TITLE: Approximation of a complex manifold by an algebraic variety
QUESTION [5 upvotes]: 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 $V_M$?

REPLY [3 votes]: 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.