I presume this is a GAGA-style result, but I cannot find a reference.


Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.

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    $\begingroup$ In particular, the analytification of a complex algebraic variety is a Stein space. $\endgroup$ – Earthliŋ May 23 at 10:31
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    $\begingroup$ @Earthliŋ: You probably mean affine algebraic variety. $\endgroup$ – Jérôme Poineau May 25 at 20:45
  • $\begingroup$ @JérômePoineau Yes, of course. Somehow "affine" got lost when I added "complex"... $\endgroup$ – Earthliŋ May 25 at 21:50

I believe the reason for this is Cartan's 'Theorem B': for a Stein manifold $\mathrm{X}$ sheaf cohomology vanishes, $\mathrm{H}^n(\mathrm{X},-)=0$ for $n\geqslant 1$, and this property characterises Stein manifolds among complex manifolds. In the same way affine schemes are characterised among (nice) schemes by cohomology vanishing (this is a theorem of Serre). This comes up in the proof of the cohomological comparison result which is part of GAGA, see for example SGA 1, Exposé XII.

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    $\begingroup$ There is no way to be completely tactful here; the vanishing condition is for coherent analytic coefficients only. It is certainly not true for all coefficients, e.g. $X= \mathbb{C}^*$ and cohomology with coefficients in $\mathbb{Z}$ is not zero, but $X$ is Stein. $\endgroup$ – Donu Arapura May 25 at 13:29
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    $\begingroup$ @DonuArapura this was understood... on the scheme side you are also not going to consider sheaves which are not even $\mathcal{O}_X$-modules or which have no finiteness properties. $\endgroup$ – Mere Scribe May 25 at 13:38
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    $\begingroup$ OK, but I'll leave the comment, since not everyone would understand the hidden assumptions. $\endgroup$ – Donu Arapura May 25 at 13:45

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