Resolution of direct image functor

Question

Let $i: X \to Y$ be an embedding of compact complex manifolds (not necessarily projective) and $E\to X$ a holomorphic vector bundle. I've seen it stated that the direct image sheaf $i_* E$ has a resolution by holomorphic vector bundles on $Y$. Is there a nice, clean way to construct this resolution? Or a good reference where this is discussed?

Answer 1

Edit: Most of my answer was total nonsense so I got rid of it but I think this paragraph is still true and still answers your question.

I'll give an answer considering them as proper varieties over $\mathbb{C}$. I'm pretty sure this should then be true in the holomorphic case as well (maybe by GAGA though I can't say I understand that).

More generally, suppose we had any morphism of complex compact manifolds $f:X \to Y$ and any coherent sheaf $\mathcal{F}$ on $X$. Again, $f$ is proper since $X$ and $Y$ are proper. Therefore, the pushforward $f_*\mathcal{F}$ is coherent on $Y$. Excersise III.6.8 of Hartshorne gurantees us the existence of a resolution by locally free sheaves and Hilbert's syzygy theorem gurantees it is finite.

For references see this MO quetsion.