@abx I don't understand the vote to close, btw. I think this is a reasonable question and it's a bit unfair to be so harsh. If you still think otherwise, can you explain to me your reason?

To spell out concretely then, $ M $ is C-M if there exist two sequences $ K \rightarrow R^n \rightarrow N \rightarrow 0 $ and $ M \rightarrow R^m \rightarrow K \rightarrow 0 $. I don't see any problems here but correct me otherwise.

But I never said it is well-defined, it's just a notation.

@Z.M I found most of the details of that relative second cohomology group and the integral (which is $ \zeta(2) $) in the paper of Francis Brown here: arxiv.org/abs/math/0606419

@Jason Starr that's what I don't understand as well. Can you please explain what to do in your first comment? I thought of it but I don't know torus actions very well.

Wonderful. So any compact Kahler manifold with $ h^{2,0} = 0 $ is projective. Your proof shows that it has nothing to do with being a threefold or $ h^{1,0} = 0 $.

I understand now. Thank you so much for your patience. I guess my exercise for now will be to prove that any CY3 with $ h^{1,0} = 0 = h^{2,0} $ is projective and then proceed.

I'm sorry, I should mention that I'm just a PhD student who is more or less ignorant about stacks. The only stack I've dealt with is the moduli stack of elliptic curves.

So my 'simply connected' assumption rules out your first two classes of examples, right? And yes, I'm mainly looking for the irreducible ones. Looks like we don't have much understanding of those yet?

@Jason Starr thank you for this information. I would like to know how this space is constructed though. For example, I know about the moduli space of curves of genus g, or the moduli of stable sheaves on a surface. There are techniques to construct such spaces, via GIT. Is there something like this for CY3s?