# Tagged Questions

**4**

votes

**2**answers

304 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 ...

**0**

votes

**0**answers

51 views

### Cohomology of Sym(E) [migrated]

Let $U,V$ be two vector bundles on some scheme $X$. Let $E$ be some bundle in $0\to U\to E\to V\to 0$. If I can show that $H^i(X; Sym(U\oplus V))=0$ for $i>0$, can we conclude the same property for ...

**-1**

votes

**0**answers

70 views

### Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$
This is the same as giving the (very ample) line bundle over the curve ...

**-3**

votes

**1**answer

254 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

**0**answers

79 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 ...

**9**

votes

**2**answers

503 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

**2**answers

96 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

**1**answer

160 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

**2**answers

169 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

**0**answers

77 views

### Subbundles of conformal blocks bundles over $\overline{\mathcal{M}}_{g,n}$

Let $\mathfrak{g}$ be a semisimple split Lie algebra over a field $k$ of characteristic zero. Let $l$ be a non-negative integer and let $P_{l}^{n}$ be the set of $n$-tuples of dominant integral ...

**2**

votes

**1**answer

116 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

**1**answer

296 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

**1**answer

110 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

**1**answer

142 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

**2**answers

201 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

**1**answer

216 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

**2**answers

172 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

**1**answer

121 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 ...

**1**

vote

**2**answers

182 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

**2**answers

270 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 ...

**0**

votes

**0**answers

53 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

**2**answers

177 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

**3**answers

434 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

**1**answer

198 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

**2**answers

93 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

**2**answers

349 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

**2**answers

191 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

**2**answers

162 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

**0**answers

327 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

**0**answers

466 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

**0**answers

66 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.
...

**15**

votes

**1**answer

424 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

**1**answer

505 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

**1**answer

320 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

**0**answers

163 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

**0**answers

71 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

**0**answers

170 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

**3**answers

254 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

**1**answer

219 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 ...

**10**

votes

**0**answers

523 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

**2**answers

362 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 ...

**1**

vote

**1**answer

217 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

**1**answer

158 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 ...

**6**

votes

**2**answers

563 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

**3**answers

818 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 ...

**2**

votes

**2**answers

388 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

**1**answer

482 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 ...

**6**

votes

**2**answers

351 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 ...

**0**

votes

**1**answer

188 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 ...

**2**

votes

**0**answers

66 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 ...