1
vote
1answer
77 views

deformations of vector bundles on curves

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of ...
1
vote
0answers
87 views

DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...
1
vote
1answer
219 views

Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
2
votes
1answer
250 views

FIltrations on a vector bundle on a curve

Let $X$ be a smooth connected projective curve of genus $g$ over an algebraically closed field. Let $E$ be a vector bundle on $X$ of rank $n$. Is it true that there exists a constand $N(g,n)$ such ...
2
votes
1answer
182 views

vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$

Let $A = \mathbb{C}[x,y,z]/(x y - z^k)$. In fact $A$ is the ring of $\mu_k$ invariants: $A = \mathbb{C}[u,v]^{\mu_k}$ where $g \in \mu_k$ acts by $g(u,v) = (g u, g^{-1} v)$. This allows one to ...
0
votes
0answers
185 views

On Bott's Formula on projective Spaces

I'm following the book of Okonek, Schneider and Spindler, Vector Bundles on Complex Projective Spaces, and they say that is a useful exercise try to prove Bott's Formula that calculates the cohomology ...
2
votes
2answers
217 views

Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...
3
votes
0answers
111 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. ...
2
votes
1answer
183 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
80 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 ...
0
votes
1answer
103 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
0
votes
0answers
90 views

torsors on quasi-split groups

Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$. Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$. We consider $G$ a quasi-split group scheme over $X$ ...
4
votes
1answer
314 views

why are principal GL(n)-bundles (Zariski-)locally trivial?

I'm in particular interested in understanding Grothendieck's argument for this in SGA 1 (page 232 in http://arxiv.org/pdf/math/0206203v2.pdf) Let $G$ be $\text{GL}_n$ over a scheme $S$ for some ...
2
votes
0answers
82 views

Moduli space of sheaves on a ribbon

In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...
0
votes
0answers
118 views

Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
4
votes
2answers
361 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 ...
-3
votes
1answer
268 views

Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given. We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$. For ...
1
vote
0answers
89 views

Degeneracy divisor of the “trace” morphism

Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...
11
votes
2answers
596 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 ...
1
vote
2answers
99 views

Extending the tautological bundle of $G(1,3)$?

It is well known that the Grassmanian of lines in $\mathbb P^3$ is isomorphic to a quadric in $\mathbb P^5$. I would like to ask if the tautological rank two bundle on the grassmanian extends to a ...
1
vote
1answer
175 views

Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?

In an answer to a MathOverflow question on the following link Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...
3
votes
2answers
176 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 ...
2
votes
1answer
124 views

stable vector bundle and space surves

I am sure this is well known, but I am not an expert...so I appreciate any help Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let ...
3
votes
1answer
312 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
128 views

How can we show that a transverse section exist

Let $c_i,d_j <n$, be a set of integers and define $$ M=\prod Gr(c_i,n),\quad N=\prod Gr(d_j,n),$$ where $Gr(k,n)$ is the grassmannian of k-planes in $\mathbb{C}^n$. Let $E_M=\oplus E_i^*$, where ...
1
vote
1answer
151 views

Do there exist equivariant sheafs that are not equivariant vector bundles?

For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...
2
votes
2answers
231 views

Connections on the Hodge bundle?

Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...
6
votes
1answer
246 views

Global sections of determinant bundle of symmetric powers

Let $E$ be a globally generated vector bundle of rank $r$ on a normal irreducible projective variety $X$. Suppose that $E$ induces a finite map $$X \to \mathbb{G}r (H^0(E), r)$$ to the Grassmannian of ...
3
votes
2answers
200 views

chern classes of push-pulled vector bundles

Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the ...
2
votes
1answer
126 views

Vector bundles on a weighted projective stack

Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack. Each vector ...
0
votes
2answers
211 views

Push-forward of a nef bundle

Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f_{*}E$. Does anyone know an example where $E$ is nef but ...
1
vote
2answers
305 views

Is the moduli space of stable vector bundles over a smooth projective curve fano?

Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space ...
1
vote
0answers
54 views

Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
3
votes
2answers
188 views

Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up. Let $X$ be a ...
5
votes
3answers
469 views

Hartshorne-Serre's correspondence in higher codimension

There's a well-known correspondence (traditionally called Hartshorne-Serre) between codimension 2 smooth subvarieties $S\subset X$ of a smooth algebraic variety $X$ and certain rank two vector bundles ...
5
votes
1answer
202 views

Looking for a special rank 2 vector bundle

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$ Question: For which $g$, there is such ...
3
votes
2answers
100 views

A question on a projective bundle $\mathbb{P}(L\oplus \mathcal{O}_X)$

Let $X$ be a complex manifold and $L$ a line bundle on it. Define $Y:=\mathbb{P}(L\oplus \mathcal{O}_X)$ be the projective bundle over $X$. Here is a statement I don't understand: The summands $L$ ...
0
votes
2answers
397 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 ...
2
votes
2answers
173 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
347 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. ...
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
533 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
345 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 ...
0
votes
0answers
167 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 $z$ ...
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 ...
3
votes
0answers
187 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 ...
1
vote
3answers
257 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 ...