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Classification of double octic Calabi-Yau threefolds defined by an arrangement of eight planes II

Sławomir Cynk, Beata Kocel-Cynk

Abstract

We present a complete classification of all arrangements of eight planes in projective threespace that give rise to double octic Calabi-Yau threefolds. Building on earlier work, we determine all 455 combinatorial types and describe the projective automorphism groups associated with each arrangement. These automorphisms allow us to identify geometrically distinguished subfamilies and elements, and to construct a rich collection of elliptic and K3 fibrations arising naturally from the singularities of the arrangements. As applications, we exhibit two one-parameter families of Calabi-Yau threefolds that each possess three pairwise non-equivalent maximally unipotent monodromy points-phenomena that are rare in known examples. We further construct two Calabi-Yau threefolds whose third cohomology decomposes, via complex multiplication, into complementary two-dimensional Hodge structures of types (1,0,1,0) and (0,1,0,1). This decomposition yields new instances of modularity not previously observed in the Calabi-Yau setting.

Classification of double octic Calabi-Yau threefolds defined by an arrangement of eight planes II

Abstract

We present a complete classification of all arrangements of eight planes in projective threespace that give rise to double octic Calabi-Yau threefolds. Building on earlier work, we determine all 455 combinatorial types and describe the projective automorphism groups associated with each arrangement. These automorphisms allow us to identify geometrically distinguished subfamilies and elements, and to construct a rich collection of elliptic and K3 fibrations arising naturally from the singularities of the arrangements. As applications, we exhibit two one-parameter families of Calabi-Yau threefolds that each possess three pairwise non-equivalent maximally unipotent monodromy points-phenomena that are rare in known examples. We further construct two Calabi-Yau threefolds whose third cohomology decomposes, via complex multiplication, into complementary two-dimensional Hodge structures of types (1,0,1,0) and (0,1,0,1). This decomposition yields new instances of modularity not previously observed in the Calabi-Yau setting.
Paper Structure (17 sections, 12 theorems, 119 equations, 1 table)

This paper contains 17 sections, 12 theorems, 119 equations, 1 table.

Table of Contents

  1. Double octic Calabi--Yau threefolds
  2. Classification
  3. Projective transformations of double octics
  4. Geometrically distinguished subfamilies and elements
  5. Geometrically distinguished elements in families of double octic Calabi--Yau manifolds.
  6. A Calabi--Yau threefold with Complex Multiplication
  7. Calabi--Yau threefold over a number field
  8. Geometrically distinguished one-parameter sub-families of double octic Calabi--Yau manifolds.
  9. Fibrations of double octic Calabi--Yau threefolds
  10. Elliptic fibrations determined by fourfold and fivefold points
  11. Elliptic fibrations determined by two skew double lines
  12. Kummer fibrations
  13. K3 fibrations by double sextics
  14. K3 fibrations by double quadrics
  15. Equations and Data
  16. ...and 2 more sections

Key Result

Theorem 1.1

For an octic arrangement $D$ there exists a resolution of singularities $X$ of the double cover of $\mathbb P^{3}$ branched along $D$ that is a smooth Calabi--Yau threefold.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.1: CSz
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3: CK-C
  • Proposition 3.1
  • ...and 18 more