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Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?

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  • $\begingroup$ At least for Riemann surfaces, this question seems helpful. $\endgroup$
    – Michael Albanese
    Nov 19, 2015 at 23:20
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    $\begingroup$ I assume you want your manifolds to be compact, otherwise it is hopeless. Then you may be interested by the papers of D. Guan, see here. Under some mild extra hypotheses, such manifold is a product of a projective one (those are well-understood) and a complex solvmanifold. Note however that there is no hope to completely classify the latter, they are just too many. $\endgroup$
    – abx
    Nov 20, 2015 at 7:26

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