Lagrangian families of Bridgeland moduli spaces from Gushel–Mukai fourfolds

Let X be a very general Gushel–Mukai (GM) variety of dimension n≥ 4 , and let Y be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of X and Y . In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an eight-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.