Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

Abstract

Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\ge 4$, 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 $8$-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.

37 pages. v4: To appear in Compos. Math. This version contains Section 1-Section 6 and Appendix A in v2. Other sections have been extended to be a different paper arXiv:2501.12964