1
vote
0answers
80 views

How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known): Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
0
votes
1answer
77 views

A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?

Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...
2
votes
1answer
80 views

Projectively flat Hermitian curvature proportional to Kähler form?

Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...
4
votes
2answers
354 views

Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better. I believe it is a theorem of Grauert that any holomorphic vector ...
1
vote
1answer
76 views

Do Hermitian metrics also split on the Riemann sphere?

Maybe this is well known, but i could not find a pointer to some literature: Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that ...
10
votes
2answers
553 views

Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base? In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
3
votes
2answers
174 views

Connectedness of a section of an algebraic bundle

Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$. There is a relatively straightforward criterium to check if the space ...
1
vote
1answer
87 views

Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant. I wonder, do there exist ...
2
votes
0answers
52 views

Formal adjoint of covariant derivative on endomorphism bundle

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$, equipped with an integrable, unitary connection $D=D'+D''$. Let $\beta\in\Lambda^{p,q}(\mathrm{End}\,E)$ be ...
3
votes
0answers
152 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
1answer
306 views

top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section. Is it be possible that $s^{-1}(0)\neq \emptyset$, ...
1
vote
1answer
90 views

Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
8
votes
1answer
346 views

Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a compatible almost complex structure $J$, such that the symplectic form determines an integer cohomology class, ie $$ [\omega] \in H^2(M, ...
1
vote
2answers
199 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 ...
5
votes
1answer
527 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 ...
6
votes
1answer
339 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
223 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 ...
10
votes
5answers
827 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
2answers
429 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
433 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
372 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
738 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 ...
3
votes
1answer
387 views

non-trivial locus of a holomorphic vector bundle

Let $X$ be a holomorphic vector bundle over $Y$ (where $Y$ is an arbitrary complex manifold, not necessary projective). Does there exist an analytic subset $Z$ of $Y$ such that the restriction of $X$ ...
0
votes
0answers
192 views

Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?
3
votes
1answer
300 views

A simple question about the degree of some vector bundle over rational curve.

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, ...
1
vote
0answers
278 views

Splitting of vector bundles on a complex torus

Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is ...
10
votes
3answers
783 views

Calculating the decomposition of a vector bundle over rational curve

Consider the rational curve (conic) given by image of the map $$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$ which lies in quintic 3-fold $X: x_1^5+\cdots+x_5^5- x_1\cdots x_5=0$. By ...
7
votes
2answers
1k views

Reference request: moduli space of vector bundles

I am trying to study the moduli of holomorphic vector bundles fast and I'm primarily interested to understand: 1) Why and were the stability is important. 2) How are the construction methods. 3) some ...
1
vote
5answers
770 views

indecomposable vector bundles having proper sub-bundles.

Over rational curve we know that any vector bundle is decomposable to direct sum of line bundles. In higher dimensions there are examples of indecomposable bundles. some indecomposable vector ...
4
votes
1answer
273 views

Is there a “simple commutation” relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ the (0,1) part of the chern connection of a vector bundle and $\delta^{'}$ the adjoint of the (1,0) part?

Hi, as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler ...
4
votes
2answers
523 views

Deformations of sheaves via automorphisms. How to express $Ext^1$?

Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). ...
3
votes
1answer
330 views

Divisors and vector bundles in various categories

I'm taking a first course on complex manifolds, and am trying to square what I hear with what I know of (real) differential geometry. Please forgive me if this question is misguided! Here are two ...
3
votes
1answer
689 views

Principal bundles and associated vector bundles, the case of the complex projective space (1,0)-forms

As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principal and their associated vector bundles. To this end I've been looking at ...
13
votes
4answers
1k views

Representations of surface groups via holomorphic connections

EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it! Background Let $E \to X$ be a ...