Given $(B,\mathscr{P},\varphi)$ simple, the set of log Calabi-Yau spaces with intersection complex $(B,\mathscr{P},\varphi)$ modulo isomorphism preserving $B$ (i.e., does not interchange irreducible components) is $H^{1}(B,i_{*}\check{\Lambda}\otimes\mathbb{C}^{\times})$. An isomorphism is said to preserve $B$ if it induces the identity on the intersection complex.