Stability conditions and moduli spaces on the Kuznetsov component of cubic fivefolds

We study the Kuznetsov component of a smooth cubic fivefold.

Using a quadric surface fibration, we construct a family of Serre-invariant Bridgeland stability conditions on the Kuznetsov component.

For every non-zero numerical class, we prove that the associated Bridgeland moduli space on the Kuznetsov component is non-empty.

When the cubic fivefold is general and the numerical class is primitive, the moduli space contains a smooth locus, on which the restriction to a general hyperplane section preserves stability.

Consequently, we obtain Lagrangian immersions into hyper-Kähler varieties arising as moduli spaces on the Kuznetsov component of cubic fourfolds, extending the work of Li-Lin-Pertusi-Zhao on cubic 3-folds.

As an example, we recover the geometric construction of Illiev-Manivel, which realizes the Fano surface of planes of the cubic fivefold as a Lagrangian subvariety in a hyper-Kähler fourfold.