Constructions and deformations of Calabi--Yau 3-folds in codimension 4

Abstract: We construct polarized Calabi--Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of $\mathbb P^2 \times \mathbb P^2$ in $\mathbb P^8$. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi--Yau 3-folds. Among other interesting results, we find a family of Calabi--Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for $\mathbb Q$-Fano 3-folds. We compute the Hodge numbers of $\mathbb P^2 \times \mathbb P^2 $ Calabi--Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer--Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi--Yau 3-folds having low codimension embeddings in weighted projective spaces.