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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?

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1 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.

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Dear @ Dori, contra what you say closed embeddings are never flat except in very exceptional or trivial circumstances. A vector bundle on $X$ will have as direct image in $Y$ a sheaf supported on the image of $X$, which prevents it from being locally free on $Y$. –  Georges Elencwajg Dec 20 '13 at 8:22
    
Oh wow you;re totally right. I was way too tired when I wrote that answer. Deleting the first half. I think the second part about the pushforward being coherent is still right correct? –  Dori Bejleri Dec 20 '13 at 15:00
    
I think this only holds for smooth submanifolds of $\mathbb CP^n$ –  Eric O. Korman Dec 20 '13 at 15:53
    
Yes you do need smooth. I assumed that you meant smooth when you said manifold. I'm not sure if you still have finite resolutions in the nonsmooth case. I think that excersise in Hartshorne will still get you a resolution it just may not terminate. I don't think you need projective. Proper should be enough. –  Dori Bejleri Dec 20 '13 at 16:09
    
On second thought, I don't think the locally factorial hypothesis of that exercise will hold for singular things so I don't know even about infinite resolutions. –  Dori Bejleri Dec 20 '13 at 16:26

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