Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Grzegorz Kapustka,Michał Kapustka

doi:10.1007/s10231-015-0476-0

Abstract

We study the geometry of $3$-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi--Yau threefolds in projective $6$-space. Moreover, we prove that this classification includes all Calabi--Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi--Yau threefolds of degree at most $14$ in $\mathbb{P}^6$.

Highlights

It is conjectured that, when 2n ≥ N, there is a finite number of smooth families of smooth n-dimensional subvarieties of PN that are not of general type
We study nondegenerate Calabi–Yau threefolds, i.e., such Calabi–Yau threefolds which are not contained in any hyperplane
We prove a technical result (Proposition 7.2) on deformation of Pfaffian varieties implying that any threefold B14 ∈ B14 appears as a smooth degeneration of the family T14 of Calabi–Yau threefolds defined by 6 × 6 Pfaffians of alternating 7 × 7 matrices of linear forms

Summary

Introduction

It is conjectured that, when 2n ≥ N , there is a finite number of smooth families of smooth n-dimensional subvarieties of PN that are not of general type. We prove that all Calabi–Yau threefolds of degree at most 14 in P6 are quasi-Buchsbaum and use the classification of the latter threefolds contained in Sect. We prove a technical result (Proposition 7.2) on deformation of Pfaffian varieties implying that any threefold B14 ∈ B14 appears as a smooth degeneration of the family T14 of Calabi–Yau threefolds defined by 6 × 6 Pfaffians of alternating 7 × 7 matrices of linear forms This proves that all families of Calabi–Yau threefolds of degree 14 which appear in the classification of Sect. Prove that the examples of degree 15 constructed in [7] are not smooth but admits three ordinary double points

Preliminaries

Quasi-Buchsbaum Calabi–Yau threefolds

The degrees of Calabi–Yau threefolds in quadrics

Classification of degree 14 Calabi–Yau threefolds in P6

Classification up to deformations