# How to judge whether an orbifold is good

My own case comes from dynamic system on compact complex manifolds. To be precise, let $$M$$ be a compact complex 3-dimentional manifold, $$W^c$$ a holomorphic foliation of M with 1-dimentional uniformly compact leaves(i.e. the leaves are compact Riemann surfaces and the holonomy groups are finite.)

It can be deduced from above that the leaves with non-trivial holonomy are finite. Then $$M/W^c$$ is a obifold with complex dimension 2 with finite singular points $$x$$ whose neighborhood $$U_x$$ is homeomorphic to $$\mathbb C^2/\Gamma_x$$ where $$\Gamma_x=\Gamma_x^s \times \Gamma_x^u$$ and $$\Gamma_x^s. Both groups $$\Gamma_x^s$$ and $$\Gamma_x^u$$ have the same order.

My question is that whether $$M$$ is good(i.e. it arises as the quotient of a simply connected manifold by a proper rigid action of a discrete group Γ) Is it purely a topological problem, or the complex structure can help us to eliminate the singulartity ?

• Curvature provides a local criterion: every non-positively curved orbifold is good. I’m uncertain how that relates to the complex structure, though.
– HJRW
Aug 9, 2021 at 12:11
• @HJRW Thanks. Are there some references about the criterion you mentioned ? Aug 10, 2021 at 4:06
• Bridson—Haefliger prove it in the much more general context of complexes of groups, in their book “Metric spaces of non-positive curvature”. I imagine it was already known in the orbifold setting, but don’t know a simpler reference.
– HJRW
Aug 10, 2021 at 11:32