Hey Misha! Long time no see! Hope you are fine. Thank you very much for your answer, I’ll read it carefully. Hope to meet you soon, best.

@BenMcKay, stated in this form unfortunately the answer is no for trivial reasons: take a non projective complex torus...

@MarkusZetto, you can still take an open ball of dimension three...

@BenMcKay, indeed, I was tacitly thinking about dimension greater than or equal to three!

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".

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.

I guess you want $X$ to be compact right?

In some cases you can define some Chern classes of $X$ by pushing forward from a resolution of singularities. This is the case for instance if your $X$ is regular in codimension $k\in\mathbb N$: then it makes sense to define $c_\ell(X)$ to be $\pi_*c_\ell(\tilde X)$, where $\pi\colon\tilde X\to X$ is a resolution of singularities.

Essentially, the $\partial\bar\partial$-lemma.