Classification of double octic Calabi–Yau threefolds with h1,2 ≤ 1 defined by an arrangement of eight planes

Abstract

In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

Periods of singular double octic Calabi–Yau threefolds and modular forms

“…The irreducible components R 0 and Q 0 have natural lifts to characteristic zero (see [6,8]); we denote them by R ∞ and Q ∞ . A key observation in the up-coming proof that the Galois action on…”

“…Additional period integrals in $$\Lambda ^{p}_{\overline{X}}$$ are related to singular points of type $$p_{4}^{0}$$. The classification in [5] shows that $$\Lambda ^{p}_{\overline{X}}$$ is the sum of $$\Lambda ^{p}_{X_{t_0}}$$ taken over all one‐parameter smoothings $$X_{t}$$ of X .…”

On the Chow ring of some special Calabi–Yau varieties

“…In case n = 2, X as in Proposition 2.3 is a K3 surface of the type studied in [35], [30], [44]. In case n = 3, Calabi-Yau varieties X as in Proposition 2.3 are special cases of so-called "double octics"; these special cases have been intensively studied, particularly their modular properties [32], [9], [10], [11], [12], [13], [19], [38].…”

Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

“…, dim(X ) − 1. Calabi-Yau varieties form a vast and actively studied area of research, also aiming for classification results such as [12,15,27,33,34] or more recently [14,16,17,38].…”