TITLE: Calabi - Yau Manifolds
QUESTION [13 upvotes]: I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one from other. But my question is the following :
"What is the most strict definition of Calabi-Yau Manifolds" 
By that I mean the definition from which all the others follow.

REPLY [6 votes]: For singular varieties , Siu introduced Calabi-Yau varieties by using numerical Kodaira dimension
Let $X_0$ be a projective variety with canonical line bundle $K\to X_0$ of Kodaira dimension $$\kappa(X_0)=\limsup\frac{\log \dim H^0(X_0,K^{\otimes \ell})}{\log\ell}$$ This can be shown to coincide with the maximal complex dimension of the image of $X_0$ under pluri-canonical maps to complex projective space, so that $\kappa(X_0)\in\{-\infty,0,1,...,m\}$. Also since in general we work on Singular K\"ahler variety we need to notion of numerical Kodaira dimension instead of Kodaira dimension.
$$\kappa_{num}(X)=\sup_{k\geq 1}\left[\limsup_{m\to \infty}\frac{\log\dim_{\mathbb C}H^0(X,mK_X+kL)}{\log m}\right]$$
where $L$ is an ample line bundle on $X$.Note that the definition of $\kappa_{num}(X)$ is independent of the choice of the ample line bundle $L$ on $X$. Siu formulated that the abundance conjecture is equivalent as
$$\kappa_{kod}(X) =\kappa_{num}(X)$$
The Minimal Model and abundance conjectures would imply that every variety of  Kodaira dimension $\kappa (X)=0$ is birational to a Calabi-Yau variety with terminal singularities.
Note that when $K_X$ is psudoeffective vanishing numerical Kodaira dimension is equivalent that $X$ is Calabi-Yau variety