Higher dimensional moduli spaces on Kuznetsov components of Fano threefolds

TL;DR

This paper investigates the structure of moduli spaces of stable objects in the Kuznetsov components of Fano threefolds, establishing non-emptiness, irreducibility, and properties of fibers, with applications to conjectures on cubic fourfolds.

Contribution

It introduces a non-emptiness criterion for moduli spaces and proves irreducibility and fiber properties for cubic threefolds, advancing understanding of Kuznetsov components in Fano threefolds.

Findings

Established non-emptiness criterion for moduli spaces.

Proved irreducibility of moduli spaces for cubic threefolds.

Showed fibers of Abel--Jacobi maps are Fano varieties and stably birational.

Abstract

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 $10 \leq d \leq 18$, and index $2$, degree $d \leq 4$. 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 [FGLZ24] concerning the existence of Lagrangian subvarieties in moduli spaces of stable objects in the Kuznetsov components of very general cubic fourfolds.