By a Calabi-Yau threefold I mean a simply connected compact Kahler threefold with trivial canonical bundle.
By an algebraic compact complex manifold, I mean one that admits a closed immersion into a complex projective space.
What are examples of non-algebraic Calabi-Yau threefolds? Or is there none?
What I know: if the Calabi-Yau threefold $ M $ has $ h^{2,0} (M) = 0 $, then it is algebraic. (Algebraicity holds for any compact Kahler manifold with the vanishing condition, btw). So any such example must have nonzero $ h^{2,0} $.
I don't know the Ricci flow viewpoint to this subject, so I'm probably missing something but I'm eager to learn.
Best Answer
All Calabi-Yau threefolds as you defined them are algebraic. In fact any Calabi-Yau manifold of complex dimension $\geq 3$ is algebraic. An important property here is that your manifold is simply connected. Otherwise there are many examples like complex tori which are not algebraic. The only proof that I am aware of is due to D. Joyce and is published in [this paper][1] and also in
Gross, Mark (ed.); Huybrechts, Daniel (ed.); Joyce, Dominic (ed.), Calabi-Yau manifolds and related geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001, Universitext. Berlin: Springer. viii, 239 p. EUR 49.95/net; sFr. 86.00; \textsterling 35.00; $ 59.95 (2003). ZBL1001.00028. [1]: https://arxiv.org/abs/math/0108088