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Let $G$ be a complex algebraic group (reductive if necessary) acting holomorphically on a complex manifold $X$. Does the closure of every $G$-orbit contain a closed orbit?

If $X$ is a complex algebraic variety and $G$ acts algebraically, this is a standard result (see [1] Proposition 1.8).

[1] A. Borel, Linear Algebraic Groups, Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991.

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No. Take a holomorphic translation vector field on the complex torus, whose orbits are translates of the dense image of some generic complex 1-dimensional linear subspace. The flow is complete, by compactness, and the group is $G=\mathbb{C}$.

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    $\begingroup$ That's a non-reductive example. $\endgroup$
    – user121777
    Commented Mar 12, 2018 at 19:47

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