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Let $\Sigma^k$ be a $k$-dimensional Stein manifold with embedding as a real manifold (let's assume that that embedding is analytic if it makes things easier) $\Sigma^k \hookrightarrow \Bbb R^{2k}$.

Main question. Is it true that $\Sigma^2$ holomorphically embeds in $\Bbb C^2$?

Addendum. Is there an example of $\Sigma^k$ without embedding into $\Bbb C^k$ with $k < 7$? ($k > 1$ by Koebe uniformization theorem; on the other side, complex $7$-sphere ($\Bbb R^{14} \setminus \Bbb R^6$) does not embed in $\Bbb C^7$)

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    $\begingroup$ Which complex structure are you equipping $\mathbb{R}^{14}\setminus\mathbb{R}^6$ with? $\endgroup$
    – Michael Albanese
    Commented Jan 11, 2021 at 19:59
  • $\begingroup$ Complex sphere, as I wrote! Level set of a nondegenrate quadratic form on $\Bbb C^8$. $\endgroup$
    – Denis T
    Commented Jan 11, 2021 at 20:02
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    $\begingroup$ Sorry, I was not familiar with the phrase "complex sphere". $\endgroup$
    – Michael Albanese
    Commented Jan 11, 2021 at 20:03
  • $\begingroup$ Surely you need a hypothesis relating the Stein structure to the embedding (as in Zippy's answer). I think that the right hypothesis is a smooth embedding, a holomorphic immersion, and a path through smooth immersions between them. The h-principle should produce such a structure from any reasonable weaker hypothesis. From such a structure, you can build a holomorphic embedding. Some fragment of this is in one of the h-principle books, maybe the full statement, maybe just an analogue of Haefliger's embedding theorem. In any event, the statement is true by the Goodwillie-Weiss manifold calculus. $\endgroup$
    – Ben Wieland
    Commented Jan 13, 2021 at 3:03

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A result of Gompf shows that, in the case $k=2$, if the complex structure on the domain induced by a smooth embedding in $C^2$ is homotopic to a Stein structure, then the embedding is isotopic to a complex embedding. "Homotopic" means homotopic through almost-complex structures.

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  • $\begingroup$ For your second question, the proof of Proposition 5.4 of this paper constructs Stein structures on a smooth 4-manifold that admits a smooth embedding in $R^4$, for which the canonical class is nonzero. Such examples can't be holomorphically embedded in $C^2$. $\endgroup$
    – Zippy
    Commented Jan 12, 2021 at 19:03

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