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 $\Gamma_x^s<O(2),\Gamma_x^y<O(2)$. 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 ?

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    $\begingroup$ Curvature provides a local criterion: every non-positively curved orbifold is good. I’m uncertain how that relates to the complex structure, though. $\endgroup$
    – HJRW
    Aug 9, 2021 at 12:11
  • $\begingroup$ @HJRW Thanks. Are there some references about the criterion you mentioned ? $\endgroup$ Aug 10, 2021 at 4:06
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    $\begingroup$ 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. $\endgroup$
    – HJRW
    Aug 10, 2021 at 11:32


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