Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds and double EPW cubes

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 the first part of the 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 8-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a very 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. In the second part, we study in detail the double EPW cube, a 6-dimensional hyperk\"ahler manifold associated with a very general GM fourfold X, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on X. We also prove that it admits a covering by Lagrangian subvarieties, constructed from the Hilbert schemes of twisted cubics on GM threefolds.