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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)=...
BinAcker's user avatar
  • 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 ...
John117's user avatar
  • 395
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 ...
John117's user avatar
  • 395
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:...
BinAcker's user avatar
  • 789
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 ...
John117's user avatar
  • 395
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 ...
BinAcker's user avatar
  • 789
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 $...
BinAcker's user avatar
  • 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.
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