Reference: Calabi-Yau toric varieties I am looking for a reference of the following fact: 


Let $N \cong \mathbb{Z}^n$ a lattice and $P \subset N_{\mathbb{R}} = N \otimes_{\mathbb{Z}} \mathbb{R}$ a convex lattice polytope. Let $P_1, \dots , P_k$ be a smooth (all $P_j$'s are elementary simplices) subdivision of $P$ consisting of convex lattice polytopes and take the fan $\Sigma$ by considering the cones over the faces of the $P_j$'s. Then the toric variety $V_{\Sigma}$ is a smooth Calabi-Yau.



Lagrangian torus fibration and mirror symmetry of Calabi-Yau hypersurface in toric variety, by Wei-Dong Ruan, see page 13, corollary 2.3.  https://arxiv.org/pdf/math/0007028.pdf
Brian Greene also has a more general discussion on this in String Theory on Calabi Yau Manifolds, see in particular sections 9.3-9.5. https://arxiv.org/pdf/hep-th/9702155.pdf
