. Let $E$ a $T$-torsor on $(X-x)\times_{k}R$. Does ...
4
votes
1answer
501 views

Why do we use the less simple convention for the definition of a vector bundle connection?

For a (smooth) vector bundle $F$ over a manifold $M$, one normally defines a connection to be a linear map $$ \nabla:\Gamma^{\infty}(V) \to \Omega^1(M) \otimes \Gamma^{\infty}(V), $$ satisfying ...
3
votes
0answers
189 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 ...
10
votes
5answers
849 views

From Topological to Smooth and Holomorphic Vector Bundles

In the last weeks I have been think of the transition from topological vector bundles to smooth and holomorphic vector bundles. This has resulted in a few questions (with a common thread) as follows: ...
1
vote
3answers
258 views

Linearly trivial bundles on hypersufaces in $\mathbb CP^n$

Recall a definition. Let $V\subset \mathbb CP^n$ be a projective variety and $E$ be a holomorphic vector bundle on it. We call $E$ linearly trivial if the restriction of $E$ to any projective line in ...
3
votes
1answer
240 views

Are rational sections of a vector bundle useful?

Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector ...
1
vote
1answer
262 views

Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?

Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a ...
4
votes
1answer
150 views

A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$

I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection ...
11
votes
0answers
642 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 ...
3
votes
2answers
383 views

Vector Bundles on normal surfaces

Let $X$ be a projective normal surface over $\mathbb{C}$. In this related question it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimension 2 subset extends ...
6
votes
3answers
452 views

How does one go from Chern--Weil to cohomology classes on BGL(n,C)?

Let's assume we start with Chern--Weil theory in the following form: Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...
3
votes
2answers
231 views

Triviality of Associated Bundles

Let $P\rightarrow M$ be a principal (right) $G$-bundle, where $G$ is a Lie group. Given a finite-dimensional representation of $G$, $V$ say, we can define the associated bundle ...
1
vote
1answer
230 views

Direct sum of two stable bundles of same slope

How to prove that the direct sum of two stable vector bundles of the same slope over a smooth curve is a semistable bundle?
0
votes
1answer
165 views

Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic?

Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the ...
4
votes
2answers
461 views

Jets of Equivariant Vector Bundles

Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
3
votes
1answer
250 views

Euler characteristics and the difference bundle construction

I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between ...
0
votes
1answer
82 views

Isotropic splitting for exact Courant algebroids

An exact Courant algebroid $E$ is one such that the sequence $0\to T^\star M\xrightarrow{\rho^\star} E^\star\simeq E\xrightarrow{\rho} TM\to 0$ is exact. Here $\rho$ is the anchor of the algebroid. ...
6
votes
2answers
613 views

Examples of excess intersection theory?

Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
13
votes
3answers
926 views

Is the category of vector bundles over a topological space abelian?

Consider the trivial bundle $V=\mathbb{R}\times\mathbb{R}$ and the map $f:V\rightarrow V$ given by $(t,x)\mapsto(t,tx)$. This has fibrewise kernels and cokernels, but the ranks jump at 0, so the ...
3
votes
1answer
452 views

1-jet bundle on vector bundle with metric connection

Background I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to ...
13
votes
2answers
874 views

Non trivial vector bundle over non-paracompact contractible space

The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of: Non trivial ...
2
votes
2answers
424 views

Moduli spaces of vector bundles and stability conditions

Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is $$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$ This induces the ...
9
votes
1answer
532 views

Rank two vector bundles on a curve of genus two

I recently learned of an interesting result of Narasimhan and Ramanan from 1969, which says that moduli space of rank two vector bundles with trivial determinant on a curve $X$ of genus two is ...
1
vote
2answers
438 views

Holomorphic bundles and maps to the Grassmannian ?

Hello, In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I ...
2
votes
2answers
445 views

Global Definition of the Dolbeault Complex of a Vector Bundle

For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from ...
6
votes
2answers
383 views

Ample vector bundles on complex tori

Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for ...
4
votes
2answers
762 views

Riemannian metrics as sections of a vector bundle

Let $\pi : E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle ...
0
votes
1answer
198 views

eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...
6
votes
2answers
399 views

Homotopy invariance of vector bundles by parallel transport: reference needed for my students.

Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are ...
1
vote
0answers
105 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
0
votes
1answer
301 views

Cotetrad, spin connection and Dirac operator

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...
2
votes
0answers
69 views

Properties of special Cayley-Bacharach bundles on a K3-surface

Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$. Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We ...
4
votes
1answer
382 views

Representations of infinite dimensional Lie algebras as vector fields on manifolds

Suppose I have e.g. the Witt algebra, $\left[l_n,l_m \right] = -(n-m)l_{n+m}$. I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the $l_n$ span the Lie ...