[Paper Review] On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

Roland Abuaf|arXiv (Cornell University)|Sep 15, 2017

Geometry and complex manifolds15 references3 citations

TL;DR

This paper establishes that a generic quartic double fivefold— a Fano fivefold of Calabi-Yau type— admits a spherical rank 6 vector bundle via matrix factorization techniques rooted in Vinberg’s $ω$-type decomposition of exceptional quaternionic representations. This leads to the key result: the homological unit of the associated CY-3 category is $\mathbb{C} \oplus \mathbb{C}[3]$, confirming a conjectural pattern seen in other Calabi-Yau-type manifolds.

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].

Motivation & Objective

• To establish that generic quartic double fivefolds are of Calabi-Yau type by showing they admit finite representations as double covers of $\mathbb{P}^5$ ramified over linear sections of the $\mathrm{Spin}_{12}$-invariant quartic in $\mathbb{P}^{31}$.
• To construct a spherical rank 6 vector bundle on a generic quartic double fivefold using matrix factorizations derived from the Vinberg-type $\mathrm{II}$ decomposition of exceptional quaternionic representations.
• To compute the homological unit of the CY-3 category associated to the derived category of a generic quartic double fivefold, using the existence of a spherical vector bundle.
• To extend the pattern observed in other Calabi-Yau-type manifolds—where the homological unit of the CY-3 category is $\mathbb{C} \oplus \mathbb{C}[3]$—to the quartic double fivefold case.

Proposed method

• Utilizes the geometry of the $\mathrm{Spin}_{12}$-invariant quartic hypersurface in $\mathbb{P}^{31}$ to show that generic quartic double fivefolds arise as double covers of $\mathbb{P}^5$ ramified over four-dimensional linear sections of this quartic.
• Applies Vinberg’s type $\mathrm{II}$ decomposition of exceptional quaternionic representations to construct matrix factorizations on the double fivefold, leveraging the structure of the representation space.
• Performs cohomological computations using Macaulay2 to verify the existence of a rank 6 vector bundle with spherical properties on the generic quartic double fivefold.
• Employs Kuznetsov’s construction of a CY-3 category from the derived category of the fivefold and uses the spherical vector bundle to compute the homological unit of this category.
• Applies the general principle that a CY-$n$ category containing a spherical object of non-zero rank has homological unit $\mathbb{C} \oplus \mathbb{C}[n]$, here with $n=3$.

Experimental results

Research questions

• RQ1 Can a generic quartic double fivefold be represented as a double cover of $\mathbb{P}^5$ ramified over a linear section of the $\mathrm{Spin}_{12}$-invariant quartic in $\mathbb{P}^{31}$?
• RQ2 Does the geometry of exceptional quaternionic representations via Vinberg’s type $\mathrm{II}$ decomposition yield matrix factorizations on quartic double fivefolds?
• RQ3 Does the derived category of a generic quartic double fivefold admit a spherical vector bundle of rank 6?
• RQ4 What is the homological unit of the CY-3 category associated to the derived category of a generic quartic double fivefold?

Key findings

• The generic quartic double fivefold admits a finite number of representations as a double cover of $\mathbb{P}^5$ ramified over a linear section of the $\mathrm{Spin}_{12}$-invariant quartic in $\mathbb{P}^{31}$.
• A spherical rank 6 vector bundle exists on a generic quartic double fivefold, constructed via matrix factorizations from the Vinberg-type $\mathrm{II}$ decomposition of exceptional quaternionic representations.
• The homological unit of the CY-3 category associated to the derived category of a generic quartic double fivefold is $\mathbb{C} \oplus \mathbb{C}[3]$.
• This result confirms a conjectural pattern observed in other Calabi-Yau-type manifolds, such as the generic cubic sevenfold, where the homological unit is also $\mathbb{C} \oplus \mathbb{C}[3]$.
• The existence of a spherical vector bundle implies that the derived category of the fivefold supports non-trivial auto-equivalences via spherical twist functors.