Higher-dimensional moduli spaces on Kuznetsov components of Fano threefolds

We study moduli spaces of stable objects in the Kuznetsov components of Fano threefolds. We prove a general non-emptiness criterion for moduli spaces, which applies to the cases of prime Fano threefolds of index 1, degree 10 ≤ d ≤ 18 and index 2, degree d ≤ 4. In the second part, we focus on cubic threefolds. We show the irreducibility of the moduli spaces, and that the general fibers of the Abel-Jacobi maps from the moduli spaces to the intermediate Jacobian are Fano varieties. When the dimension is sufficiently large, we further show that the general fibers of the Abel-Jacobi maps are stably birational equivalent to each other. As an application of our methods, we prove Conjecture A.1 in [S. Feyzbakhsh, H. Guo, Z. Liu and S. Zhang, Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds, Compos. Math. 161 (2025), 8, 2091-2135] concerning the existence of Lagrangian subvarieties in moduli spaces of stable objects in the Kuznetsov components of very general cubic fourfolds.