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-Negård 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 $\rho =1$).

23 pages