6

Is there a closed complex manifold into which every closed complex surface embeds?

  • 1
    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 $\mathbb{P}^5$. – Nick L Sep 28 '20 at 10:53
  • 2
    Indeed. Non-algebraic closed surfaces however cannot embed into a projective variety. –  Sep 28 '20 at 10:55
  • 1
    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? – Michael Albanese Sep 28 '20 at 13:44
  • I guess you are right that there may be no such manifold of any dimension. –  Sep 28 '20 at 14:24
  • 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. – David Roberts Sep 28 '20 at 23:12

0 Answers0