All Questions
8 questions
1
vote
0
answers
226
views
Resolution of the pushforward of a vector bundle
Let $i:Z\hookrightarrow X$ be a subvariety of a compact Kahler manifold. Assume that $Z$ can be realize as the zero locus of a section $s$ of a holomorphic vector bundle $E\to X$ of rank $r$. The ...
- 789
2
votes
0
answers
168
views
Criteria for a sheaf to be locally free over subvariety
Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.
Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:...
- 789
5
votes
0
answers
269
views
Line bundle whose pushforward is a complex of vector bundles
If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...
- 789
4
votes
1
answer
435
views
Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
- 395
1
vote
1
answer
427
views
Flat familiy of coherent sheaves over a scheme
I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
- 2,808
2
votes
1
answer
399
views
Locally free sheaves and vector bundles over smooth connected projective curve
Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
- 2,300
0
votes
0
answers
185
views
Recipe for resolving a coherent sheaf
Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...
- 789
0
votes
0
answers
635
views
A coherent sheaf is a vector bundle over subvariety?
Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.
- 789