A Calabi-Yau manifold $X$ is defined to be a compact Kahler manifold with $c_1(X)=0$. We also assume it is simple connected in the following.
It seems that in this paper (see the bottom of the first page and the beginning of the second page), the authors claimed that "simple connected Kahler Calabi-Yau threefold is projective".
I was not able to figure this out, could anyone give me a hint?
This is equivalent by the Calabi conjecture to the existence of a Kahler metric on $M$ with vanishing Ricci curvature. Then you can apply formulas relating two different kinds of Laplacians to see that holomorphic $(k,0)$-forms are parallel, and which because the holonomy is $SU(3)$ implies that $b^{2,0}=0$. Now the Kahler cone is open in $H^{1,1}$, and because of the above vanishing Betti number, it's open in $H^2$. $H^2(X;\Bbb Q)$ is dense here, and thus you can find a rational Kahler form. Taking an appropriate multiple, we can choose the Kahler form to be integral, apply the Kodaira embedding theorem, and we're done.