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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>10</m:mn> <m:mo>≤</m:mo> <m:mi>d</m:mi> <m:mo>≤</m:mo> <m:mn>18</m:mn> </m:mrow> </m:math> 10≤ d≤ 18 and index 2, degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≤</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> 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.