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When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual case one wants to consider is usually that of Calabi-Yau-manifolds.

It is usually not explained why the former case includes the latter, although it seems to be used very often that it does. My question is therefore: Can every (possibly non-compact) Calabi-Yau be embedded into projective space, giving a smooth quasiprojective variety (I read that it sometimes can't be included as a projective one)? I assume that to show this one would have to (like for Fano manifolds) construct an ample line bundle over it, as that would already imply the result. If it is possible, how can it be shown, and more importantly if not, why is Orlov's approach even justified?

Maybe a few further remarks: It seems like to embed a complex manifold $X$ into projective space, it must neccessarily be Kähler, as the Kähler structure of $\mathbb{C} P^n$ restricts to one on $X$, but this is obviously fulfilled here. Also, I read the article on ncatlab about Calabi-Yau-varieties, and to me it seems like it also supposes that these are equivalent to Calabi-Yau manifolds, although it only describes the analytification direction.

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    $\begingroup$ A quick remark, for the compact case: if your definition of Calabi-Yau manifold is the strongest one, namely holonomy equal to SU(n) or, equivalently, simples connected, trivial canonical bundle and no holomorphic p-forms except for bottom and top degree, then it is automatically projective, by Kodaira's criterion. $\endgroup$
    – diverietti
    Commented Jun 19, 2020 at 8:56
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    $\begingroup$ So, which is your definition of Calabi-Yau? I am curious! P.S. The Kodaira criterion I am speaking about in this situation is "compact Kähler manifold with $H^2(X,\mathcal O_X)=0$ is projective". $\endgroup$
    – diverietti
    Commented Jun 19, 2020 at 9:15
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    $\begingroup$ Your definition allows every open subset of a Calabi-Yau manifold to be also a Calabi-Yau manifold. For example, the unit ball in complex Euclidean space is a Calabi-Yau manifold in your sense, but not a quasi projective variety, as its closure in projective space is not a projective variety. So you need a better definition, with some sort of completeness or inextensibility, at the least. $\endgroup$
    – Ben McKay
    Commented Jun 19, 2020 at 9:32
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    $\begingroup$ K3 surfaces are not necessarily embeddable in projective space, but they are Calabi Yau in most senses. $\endgroup$
    – Ben McKay
    Commented Jun 19, 2020 at 9:33
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    $\begingroup$ @BenMcKay, indeed, I was tacitly thinking about dimension greater than or equal to three! $\endgroup$
    – diverietti
    Commented Jun 19, 2020 at 10:35

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