Examples of Calabi-Yau 3-folds of $\mathbb{P}^7$ with $\rho=1$

Abstract

We give some examples of Calabi-Yau 3-folds with $\rho=1$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of $\mathbb{P}^7$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field $\mathbb{F}_{p}$. Three of our examples are new. These examples are built out of Gulliksen-Neg\r{a}rd and Kustin-Miller complexes of locally free sheaves. Finally, we give two new examples of Calabi-Yau 3-folds of $\mathbb{P}^6$ of degree 14 and 15 that are not deformation-equivalent to the previously known examples of these degrees in $\mathbb{P}^6$ (even though they share the same invariants $(H^3, c_2\cdot H, c_3)$ and $\rho=1$).