What are some natural notions of distance $d$ between two complex manifolds of dimension $n$? For any of these notions what are the current best results on approximation of a complex manifold $M$ by a complex algebraic variety $V_M$? What happens if we impose the condition that $V_M$ must be non-singular? Can the precision of approximation be measured in terms of algebraic invariants of $M$ and $V_M$?
1 Answer
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You ask several questions, but I will only really address the last, or at least my interpretation of it. In the real setting, any compact manifold is diffeomorphic to a real algebraic variety by work of Nash-Tognolli. In the complex setting, such a result is not possible. For example, any finitely presented group is the fundamental group of a compact complex manifold (Taubes), but this is far from true for nonsingular algebraic varieties. For instance, if $\Gamma$ is a nonvirtually nilpotent solvable group, then it won't be the fundamental group of a nonsingular algebraic variety. So I'm really sure what approximation would be mean in this case.
answered Mar 15, 2021 at 17:25
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$\begingroup$ Thanks for mentioning the real result. That's what made me ask about the complex counterpart. Can you please explain more how contraint on the fundamental group makes approximation difficult or meaningless? $\endgroup$– AfaCommented Mar 15, 2021 at 17:47
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$\begingroup$ I guessed that you might have been motivated by the real version. I would imagine for any reasonable notion of approximation, if spaces are sufficiently close, then they should share topological invariants such as Betti numbers and fundamental group. $\endgroup$ Commented Mar 15, 2021 at 18:16
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$\begingroup$ The complex K3 surfaces form a 20 dimensional complex space and the algebraic ones, alre located on a codimension 1 subvariety. In this case at least you can speak of a distance by measuring how far you are away from the divisor of algebraic ones. $\endgroup$ Commented Mar 15, 2021 at 21:47
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$\begingroup$ @LiviuNicolaescu Is that true? Aren't the points defining algebraic K3 surfaces dense in the moduli of all K3 surfaces? For example look at Siu's paper on Kähler metrics on K3 surfaces. $\endgroup$– FarisCommented Mar 16, 2021 at 12:33
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$\begingroup$ @Faris I will let the algebraic geometers on this site (I'm not one) confirm or infirm what I wrote. $\endgroup$ Commented Mar 16, 2021 at 13:32