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Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth manifolds. Is $L$ also a limit in $\textrm{Diff}_\mathbb{C}$? In other words, does $L$ admit a complex structure such that the co-cone is holomorphic and universal? (I think universality would be straightforward.)

Strictly speaking, the answer is "no." For example, given two complex manifolds, we could glue them together in any smooth way using a diagram with very many discrete point manifolds mapping to the glued points, and of course this diagram would be holomorphic. But it feels like something along these lines should be true for a sufficiently nice class of diagrams. For example, what if we ask for a finite diagram with a suitable notion of "small" manifold?

Positive insights or discouraging counterexamples welcome.

(N.B.: The particular colimit I'm working with I can argue is complex from the details. However, I'd like to know if there's a general fact. For those who are interested, I'm generating a Lie groupoid from a local Lie groupoid via composable sequences modulo the "small" relations provided by the local groupoid. In my case I know the result is smooth, and the local Lie groupoid is holomorphic, so...?)

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    $\begingroup$ An easy observation is that the inclusion creates coproducts and coequalisers whose quotient morphism is a local homeomorphism. $\endgroup$
    – Zhen Lin
    Jun 24, 2015 at 18:30
  • $\begingroup$ Note that this fails horribly if you use topological manifolds rather than smooth manifolds. For instance, a coequalizer that collapses a small ball to a point exists in topological manifolds. $\endgroup$
    – Eric Wofsey
    Jun 25, 2015 at 0:35

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