Gepner Type Stability Condition via Orlov/Kuznetsov Equivalence

Abstract

We show the existence of Gepner type Bridgeland stability conditions on the triangulated categories of graded matrix factorizations associated with homogeneous polynomials which define general cubic fourfolds containing a plane. The key ingredient is to describe the grade shift functor of matrix factorizations in terms of sheaves of Clifford algebras on the projective plane under Orlov/Kuznetsov equivalence.

The K3 category of a cubic fourfold

“…For example, to the best of my knowledge no , which is not equivalent to the derived category of some twisted K3 surface , has yet been endowed with a bounded t-structure, let alone a stability condition. See [Tod14, Tod13] for a discussion of special stability conditions on certain of the form .…”

Lagrangian embeddings of cubic fourfolds containing a plane

“…Finally, we comment on two recent papers on cubic fourfolds. One is the work by Toda [Tod16] on Bridgeland stability conditions on . By Orlov’s theorem [Orl09], the triangulated category is equivalent to the triangulated category of graded matrix factorizations of the defining polynomial of .…”

“…By Orlov’s theorem [Orl09], the triangulated category is equivalent to the triangulated category of graded matrix factorizations of the defining polynomial of . The investigation of Bridgeland stability conditions on is related to the existence problem of Gepner-type stability conditions on , which is treated in [Tod16]. However, it is also difficult to construct the heart of a bounded t-structure on .…”

The derived category of a non-generic cubic fourfold containing a plane

“…Some recent results involving cubic fourfolds containing a plane that make use of the tool of quadric fibrations can be found in [3], [5], [11], [22], [23], [25] and [27].…”

“…A geometrical meaning to the Kuznetsov component T Y has already been established in [27] for cubic forufolds containing a plane and in [6] for all cubic fourfolds.…”