Is there a closed complex manifold into which every closed complex surface embeds?
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1$\begingroup$ Every complex algebraic surface embeds in $\mathbb{CP}^5$ see Beauville's "Complex Algebraic Surfaces" Proposition 4.5. It is proven by choosing an embedding in some projective space then generically projecting away from points, for dimension reasons you can always keep doing this untill you get to $\mathbb{P}^5$. $\endgroup$– Nick LCommented Sep 28, 2020 at 10:53
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2$\begingroup$ Indeed. Non-algebraic closed surfaces however cannot embed into a projective variety. $\endgroup$– user164740Commented Sep 28, 2020 at 10:55
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1$\begingroup$ This is similar in spirit to this question. One difference here is that it is not clear (to me) that there is a compact complex manifold into which every complex surface embeds, let alone one of dimension five. Do you know if there is such a manifold? $\endgroup$– Michael AlbaneseCommented Sep 28, 2020 at 13:44
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$\begingroup$ I guess you are right that there may be no such manifold of any dimension. $\endgroup$– user164740Commented Sep 28, 2020 at 14:24
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$\begingroup$ I asked Nick Buchdahl about this and he said it's unlikely, since not all complex surfaces are classified. If this were true, he said, we'd know more about class VII surfaces than we currently do. $\endgroup$– David Roberts ♦Commented Sep 28, 2020 at 23:12
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