All Questions

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)=...

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:...

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 ...

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 $...

Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset? Thanks in advance.