Stability Conditions and Moduli Spaces on Kuznetsov Component of Cubic Fivefolds

We study the Kuznetsov component of cubic fivefolds via their quadric fibration model, and construct a family of Serre-invariant Bridgeland stability conditions on it. For every primitive numerical class, we prove that the associated Bridgeland moduli space contains a non-empty smooth locus, on which the restriction to a general hyperplane section preserves stability. As a consequence, we obtain Lagrangian immersions into hyper-Kähler varieties arising as moduli spaces on the Kuznetsov component of cubic fourfolds, generalizing a 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.