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-K\"ahler 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-K\"ahler fourfold.