On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

TL;DR

This paper explores the geometry of quartic double fivefolds through exceptional quaternionic representations, establishing their relation to Fano manifolds, vector bundles, and derived categories, revealing new structural insights.

Contribution

It demonstrates that generic quartic double fivefolds can be represented as double covers ramified along specific linear sections and proves the existence of a special spherical vector bundle on them.

Findings

Representation of quartic double fivefolds as double covers of P^5

Existence of a spherical rank 6 vector bundle on these fivefolds

Homological unit of the associated CY-3 category is C ⊕ C[3]

Abstract

We study quartic double fivefolds from the perspective of Fano manifolds of Calabi-Yau type and that of exceptional quaternionic representations. We first prove that the generic quartic double fivefold can be represented, in a finite number of ways, as a double cover of P^5 ramified along a linear section of the Sp 12-invariant quartic in P^31. Then, using the geometry of the Vinberg's type II decomposition of some exceptional quaternionic representations, and backed by some cohomological computations performed by Macaulay2, we prove the existence of a spherical rank 6 vector bundle on such a generic quartic double fivefold. We finally use the existence this vector bundle to prove that the homological unit of the CY-3 category associated by Kuznetsov to the derived category of a generic quartic double fivefold is C $\oplus$ C[3].